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What Is a “literature Value” in Chemistry?

In chemistry, a literature value is any value that is necessary to carry out an experiment in a laboratory. This may include physical data, instructions for synthesis, reactions, concepts and techniques. Literature values may be found in references or in journals.

In the laboratory, researchers work with theoretical and experimental literature values. The theoretical literature value is the value that is expected based on the reactions taking place. The experimental values are what are actually derived after completing the experiment, and these can be attributed to a variety of factors, including the nature of the chemicals, as well as equipment and experimenter technique. Comparisons of experimental and theoretical values may be made.

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What is the difference between Accepted Value vs. Experimental Value?

#"Error" = "|experimental value - accepted value|"#

The difference is usually expressed as percent error .

#"% error" = "|experimental value - accepted value|"/"experimental value" × 100 %#

For example, suppose that you did an experiment to determine the boiling point of water and got a value of 99.3 °C.

Your experimental value is 99.3 °C.

The theoretical value is 100.0 °C.

The experimental error is #"|99.3 °C - 100.0 °C| = 0.7 °C"#

The percent error is #"|99.3 °C - 100.0 °C|"/"100.0 °C" = "0.7 °C"/"100.0 °C" × 100% = 0.7 %#

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How to Calculate Experimental Error in Chemistry

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Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
B.A., Physics and Mathematics, Hastings College

Error is a measure of accuracy of the values in your experiment. It is important to be able to calculate experimental error, but there is more than one way to calculate and express it. Here are the most common ways to calculate experimental error:

Error Formula

In general, error is the difference between an accepted or theoretical value and an experimental value.

Error = Experimental Value - Known Value

Relative Error Formula

Relative Error = Error / Known Value

Percent Error Formula

% Error = Relative Error x 100%

Example Error Calculations

Let's say a researcher measures the mass of a sample to be 5.51 grams. The actual mass of the sample is known to be 5.80 grams. Calculate the error of the measurement.

Experimental Value = 5.51 grams Known Value = 5.80 grams

Error = Experimental Value - Known Value Error = 5.51 g - 5.80 grams Error = - 0.29 grams

Relative Error = Error / Known Value Relative Error = - 0.29 g / 5.80 grams Relative Error = - 0.050

% Error = Relative Error x 100% % Error = - 0.050 x 100% % Error = - 5.0%

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How to find the literature value (known value) of theoretical yield and percentage yield of adipic acid?

In a given lab practical for the oxidation of cyclohexene to adipic acid using KMnO 4 , we have been tasked with finding the literature value or the known value of the theoretical yield and percentage yield of the adipic acid. As for the experimental value, 1.24 mL of cyclohexene was used, and the amount of adipic acid yielded after the experiment was 1.4g, therefore having a theoretical yield of 2.207 g and a percentage yield of 63%. Is there a way I can find the literature value (either known or by calculation from known values) of the theoretical yield and percentage yield of Adipic Acid?

1 Expert Answer

Bharat K. answered • 10/02/23

EXPERIENCE HIGH SCHOOL TEACHER

To find the literature value of the theoretical yield and percentage yield of adipic acid, you can follow these steps:

1. Conduct a literature search: Start by conducting a literature search using academic databases, scientific journals, or chemical databases. Look for studies or articles that specifically mention the synthesis or production of adipic acid. Pay attention to the experimental procedures and the reported yields.

2. Calculate the theoretical yield: Once you find a reliable source that provides the experimental procedure and the yield of adipic acid, you can calculate the theoretical yield. The theoretical yield is the maximum possible amount of product that can be obtained from a given amount of starting material, assuming complete conversion and perfect reaction conditions. You can use the balanced chemical equation for the synthesis of adipic acid to calculate the amount of adipic acid that should be produced.

3. Calculate the percentage yield: To calculate the percentage yield, you need the experimental yield (the actual amount of adipic acid obtained in your own experiment) and the theoretical yield. The percentage yield is calculated using the formula: (experimental yield / theoretical yield) x 100%.

Keep in mind that the literature values may vary depending on the specific experimental conditions, starting materials, and reaction methods used in different studies. Therefore, it is important to consult multiple reliable sources to obtain a range of literature values for the theoretical yield and percentage yield of adipic acid.

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Literary Theory and Criticism

Home › Experimental Novels › Experimental Novels and Novelists

Experimental Novels and Novelists

By NASRULLAH MAMBROL on March 6, 2019 • ( 2 )

Literature is forever transforming. A new literary age is new precisely because its important writers do things differently from their predecessors. Thus, it could be said that almost all significant literature is in some sense innovative or experimental at its inception but inevitably becomes, over time, conventional. Regarding long fiction, however, the situation is a bit more complex.

It is apparent that, four centuries after Miguel de Cervantes wrote what is generally recognized the first important novel, Don Quixote de la Mancha (1605, 1615), readers have come to accept a certain type of long fiction as most conventional and to regard significant departures from this type as experimental. This most conventional variety is the novel of realism as practiced by nineteenth century giants such as Gustave Flaubert, Leo Tolstoy, Charles Dickens, and George Eliot .

The first task in surveying contemporary experiments in long fiction, therefore, is to determine what “conventional” means in reference to the novel of realism. Most nineteenth century novelists considered fiction to be an imitative form; that is, it presents in words a representation of reality. The underlying assumption of these writers and their readers was that there is a shared single reality, perceived by all—unless they are mad, ill, or hallucinatory—in a similar way. This reality is largely external and objectively verifiable. Time is orderly and moves forward. The novel that reflects this view of reality is equally orderly and accountable. The point of view is frequently, though not always, omniscient (all-knowing): The narrators understand all and tell their readers all they need to know to understand a given situation. The virtues of this variety of fiction are clarity of description and comprehensiveness of analysis.

After reading Flaubert’s Madame Bovary (1857; English translation, 1886), one can be confident that he or she knows something about Emma Bovary’s home, village, and manner of dress; knows her history, her motivations, and the way she thought; and knows what others thought of her. Not knowing would be a gap in the record; not knowing would mean, according to standards against which readers have judged “conventional” novels, a failure of the author.

Modernism and its Followers

Early in the twentieth century, a disparate group of novelists now generally referred to as modernists— James Joyce , Virginia Woolf , William Faulkner , Franz Kafka , Marcel Proust , and others—experimented with or even abandoned many of the most hallowed conventions of the novel of realism. These experiments were motivated by an altered perception of reality. Whereas the nineteenth century assumption was that reality is external, objective, and measurable, the modernists believed reality to be also internal, subjective, and dependent upon context. Reflecting these changing assumptions about reality, point of view in the modernist novel becomes more often limited, shifting, and even unreliable rather than omniscient.

This subjectivity reached its apogee in one of the great innovations of modernism, the point-of-view technique dubbed “stream of consciousness,” which plunges the reader into a chaos of thoughts arrayed on the most tenuous of organizing principles—or so it must have seemed to the early twentieth century audience accustomed to the orderly fictional worlds presented by the nineteenth century masters.

Once reality is acknowledged to be inner and subjective, all rules about structure in the novel are abandoned. The most consistent structuring principle of premodernist novelists—the orderly progression of time—was rejected by many modernists. Modern novels do not “progress” through time in the conventional sense; instead, they follow the inner, subjective, shifting logic of a character’s thoughts. Indeed, the two great innovations of modernist fiction—stream of consciousness and nonchronological structure—are inseparable in the modern novel.

Among the most famous and earliest practitioners of these techniques were Joyce (especially Ulysses , 1922, and Finnegans Wake , 1939), Woolf (especially Jacob’s Room , 1922; Mrs. Dalloway , 1925; and To the Lighthouse , 1927), and Faulkner (especially The Sound and the Fury , 1929; As I Lay Dying , 1930). Many of the experimental works of post-World War II long fiction extended these techniques, offering intensely subjective narrative voices and often extreme forms of stream of consciousness, including disruptions of orderly time sequences.

In La traición de Rita Hayworth (1968; Betrayed by Rita Hayworth , 1971), Manuel Puig employs (among other techniques) the words of several sets of characters in different rooms of a house without identifying the speakers or providing transitions to indicate a change in speaker. The effect may be experienced by the reader as a strange solipsistic cacophony, or something like a disjointed choral voice; in fact, the technique is a variation on stream of consciousness and not so very different from the tangle of interior monologues in Faulkner ’s novels.

Tim O’Brien’s novel of the Vietnam War, Going After Cacciato (1978, revised 1989), is another example of a work that makes fresh use of a modernist strategy. Here, reality at first seems more external and hence clearer than in Betrayed by Rita Hayworth . The bulk of the action concerns a rifle squad that follows a deserter, Cacciato, out of Vietnam and across Asia and Europe until he is finally surrounded in Paris, where he once again escapes. The chapters that make up this plot, however, are interspersed with generally shorter chapters recounting the experiences of the point-of-view character, Paul Berlin, at home and in Vietnam. In another set of short chapters, Berlin waits out a six-hour guard shift in an observation post by the sea. The most orderly part of the novel is the pursuit of Cacciato, which moves logically through time and place. The perceptive reader eventually realizes, however, that the pursuit of Cacciato is a fantasy conjured up by Berlin, whose “real” reality is the six hours in the observation post, where his thoughts skip randomly from the present to the past (in memories) to a fantasy world. As in the best modernist tradition, then, the structure of Going After Cacciato reflects an inner, subjective reality.

The post-World War II writer who most famously and provocatively continued the modernist agenda in long fiction was Samuel Beckett, especially in his trilogy: Molloy (1951; English translation, 1955), Malone meurt (1951; Malone Dies , 1956), and L’Innommable (1953; The Unnamable , 1958). In each successive novel, external reality recedes as the narrative voice becomes more inward-looking. In Molloy , for example, the title character searches for his mother, but he is lost from the beginning. He can find neither her (if she truly exists) nor his way back home, wherever that is—nor can he be sure even of the objective reality of recent experience. In one passage Molloy notes that he stayed in several rooms with several windows, but then he immediately conjectures that perhaps the several windows were really only one, or perhaps they were all in his head. The novel is filled with “perhapses” and “I don’t knows,” undermining the reader’s confidence in Beckett’s words.

The subjectivity and uncertainty are intensified in Malone Dies . At least in Molloy, the protagonist was out in the world, lost in a countryside that appears to be realistic, even if it is more a mindscape than a convincing geographic location. In Malone Dies , the protagonist spends most of his days immobile in what he thinks is a hospital, but beyond this nothing—certainly not space or time—is clear. As uncertain of their surroundings as Molloy and Malone are, they are fairly certain of their own reality; the unnamed protagonist of the final volume of the trilogy, The Unnamable , does not know his reality. His entire labyrinthine interior monologue is an attempt to find an identity for himself and a definition of his world, the one depending upon the other. In those attempts, however, he fails, and at no time does the reader have a confident sense of time and place in reference to the protagonist and his world.

The New Novel

With The Unnamable , long fiction may seem to have come a great distance from the modernist novel, but in fact Beckett was continuing the modernist practice of locating reality inside a limited and increasingly unreliable consciousness. Eventually, voices cried out against the entire modernist enterprise. Among the earliest and most vocal of those calling for a new fiction—for le nouveau roman, or a new novel—was a group of French avantgarde writers who became known as the New Novelists. However, as startlingly innovative as their fiction may at first appear, they often were following in the footsteps of the very modernists they rejected.

Among the New Novelists (sometimes to their dismay) were Michel Butor, Nathalie Sarraute, and Claude Simon. Even though Simon won the Nobel Prize in Literature, probably the most famous (or infamous) of the New Novelists was Alain Robbe-Grillet.

Alain Robbe-Grillet

Robbe-Grillet decried what he regarded as outmoded realism and set forth the program for a new fiction in his Pour un nouveau roman (1963; For a New Novel: Essays on Fiction , 1965). His own career might offer the best demonstration of the movement from old to new. His first published novel, Les Gommes (1953; The Erasers, 1964), while hardly Dickensian, was not radically innovative. With Le Voyeur (1955; The Voyeur , 1958), however, his work took a marked turn toward the experimental, and with La Jalousie (1957; Jealousy, 1959) and Dans le labyrinthe (1959; In the Labyrinth, 1960), the New Novel came to full flower.

The most famous technical innovation of the New Novelists was the protracted and obsessive descriptions of objects—a tomato slice or box on a table or a picture on a wall—often apparently unrelated to theme or plot. The use of this device led some critics to speak of the “objective” nature of the New Novel, as if the technique offered the reader a sort of photographic clarity. On the contrary, in the New Novel, little is clear in a conventional sense. Robbe-Grillet fills his descriptions with “perhapses” and “apparentlys” along with other qualifiers, and the objects become altered or metamorphosed over time. After Robbe-Grillet’s early novels, time is rarely of the conventional earlier-to-later variety but instead jumps and loops and returns.

One example of the transforming nature of objects occurs in The Voyeur , when a man on a boat peers obsessively at the figure-eight scar left by an iron ring flapping against a seawall. Over the course of the novel the figure-eight pattern becomes a cord in a salesman’s suitcase, two knotholes side by side on a door, a bicycle, a highway sign, two stacks of plates, and so on—in more than a dozen incarnations. Moreover, Robbe-Grillet’s objects are not always as “solid.” A painting on a wall (In the Labyrinth) or a photograph in a newspaper lying in the gutter ( La Maison de rendez-vous , 1965; English translation, 1966) may become “animated” as the narrative eye enters it, and the action will transpire in what was, a paragraph before, only ink on paper or paint on canvas.

Such techniques indeed seemed radically new, far afield of the novels of Joyce and Faulkner. However, it is generally the case with the New Novelists, especially with Robbe-Grillet, that this obsessive looking, these distortions and uncertainties and transformations imposed on what might otherwise be real surfaces, have their origins in a narrative consciousness that warps reality according to its idiosyncratic way of seeing. The point-of-view character of In the Labyrinth is a soldier who is likely feverish and dying; in The Voyeur , a psychotic murderer; in Jealousy, a jealous husband who quite possibly has committed an act of violence or contemplates doing so. In all cases the reader has even more trouble arriving at definitive conclusions than is the case with the presumably very difficult novels of Joyce and Faulkner.

Ultimately, the New Novelists’ program differs in degree more than in kind from the modernist assumption that reality is subjective and that fictional structures should reflect that subjectivity. As famous and frequently discussed as the New Novelists have been, their fiction has had relatively little influence beyond France, and when literary theorists define “postmodernism” (that is, the literary expression that has emerged after, and is truly different from, modernism), they rarely claim the New Novel as postmodern.

Metafiction

A far more significant departure from modernism occurred when writers began to reject the notion that had been dominant among novelists since Miguel de Cervantes: that it is the chief duty of the novelist to be realistic, and the more realistic the fiction the worthier it is. This breakthrough realization—that realism of whatever variety is no more than a preference for a certain set of conventions—manifests itself in different ways in fiction. In metafiction, also known as self-mimesis or selfreferential fiction, the author (or his or her persona), deliberately reminds the reader that the book is a written entity; in the traditional novel, however, the reader is asked to suspend his or her disbelief.

Often the metafictive impulse appears as little more than an intensification of the first-person-omniscient narrator, the “intrusive author” disparaged by Henry James but favored by many nineteenth century writers. Rather than employing an “I” without an identity, as in William Makepeace Thackeray’s Vanity Fair: A Novel Without a Hero (1847-1848), metafiction makes clear that the “I” is the novel’s author. Examples of this technique include the novels of José Donoso in Casa de campo (1978; A House in the Country, 1984) and of Luisa Valenzuela in Cola de lagartija (1983; The Lizard’s Tail, 1983).

In other novels, the metafictive impulse is more radical and transforming. When Donald Barthelme stops the action halfway through Snow White (1967), for example, and requires the reader to answer a fifteen-question quiz on the foregoing, the readers’ ability to “lose themselves” in the novel’s virtual world is hopelessly and hilariously destroyed. Another witty but vastly different metafictive novel is Italo Calvino ’s Se una notte d’inverno un viaggiatore (1979; If on a Winter’s Night a Traveler , 1981), in which the central character, Cavedagna, purchases a novel called If on a Winter’s Night a Traveler by an author named “Italo Calvino.” Cavedagna finds that his copy is defective: The first thirty-two pages are repeated again and again, and the text is not even that of Calvino; it is instead the opening of a Polish spy novel. The remainder of the book concerns Cavedagna’s attempts to find the rest of the spy novel, his blossoming romance with a woman on the same mission, and a rambling intrigue Calvino would surely love to parody had he not invented it. Furthermore, the novel is constructed around a number of openings of other novels that never, for a variety of reasons, progress past the first few pages.

Metafiction is used to represent the impossibility of understanding the global world, particularly the complexity of politics and economics. Critically acclaimed metafictive novels include Australian Peter Carey’s Illywhacker (1985), American Mark Z. Danielewski’s House of Leaves (2000), Canadian Yann Martel’s Life of Pi (2001), and South-African Nobel laureate J. M. Coetzee’s Slow Man (2005). Junot Díaz won the Pulitzer Prize and the National Book Critics Circle Award for his first novel, The Brief Wondrous Life of Oscar Wao (2007). The novel is ostensibly a love story about a Dominican American man in New Jersey, but through a series of extended footnotes and asides the narrator relates and comments on the history of the brutal dictatorship of Rafael Trujillo in the Dominican Republic. The novel has different narrators and settings, as well as occasional messages from the author, and is filled with wordplay and lively slang in English and Spanish.

Fiction as Artifice

One might well ask if metafiction is not too narrow an endeavor to define an age (for example, postmodernism). The answer would be yes, even If on a Winter’s Night a Traveler , for example, might more properly be described as a novel whose subject is reading a novel rather than writing one. Metafiction is best considered one variation of a broader, more pervasive impulse in post-World War II long fiction: fiction-as-artifice. Rather than narrowly focusing on the process of writing fiction (metafiction), in fiction-as-artifice the author directly attacks the conventions of realism or acknowledges that all writing is a verbal construct bearing the most tenuous relationship to actuality.

One of the earliest examples of fiction-as-artifice in the post-World War II canon is Raymond Queneau’s Exercices de style (1947; Exercises in Style , 1958). The title states where Queneau’s interests lie: in technique and in the manipulation of language, rather than in creating an illusion of reality. The book comprises ninety-nine variations on a brief scene between two strangers on a Parisian bus. In each retelling of the incident, Queneau uses a different dialect or style (“Notation” and “Litotes”). The almost endless replication of the single scene forces the reader to see that scene as a verbal construct rather than an approximation of reality. Such “pure” manifestations of fiction-as-artifice as Exercises in Style are relatively rare. More often, fiction-as-artifice is a gesture employed intermittently, side by side with realist techniques. The interplay of the two opposing strategies create a delightful aesthetic friction.

One of the most famous and provocative examples of fiction-as-artifice is Vladimir Nabokov ’s Pale Fire (1962). The structure of the work belies all traditional conventions of the novel. Pale Fire opens with an “editor’s” introduction, followed by a long poem, hundreds of pages of annotations, and an index. The reader discovers, however, that this apparatus tells a hilarious and moving story of political intrigue, murder, and madness. Does Pale Fire , ultimately, underscore the artifices of fiction or, instead, demonstrate how resilient is the writer’s need to tell a story and the reader’s need to read one? Either way, Pale Fire is one of the most inventive and fascinating novels of any genre.

The same questions could be asked of Julio Cortázar’s Rayuela (1963; Hopscotch, 1966), a long novel comprising scores of brief, numbered sections, which, the reader is advised in the introductory “Instructions,” can be read in a number of ways: in the order presented, in a different numbered sequence suggested by the author, or perhaps, if the reader prefers, by “hopscotching” through the novel.

A similar strategy is employed in Milorad Pavic’s Hazarski renik: Roman leksikon u 100,000 reci (1984; Dictionary of the Khazars: A Lexicon Novel in 100,000 Words , 1988). The work is constructed as a dictionary with many brief sections, alphabetized by heading and richly cross-referenced. The reader may read the work from beginning to end, alphabetically, or may follow the cross-references. An added inventive complication is the Dictionary of the Khazars’s two volumes, one “male” and the other “female.” The volumes are identical except for one brief passage, which likely alters the reader’s interpretation of the whole.

Although fiction-as-artifice is European in origin— indeed, it can be traced back to Laurence Sterne’s Tristram Shandy (1759-1767)—its most inventive and varied practitioner is the American writer John Barth . In work after work, Barth employs, parodies, and lays bare for the reader’s contemplation the artifices of fiction.

In his unified collection Lost in the Funhouse (1968), for example, Barth narrates the history—from conception through maturity and decline—of a man, of humankind, and of fiction itself. The story’s telling, however, highlights the artificiality of writing. The first selection (it cannot be called a “story”) of the novel, “Frame Tale,” is a single, incomplete sentence—“Once upon a time there was a story that began”—which is designed to be cut out and pasted together to form a Möbius strip. When assembled, the strip leads to the complete yet infinite and never-ending sentence “Once upon a time there was a story that began Once upon a time. . . .” The novel’s title story, “ Lost in the Funhouse, ” contains graphs illustrating the story’s structure. “Glossolalia” is formed from six brief sections all written in the rhythms of the Lord’s Prayer . In “ Menelaid ,” the dialogue is presented in a dizzying succession of quotation marks within quotation marks within quotation marks. Barth’s experiments in Lost in the Funhouse are continued and intensified in later novels, especially Chimera (1972) and Letters (1979).

In The Broom of the System (1987) by David Foster Wallace, the protagonist, Lenore Stonecipher Beadsman, feels sometimes that she is just a character in a novel. Wallace’s Infinite Jest (1996) is a massive work about a future North America where people become so engrossed in watching a film called Infinite Jest that they lose all interest in other activities. Wallace’s fiction moves back and forth in time without warning and combines wordplay, long sentences, footnotes and endnotes, transcripts, and acronyms to create a disjointed postmodern language.

Mark Dunn’s Ella Minnow Pea: A Progressively Lipogrammatic Epistolary Fable ( 2001) is an epistolary novel about a fictional island off the coast of South Carolina, where the government forbids the use of one letter of the alphabet, at a given time; in effect, the government is parsing the alphabet. The novel is a collection of letters and notes from inhabitants, often less than a page in length, written with a diminishing set of alphabet letters.

Fiction or Nonfiction?

Even at his most experimental, however, Barth never abandons his delight in storytelling. Indeed, virtually all the long fiction addressed thus far show innovations in certain technical strategies but do not substantially challenge the reader’s concept of what is “fictional.” A number of other writers, however, while not always seeming so boldly experimental in technique, have blurred the distinctions between fiction and nonfiction and thus perhaps represent a more fundamental departure from the conventional novel.

The new journalists—such as Truman Capote ( In Cold Blood , 1966), Norman Mailer ( The Armies of the Night: History as a Novel, the Novel as History , 1968), Tom Wolfe ( The Electric Kool-Aid Acid Test , 1968), and Hunter S. Thompson with his Fear and Loathing series (beginning in 1972)—blur the lines between fiction and nonfiction by using novelistic techniques to report facts. However, the subtitle of Mailer’s work notwithstanding, the reader rarely is uncertain what side of the fictionnonfiction line these authors occupy. The same cannot be said for Don DeLillo (Libra, 1988). For his interpretation of the Kennedy assassination, DeLillo spent countless hours researching the voluminous reports of the Warren Commission and other historical documents. With this factual material as the basis for the novel and with assassin Lee Harvey Oswald as the central character, the degree to which Libra can be considered fictional as opposed to nonfictional remains a challenging question.

The question is even more problematic in reference to Maxine Hong Kingston’s The Woman Warrior: Memoirs of a Girlhood Among Ghosts (1976). Kingston conducted her research for her memoir not in library stacks but by plumbing her own memory, especially of stories told her by her mother. At times, Kingston not only is imaginatively enhancing reconstructed scenes but also is inventing details. Is this a work of autobiography or a kind of fiction?

Publishers had trouble classifying Nicholson Baker ’s The Mezzanine (1988) and W. G. Sebald’s Die Ausgewanderten (1992; The Emigrants , 1996). The reader is fairly certain that the point-of-view character in The Mezzanine is fictional, but in what sense is his experience fictional? The work, made up of essaylike contemplations of whatever the persona’s eye falls on as he goes about his business on a mezzanine, recalls in some ways the intensely detailed descriptions of the New Novelists but with even less of an apparent conflict or movement toward climax one expects in fiction.

Sebald’s work is in some ways even odder. His short biographies of a selection of dislocated Europeans have a documentary feel—complete with photographs. The photographs, however, have a vagueness about them that makes them seem almost irrelevant to their subjects, and the reader has the uneasy impression that the book may well be a fabrication.

The distinction between fiction and nonfiction was brought into new relief in 2003, when James Frey published A Million Little Pieces , his memoir of escaping drug addiction. The memoir was aggressively gritty in its detail of the author’s struggles, and Frey was widely praised for his courage and honesty in revealing his own mistakes and weaknesses. In 2005 the book was named to Oprah’s Book Club , and soon after topped nonfiction best-seller lists. In 2006, however, much of the material in the “memoir” was found to have been fabricated. The resulting clamor from talk-show host Oprah Winfrey, Frey’s publishers, and readers led to a lively and interesting public debate over art and truth. Frey claimed that the work presented the “essential truth” of his life. Many readers continued to value the book as an honest accounting of what life is like for some addicts, but readers who felt that they had been defrauded were offered a refund. The Brooklyn Public Library, for example, moved the book to its fiction section.

Just as Baker and Sebald call into question what earlier generations would have thought too obvious to debate—the difference between fiction and nonfiction— one consistent impulse among experimenters in long fiction has been the question, What is necessary in fiction and what is merely conventional? Their efforts to test this question have brought readers some of the most provocative and entertaining works of fiction in the late twentieth and early twenty-first centuries.

BS Johnson with pages from his 1968 novel-in-a-box The Unfortunates , which could be shuffled and read in any order.

Bibliography Currie, Mark, ed. Metafiction. London: Longman, 1995. Collection of articles on experimental themes and techniques. Friedman, Ellen G., and Miriam Fuchs, eds. Breaking the Sequence: Women’s Experimental Fiction. Princeton, N.J.: Princeton University Press, 1989. Levitt, Morton. The Rhetoric of Modernist Fiction from a New Point of View. Hanover, N.H.: University Press of New England, 2006. Robbe-Grillet, Alain. For a New Novel: Essays on Fiction. 1965. New ed. Translated by Richard Howard. Evanston, Ill.: Northwestern University Press, 1996. Seltzer, Alvin J. Chaos in the Novel: The Novel in Chaos. New York: Schocken Books, 1974. Shiach, Morag, ed. The Cambridge Companion to the Modernist Novel. New York: Cambridge University Press, 2007. Waugh, Patricia. Metafiction: The Theory and Practice of Self-Conscious Fiction. New York: Routledge, 2005. Rollyson, Carl. Critical Survey Of Long Fiction. 4th ed. New Jersey: Salem Press, 2010.

Student’s t-Test – a significant difference?

The ambition is that this article should be self-contained. However some concepts, such as standard deviation, are discussed in more detail in the previous article .

What is a significance test?

You’ve reached the point towards the end of a lab report, and you need to compare your experimentally derived mean value with a literature value. Then you need to comment on the ‘significance’ of the difference between the two. So how do you do that?

This article looks at the Student’s t-Test; a method to assess if the difference between two sets of data is ‘significantly different’.

It’s been written as a practical guide, so I’m not going to include some of the more ‘gory’ statistical details. If you want to learn more about significance testing in a broader sense, I suggest Chapter 4 of “Practical Statistics for the Analytical Scientist” by Ellison, Barwick, and Duguid Farrant. For Edinburgh students, it’s available as a free e-book via the University of Edinburgh intranet.

Quality testing at Guinness

The Student’s t-Test is a significance test, developed by William Sealy Gosset in the 1900s. While working at Guinness, he developed the test as a cheap way of monitoring the quality of the stout.

At the time, Guinness forbade employees from publishing results openly, so it was necessary for Gosset to use a pseudonym for his test.

It’s called the Student’s t-Test, rather than Gosset’s t-Test for this reason.

So what is it, and why do we use it?

The aim of the Student’s t-Test is to compare an experimental value, (determined as the mean of a series of measurements), with a ‘literature value’ taken from a paper, or some value quoted in your laboratory manual. This comparison shows how closely your results compare with published data.

To demonstrate how the Student’s t-Test is used, I’m going to use the same example of ‘activation energy’ discussed in the previous article All About Errors. (See Table 1 below.)

The purpose of significance testing is to calculate the probability that observed differences occur as a result of random fluctuations. In other words, the likelihood of two values being different due to chance alone. So it’s necessary to define this ‘likeliness’ as a probability, called the significance level, α .

Most commonly (and for the sake of simplicity), the significance level is set to be 0.05. α = 0.05 means that there is a 5% probability of the two values being different due to random fluctuations. The significance level is also known as the confidence level ( CL ), essentially the ‘unlikeliness’ of a difference arising due to only chance.

Table 1 shows the results of 12 measurements of activation energy ( E a ) for some reaction. The literature value of the activation energy for this reaction is 38.942 kJ/mol.

Table 1. Sample measurements of E a (kJ/mol)

	(kJ/mol)
1	38.542
2	38.751
3	39.016
4	38.768
5	38.943
6	38.426
7	39.124
8	38.546
9	38.864
10	38.687
11	38.579
12	38.798

Student’s t-Test statistic

3 metrics are required to calculate the t-Test statistic, t :

1. Mean activation energy: x̄ = 38.753 kJ/mol.

2. Standard deviation: s = 0.210 kJ/mol. (Deriving the mean and standard deviation is discussed in All About Errors .)

3. Literature value: μ = 38.942 kJ/mol.

Where, n , is the number of values in the data set, since there are 12 measurements in Table 1, n = 12 . This means that for the given data set and literature value, the t-Test statistic is:

(Eqn 3)

The critical value

The value of the t-Test statistic is meaningless without something to compare it to. This is where the significance level comes in; the significance level is used to find the critical value. The critical value can be found from ‘statistical tables’ located at the back of statistics textbooks or online.

The easiest way to find the critical value is to use Microsoft Excel with the formula “ =TINV( α , ν) ” using the significance level desired ( α ) with the degrees of freedom ( ν ). Degrees of freedom ( v ) is the number of values in a data set ( n ) minus one. (Note that the symbol for degrees of freedom is the greek letter pronounced nu , not the letter ‘v’.)

The Microsoft Excel formula will return the critical value for the particular α and ν . For α = 0.05 and ν = 11 , as is the case in our activation energy example, the critical value is:

An example of how to use the Microsoft Excel formula to find this value is shown in Figure 1.

Figure 1. How to use the function in Microsoft Excel to find the critical value for a given significance level and degrees of freedom.

Comparing the t-Test statistic with critical value

When comparing the calculated t-Test statistic with the critical value, there are two possible outcomes:

t-Test statistic < critical value

The experimentally determined value is not significantly different from the literature value. The difference between the two values is due to chance.

t-Test statistic ≥ critical value

The experimentally determined value is significantly different from the literature value. This means that you should consider experimental errors that could give rise to this difference.

So is our difference significant?

As can be seen from Equations 3 and 5 in our example of activation energy, the t-Test statistic is greater than the critical value. This indicates the difference between the experimental mean and literature value is not due to chance, and caused by some experimental or systematic error which should be accounted for. This could be due to poor experimental practice, low precision in equipment, or some other factor.

It should be noted that the Student’s t-Test is just one of a range of significance tests, the use of which depends on the type of “distribution” under study (this is some of the more ‘gory’ detail discussed in the book “Practical Statistics for the Analytical Scientist” mentioned earlier).

So, that’s a brief guide showing how it’s possible to assess if a difference between two data sets is significant. This is particularly useful when comparing the mean value, determined experimentally with some literature value.

Hope you found it helpful, and if you’d like me to cover any other topics just let us know., share this:.

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Expected accuracy of experimental IR spectra compared to literature reference

I have a table from a journal which indicated signals of an alpha chitin that I used as theoretical values. I did my own analysis to identify a chitin sample and it had signals that slightly varied from the theoretical (ex. $3106\ \mathrm{cm}^{-1}$ symmetrical N-H vibration and I got $3103\ \mathrm{cm}^{-1}$). Should I consider this? And what caused the variation?

organic-chemistry
spectroscopy
ir-spectroscopy

3 $\begingroup$ seems pretty close to me, only 3 in 3100; what is the resolution with which the spectra was taken, are the lines overlapping , was the sample environment the same etc? These may explain the very small difference as may slight changes in calibration. $\endgroup$ – porphyrin Commented May 3, 2017 at 14:14
$\begingroup$ Hello @porphyrin, the lines weren't overlapping and the sample environment were the same. $\endgroup$ – Byte Commented May 4, 2017 at 14:01
$\begingroup$ It is a very good agreement, most likely within experimental error. $\endgroup$ – porphyrin Commented May 4, 2017 at 15:27

In addition to @porphyrin's comment, the interaction of your sample with the surrounding may overall cause some changes in shape and place of the absorption bands; namely if your analysis

follows the classical sample preparation, grinding your compound altogether with excess of KCl to press a pellet (ionic environment) / mixed in nujol (apolar environment) that is then characterised in transmission
is an analysis by ATR with single or multiple reflection where your sample basically is a neat film on top of your optical window; either probe head or between crystal and anvil. (The ATR correction offered by the spectrometer's software, not every time active, basically will account for the penetration depth varying by the wavelength used, hence influencing rather the displayed %transmittance, then the position of the barycentre of the absorption band in question.)

If you aim for a qualitative IR analysis of a small organic molecule, I would not worry too much if the reported IR band was reported at 3106 cm$^{-1}$ and is now spot at 3103 cm$^{-1}$. The iS10 ATR-FT-IR spectrometer by Nicolet/ThermoScientific, for example, provides a data spacing of around 0.7 cm$^{-1}$. Performing a library search in an electronic database to identify / narrow which compound the sample in question is hopefully would not be affected, either.

(A quantification , like determining the composition of gasoline by PCA , would of course need some calibration.)

The original question however may be read as if you aim to monitor the (partial) digestion of larger proteins and polysaccharides, to say something about their secondary structures. A shift of 10...20 cm$^{-1}$ of an amide absorption band may be relevant; yet then more often the ones around 1600...1700 cm$^{-1}$ are taken into consideration.

$\begingroup$ Hello @buttonwood, I used a ThermoScientific FT-IR spectrometer with an old OMNIC software and proceeded with the KBr pellet method. I was aiming for a qualitative analysis and got confused with the slight difference. Thanks for clarifying it though! $\endgroup$ – Byte Commented May 4, 2017 at 14:03

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Chapter 3

Experimental Errors and

Error Analysis

This chapter is largely a tutorial on handling experimental errors of measurement. Much of the material has been extensively tested with science undergraduates at a variety of levels at the University of Toronto.

Whole books can and have been written on this topic but here we distill the topic down to the essentials. Nonetheless, our experience is that for beginners an iterative approach to this material works best. This means that the users first scan the material in this chapter; then try to use the material on their own experiment; then go over the material again; then ...

provides functions to ease the calculations required by propagation of errors, and those functions are introduced in Section 3.3. These error propagation functions are summarized in Section 3.5.

3.1 Introduction

3.1.1 The Purpose of Error Analysis

For students who only attend lectures and read textbooks in the sciences, it is easy to get the incorrect impression that the physical sciences are concerned with manipulating precise and perfect numbers. Lectures and textbooks often contain phrases like:

For an experimental scientist this specification is incomplete. Does it mean that the acceleration is closer to 9.8 than to 9.9 or 9.7? Does it mean that the acceleration is closer to 9.80000 than to 9.80001 or 9.79999? Often the answer depends on the context. If a carpenter says a length is "just 8 inches" that probably means the length is closer to 8 0/16 in. than to 8 1/16 in. or 7 15/16 in. If a machinist says a length is "just 200 millimeters" that probably means it is closer to 200.00 mm than to 200.05 mm or 199.95 mm.

We all know that the acceleration due to gravity varies from place to place on the earth's surface. It also varies with the height above the surface, and gravity meters capable of measuring the variation from the floor to a tabletop are readily available. Further, any physical measure such as can only be determined by means of an experiment, and since a perfect experimental apparatus does not exist, it is impossible even in principle to ever know perfectly. Thus, the specification of given above is useful only as a possible exercise for a student. In order to give it some meaning it must be changed to something like:

Two questions arise about the measurement. First, is it "accurate," in other words, did the experiment work properly and were all the necessary factors taken into account? The answer to this depends on the skill of the experimenter in identifying and eliminating all systematic errors. These are discussed in Section 3.4.

The second question regards the "precision" of the experiment. In this case the precision of the result is given: the experimenter claims the precision of the result is within 0.03 m/s

1. The person who did the measurement probably had some "gut feeling" for the precision and "hung" an error on the result primarily to communicate this feeling to other people. Common sense should always take precedence over mathematical manipulations.

2. In complicated experiments, error analysis can identify dominant errors and hence provide a guide as to where more effort is needed to improve an experiment.

3. There is virtually no case in the experimental physical sciences where the correct error analysis is to compare the result with a number in some book. A correct experiment is one that is performed correctly, not one that gives a result in agreement with other measurements.

4. The best precision possible for a given experiment is always limited by the apparatus. Polarization measurements in high-energy physics require tens of thousands of person-hours and cost hundreds of thousand of dollars to perform, and a good measurement is within a factor of two. Electrodynamics experiments are considerably cheaper, and often give results to 8 or more significant figures. In both cases, the experimenter must struggle with the equipment to get the most precise and accurate measurement possible.

3.1.2 Different Types of Errors

As mentioned above, there are two types of errors associated with an experimental result: the "precision" and the "accuracy". One well-known text explains the difference this way:

" " E.M. Pugh and G.H. Winslow, p. 6.

The object of a good experiment is to minimize both the errors of precision and the errors of accuracy.

Usually, a given experiment has one or the other type of error dominant, and the experimenter devotes the most effort toward reducing that one. For example, in measuring the height of a sample of geraniums to determine an average value, the random variations within the sample of plants are probably going to be much larger than any possible inaccuracy in the ruler being used. Similarly for many experiments in the biological and life sciences, the experimenter worries most about increasing the precision of his/her measurements. Of course, some experiments in the biological and life sciences are dominated by errors of accuracy.

On the other hand, in titrating a sample of HCl acid with NaOH base using a phenolphthalein indicator, the major error in the determination of the original concentration of the acid is likely to be one of the following: (1) the accuracy of the markings on the side of the burette; (2) the transition range of the phenolphthalein indicator; or (3) the skill of the experimenter in splitting the last drop of NaOH. Thus, the accuracy of the determination is likely to be much worse than the precision. This is often the case for experiments in chemistry, but certainly not all.

Question: Most experiments use theoretical formulas, and usually those formulas are approximations. Is the error of approximation one of precision or of accuracy?

3.1.3 References

There is extensive literature on the topics in this chapter. The following lists some well-known introductions.

D.C. Baird, (Prentice-Hall, 1962)

E.M. Pugh and G.H. Winslow, (Addison-Wesley, 1966)

J.R. Taylor, (University Science Books, 1982)

In addition, there is a web document written by the author of that is used to teach this topic to first year Physics undergraduates at the University of Toronto. The following Hyperlink points to that document.

3.2 Determining the Precision

3.2.1 The Standard Deviation

In the nineteenth century, Gauss' assistants were doing astronomical measurements. However, they were never able to exactly repeat their results. Finally, Gauss got angry and stormed into the lab, claiming he would show these people how to do the measurements once and for all. The only problem was that Gauss wasn't able to repeat his measurements exactly either!

After he recovered his composure, Gauss made a histogram of the results of a particular measurement and discovered the famous Gaussian or bell-shaped curve.

Many people's first introduction to this shape is the grade distribution for a course. Here is a sample of such a distribution, using the function .

We use a standard package to generate a Probability Distribution Function ( ) of such a "Gaussian" or "normal" distribution. The mean is chosen to be 78 and the standard deviation is chosen to be 10; both the mean and standard deviation are defined below.

We then normalize the distribution so the maximum value is close to the maximum number in the histogram and plot the result.

In this graph,

Finally, we look at the histogram and plot together.

We can see the functional form of the Gaussian distribution by giving symbolic values.

In this formula, the quantity , and . The is sometimes called the . The definition of is as follows.

Here is the total number of measurements and is the result of measurement number .

The standard deviation is a measure of the width of the peak, meaning that a larger value gives a wider peak.

If we look at the area under the curve from graph, we find that this area is 68 percent of the total area. Thus, any result chosen at random has a 68% change of being within one standard deviation of the mean. We can show this by evaluating the integral. For convenience, we choose the mean to be zero.

Now, we numericalize this and multiply by 100 to find the percent.

The only problem with the above is that the measurement must be repeated an infinite number of times before the standard deviation can be determined. If is less than infinity, one can only estimate measurements, this is the best estimate.

The major difference between this estimate and the definition is the . This is reasonable since if = 1 we know we can't determine

Here is an example. Suppose we are to determine the diameter of a small cylinder using a micrometer. We repeat the measurement 10 times along various points on the cylinder and get the following results, in centimeters.

The number of measurements is the length of the list.

The average or mean is now calculated.

Then the standard deviation is to be 0.00185173.

We repeat the calculation in a functional style.

Note that the package, which is standard with , includes functions to calculate all of these quantities and a great deal more.

We close with two points:

1. The standard deviation has been associated with the error in each individual measurement. Section 3.3.2 discusses how to find the error in the estimate of the average.

2. This calculation of the standard deviation is only an estimate. In fact, we can find the expected error in the estimate,

As discussed in more detail in Section 3.3, this means that the true standard deviation probably lies in the range of values.

Viewed in this way, it is clear that the last few digits in the numbers above for function adjusts these significant figures based on the error.

is discussed further in Section 3.3.1.

3.2.2 The Reading Error

There is another type of error associated with a directly measured quantity, called the "reading error". Referring again to the example of Section 3.2.1, the measurements of the diameter were performed with a micrometer. The particular micrometer used had scale divisions every 0.001 cm. However, it was possible to estimate the reading of the micrometer between the divisions, and this was done in this example. But, there is a reading error associated with this estimation. For example, the first data point is 1.6515 cm. Could it have been 1.6516 cm instead? How about 1.6519 cm? There is no fixed rule to answer the question: the person doing the measurement must guess how well he or she can read the instrument. A reasonable guess of the reading error of this micrometer might be 0.0002 cm on a good day. If the experimenter were up late the night before, the reading error might be 0.0005 cm.

An important and sometimes difficult question is whether the reading error of an instrument is "distributed randomly". Random reading errors are caused by the finite precision of the experiment. If an experimenter consistently reads the micrometer 1 cm lower than the actual value, then the reading error is not random.

For a digital instrument, the reading error is ± one-half of the last digit. Note that this assumes that the instrument has been properly engineered to round a reading correctly on the display.

3.2.3 "THE" Error

So far, we have found two different errors associated with a directly measured quantity: the standard deviation and the reading error. So, which one is the actual real error of precision in the quantity? The answer is both! However, fortunately it almost always turns out that one will be larger than the other, so the smaller of the two can be ignored.

In the diameter example being used in this section, the estimate of the standard deviation was found to be 0.00185 cm, while the reading error was only 0.0002 cm. Thus, we can use the standard deviation estimate to characterize the error in each measurement. Another way of saying the same thing is that the observed spread of values in this example is not accounted for by the reading error. If the observed spread were more or less accounted for by the reading error, it would not be necessary to estimate the standard deviation, since the reading error would be the error in each measurement.

Of course, everything in this section is related to the precision of the experiment. Discussion of the accuracy of the experiment is in Section 3.4.

3.2.4 Rejection of Measurements

Often when repeating measurements one value appears to be spurious and we would like to throw it out. Also, when taking a series of measurements, sometimes one value appears "out of line". Here we discuss some guidelines on rejection of measurements; further information appears in Chapter 7.

It is important to emphasize that the whole topic of rejection of measurements is awkward. Some scientists feel that the rejection of data is justified unless there is evidence that the data in question is incorrect. Other scientists attempt to deal with this topic by using quasi-objective rules such as 's . Still others, often incorrectly, throw out any data that appear to be incorrect. In this section, some principles and guidelines are presented; further information may be found in many references.

First, we note that it is incorrect to expect each and every measurement to overlap within errors. For example, if the error in a particular quantity is characterized by the standard deviation, we only expect 68% of the measurements from a normally distributed population to be within one standard deviation of the mean. Ninety-five percent of the measurements will be within two standard deviations, 99% within three standard deviations, etc., but we never expect 100% of the measurements to overlap within any finite-sized error for a truly Gaussian distribution.

Of course, for most experiments the assumption of a Gaussian distribution is only an approximation.

If the error in each measurement is taken to be the reading error, again we only expect most, not all, of the measurements to overlap within errors. In this case the meaning of "most", however, is vague and depends on the optimism/conservatism of the experimenter who assigned the error.

Thus, it is always dangerous to throw out a measurement. Maybe we are unlucky enough to make a valid measurement that lies ten standard deviations from the population mean. A valid measurement from the tails of the underlying distribution should not be thrown out. It is even more dangerous to throw out a suspect point indicative of an underlying physical process. Very little science would be known today if the experimenter always threw out measurements that didn't match preconceived expectations!

In general, there are two different types of experimental data taken in a laboratory and the question of rejecting measurements is handled in slightly different ways for each. The two types of data are the following:

1. A series of measurements taken with one or more variables changed for each data point. An example is the calibration of a thermocouple, in which the output voltage is measured when the thermocouple is at a number of different temperatures.

2. Repeated measurements of the same physical quantity, with all variables held as constant as experimentally possible. An example is the measurement of the height of a sample of geraniums grown under identical conditions from the same batch of seed stock.

For a series of measurements (case 1), when one of the data points is out of line the natural tendency is to throw it out. But, as already mentioned, this means you are assuming the result you are attempting to measure. As a rule of thumb, unless there is a physical explanation of why the suspect value is spurious and it is no more than three standard deviations away from the expected value, it should probably be kept. Chapter 7 deals further with this case.

For repeated measurements (case 2), the situation is a little different. Say you are measuring the time for a pendulum to undergo 20 oscillations and you repeat the measurement five times. Assume that four of these trials are within 0.1 seconds of each other, but the fifth trial differs from these by 1.4 seconds ( , more than three standard deviations away from the mean of the "good" values). There is no known reason why that one measurement differs from all the others. Nonetheless, you may be justified in throwing it out. Say that, unknown to you, just as that measurement was being taken, a gravity wave swept through your region of spacetime. However, if you are trying to measure the period of the pendulum when there are no gravity waves affecting the measurement, then throwing out that one result is reasonable. (Although trying to repeat the measurement to find the existence of gravity waves will certainly be more fun!) So whatever the reason for a suspect value, the rule of thumb is that it may be thrown out provided that fact is well documented and that the measurement is repeated a number of times more to convince the experimenter that he/she is not throwing out an important piece of data indicating a new physical process.

3.3 Propagation of Errors of Precision

3.3.1 Discussion and Examples

Usually, errors of precision are probabilistic. This means that the experimenter is saying that the actual value of some parameter is within a specified range. For example, if the half-width of the range equals one standard deviation, then the probability is about 68% that over repeated experimentation the true mean will fall within the range; if the half-width of the range is twice the standard deviation, the probability is 95%, etc.

If we have two variables, say and , and want to combine them to form a new variable, we want the error in the combination to preserve this probability.

The correct procedure to do this is to combine errors in quadrature, which is the square root of the sum of the squares. supplies a function.

For simple combinations of data with random errors, the correct procedure can be summarized in three rules. will stand for the errors of precision in , , and , respectively. We assume that and are independent of each other.

Note that all three rules assume that the error, say , is small compared to the value of .

z = x * y

then

In words, the fractional error in is the quadrature of the fractional errors in and .

z = x + y

z = x - y

then

In words, the error in is the quadrature of the errors in and .

then

or equivalently

includes functions to combine data using the above rules. They are named , , , , and .

Imagine we have pressure data, measured in centimeters of Hg, and volume data measured in arbitrary units. Each data point consists of { , } pairs.

We calculate the pressure times the volume.

In the above, the values of and have been multiplied and the errors have ben combined using Rule 1.

There is an equivalent form for this calculation.

Consider the first of the volume data: {11.28156820762763, 0.031}. The error means that the true value is claimed by the experimenter to probably lie between 11.25 and 11.31. Thus, all the significant figures presented to the right of 11.28 for that data point really aren't significant. The function will adjust the volume data.

Notice that by default, uses the two most significant digits in the error for adjusting the values. This can be controlled with the option.

For most cases, the default of two digits is reasonable. As discussed in Section 3.2.1, if we assume a normal distribution for the data, then the fractional error in the determination of the standard deviation , and can be written as follows.

Thus, using this as a general rule of thumb for all errors of precision, the estimate of the error is only good to 10%, ( one significant figure, unless is greater than 51) . Nonetheless, keeping two significant figures handles cases such as 0.035 vs. 0.030, where some significance may be attached to the final digit.

You should be aware that when a datum is massaged by , the extra digits are dropped.

By default, and the other functions use the function. The use of is controlled using the option.

The number of digits can be adjusted.

To form a power, say,

we might be tempted to just do

function.

Finally, imagine that for some reason we wish to form a combination.

We might be tempted to solve this with the following.

then the error is

Here is an example solving . We shall use and below to avoid overwriting the symbols and . First we calculate the total derivative.

Next we form the error.

Now we can evaluate using the pressure and volume data to get a list of errors.

Next we form the list of pairs.

The function combines these steps with default significant figure adjustment.

The function can be used in place of the other functions discussed above.

In this example, the function will be somewhat faster.

There is a caveat in using . The expression must contain only symbols, numerical constants, and arithmetic operations. Otherwise, the function will be unable to take the derivatives of the expression necessary to calculate the form of the error. The other functions have no such limitation.

3.3.1.1 Another Approach to Error Propagation: The and Datum

value error

Data[{{789.7, 2.2}, {790.8, 2.3}, {791.2, 2.3}, {792.6, 2.4}, {791.8, 2.5},
{792.2, 2.5}, {794.7, 2.6}, {794., 2.6}, {794.4, 2.7}, {795.3, 2.8},
{796.4, 2.8}}]Data[{{789.7, 2.2}, {790.8, 2.3}, {791.2, 2.3}, {792.6, 2.4}, {791.8, 2.5},

{792.2, 2.5}, {794.7, 2.6}, {794., 2.6}, {794.4, 2.7}, {795.3, 2.8},

{796.4, 2.8}}]

The wrapper can be removed.

{{789.7, 2.2}, {790.8, 2.3}, {791.2, 2.3}, {792.6, 2.4}, {791.8, 2.5},
{792.2, 2.5}, {794.7, 2.6}, {794., 2.6}, {794.4, 2.7}, {795.3, 2.8}, {796.4, 2.8}}{{789.7, 2.2}, {790.8, 2.3}, {791.2, 2.3}, {792.6, 2.4}, {791.8, 2.5},

{792.2, 2.5}, {794.7, 2.6}, {794., 2.6}, {794.4, 2.7}, {795.3, 2.8}, {796.4, 2.8}}

The reason why the output of the previous two commands has been formatted as is that typesets the pairs using ± for output.

A similar construct can be used with individual data points.

Datum[{70, 0.04}]Datum[{70, 0.04}]

Just as for , the typesetting of uses

The and constructs provide "automatic" error propagation for multiplication, division, addition, subtraction, and raising to a power. Another advantage of these constructs is that the rules built into know how to combine data with constants.

The rules also know how to propagate errors for many transcendental functions.

This rule assumes that the error is small relative to the value, so we can approximate.

or arguments, are given by .

We have seen that typesets the and constructs using ±. The function can be used directly, and provided its arguments are numeric, errors will be propagated.

One may typeset the ± into the input expression, and errors will again be propagated.

The ± input mechanism can combine terms by addition, subtraction, multiplication, division, raising to a power, addition and multiplication by a constant number, and use of the . The rules used by for ± are only for numeric arguments.

This makes different than

3.3.1.2 Why Quadrature?

Here we justify combining errors in quadrature. Although they are not proofs in the usual pristine mathematical sense, they are correct and can be made rigorous if desired.

First, you may already know about the "Random Walk" problem in which a player starts at the point = 0 and at each move steps either forward (toward + ) or backward (toward - ). The choice of direction is made randomly for each move by, say, flipping a coin. If each step covers a distance , then after steps the expected most probable distance of the player from the origin can be shown to be

Thus, the distance goes up as the square root of the number of steps.

Now consider a situation where measurements of a quantity are performed, each with an identical random error . We find the sum of the measurements.

, it is equally likely to be + as - , and which is essentially random. Thus, the expected most probable error in the sum goes up as the square root of the number of measurements.

This is exactly the result obtained by combining the errors in quadrature.

Another similar way of thinking about the errors is that in an abstract linear error space, the errors span the space. If the errors are probabilistic and uncorrelated, the errors in fact are linearly independent (orthogonal) and thus form a basis for the space. Thus, we would expect that to add these independent random errors, we would have to use Pythagoras' theorem, which is just combining them in quadrature.

3.3.2 Finding the Error in an Average

The rules for propagation of errors, discussed in Section 3.3.1, allow one to find the error in an average or mean of a number of repeated measurements. Recall that to compute the average, first the sum of all the measurements is found, and the rule for addition of quantities allows the computation of the error in the sum. Next, the sum is divided by the number of measurements, and the rule for division of quantities allows the calculation of the error in the result ( the error of the mean).

In the case that the error in each measurement has the same value, the result of applying these rules for propagation of errors can be summarized as a theorem.

Theorem: If the measurement of a random variable is repeated times, and the random variable has standard deviation , then the standard deviation in the mean is

Proof: One makes measurements, each with error .

{x1, errx}, {x2, errx}, ... , {xn, errx}

We calculate the sum.

sumx = x1 + x2 + ... + xn

We calculate the error in the sum.

This last line is the key: by repeating the measurements times, the error in the sum only goes up as [ ].

The mean

Applying the rule for division we get the following.

This completes the proof.

The quantity called

Here is an example. In Section 3.2.1, 10 measurements of the diameter of a small cylinder were discussed. The mean of the measurements was 1.6514 cm and the standard deviation was 0.00185 cm. Now we can calculate the mean and its error, adjusted for significant figures.

Note that presenting this result without significant figure adjustment makes no sense.

The above number implies that there is meaning in the one-hundred-millionth part of a centimeter.

Here is another example. Imagine you are weighing an object on a "dial balance" in which you turn a dial until the pointer balances, and then read the mass from the marking on the dial. You find = 26.10 ± 0.01 g. The 0.01 g is the reading error of the balance, and is about as good as you can read that particular piece of equipment. You remove the mass from the balance, put it back on, weigh it again, and get = 26.10 ± 0.01 g. You get a friend to try it and she gets the same result. You get another friend to weigh the mass and he also gets = 26.10 ± 0.01 g. So you have four measurements of the mass of the body, each with an identical result. Do you think the theorem applies in this case? If yes, you would quote = 26.100 ± 0.01/ [4] = 26.100 ± 0.005 g. How about if you went out on the street and started bringing strangers in to repeat the measurement, each and every one of whom got = 26.10 ± 0.01 g. So after a few weeks, you have 10,000 identical measurements. Would the error in the mass, as measured on that $50 balance, really be the following?

The point is that these rules of statistics are only a rough guide and in a situation like this example where they probably don't apply, don't be afraid to ignore them and use your "uncommon sense". In this example, presenting your result as = 26.10 ± 0.01 g is probably the reasonable thing to do.

3.4 Calibration, Accuracy, and Systematic Errors

In Section 3.1.2, we made the distinction between errors of precision and accuracy by imagining that we had performed a timing measurement with a very precise pendulum clock, but had set its length wrong, leading to an inaccurate result. Here we discuss these types of errors of accuracy. To get some insight into how such a wrong length can arise, you may wish to try comparing the scales of two rulers made by different companies — discrepancies of 3 mm across 30 cm are common!

If we have access to a ruler we trust ( a "calibration standard"), we can use it to calibrate another ruler. One reasonable way to use the calibration is that if our instrument measures and the standard records , then we can multiply all readings of our instrument by / . Since the correction is usually very small, it will practically never affect the error of precision, which is also small. Calibration standards are, almost by definition, too delicate and/or expensive to use for direct measurement.

Here is an example. We are measuring a voltage using an analog Philips multimeter, model PM2400/02. The result is 6.50 V, measured on the 10 V scale, and the reading error is decided on as 0.03 V, which is 0.5%. Repeating the measurement gives identical results. It is calculated by the experimenter that the effect of the voltmeter on the circuit being measured is less than 0.003% and hence negligible. However, the manufacturer of the instrument only claims an accuracy of 3% of full scale (10 V), which here corresponds to 0.3 V.

Now, what this claimed accuracy means is that the manufacturer of the instrument claims to control the tolerances of the components inside the box to the point where the value read on the meter will be within 3% times the scale of the actual value. Furthermore, this is not a random error; a given meter will supposedly always read too high or too low when measurements are repeated on the same scale. Thus, repeating measurements will not reduce this error.

A further problem with this accuracy is that while most good manufacturers (including Philips) tend to be quite conservative and give trustworthy specifications, there are some manufacturers who have the specifications written by the sales department instead of the engineering department. And even Philips cannot take into account that maybe the last person to use the meter dropped it.

Nonetheless, in this case it is probably reasonable to accept the manufacturer's claimed accuracy and take the measured voltage to be 6.5 ± 0.3 V. If you want or need to know the voltage better than that, there are two alternatives: use a better, more expensive voltmeter to take the measurement or calibrate the existing meter.

Using a better voltmeter, of course, gives a better result. Say you used a Fluke 8000A digital multimeter and measured the voltage to be 6.63 V. However, you're still in the same position of having to accept the manufacturer's claimed accuracy, in this case (0.1% of reading + 1 digit) = 0.02 V. To do better than this, you must use an even better voltmeter, which again requires accepting the accuracy of this even better instrument and so on, ad infinitum, until you run out of time, patience, or money.

Say we decide instead to calibrate the Philips meter using the Fluke meter as the calibration standard. Such a procedure is usually justified only if a large number of measurements were performed with the Philips meter. Why spend half an hour calibrating the Philips meter for just one measurement when you could use the Fluke meter directly?

We measure four voltages using both the Philips and the Fluke meter. For the Philips instrument we are not interested in its accuracy, which is why we are calibrating the instrument. So we will use the reading error of the Philips instrument as the error in its measurements and the accuracy of the Fluke instrument as the error in its measurements.

We form lists of the results of the measurements.

We can examine the differences between the readings either by dividing the Fluke results by the Philips or by subtracting the two values.

The second set of numbers is closer to the same value than the first set, so in this case adding a correction to the Philips measurement is perhaps more appropriate than multiplying by a correction.

We form a new data set of format { }.

We can guess, then, that for a Philips measurement of 6.50 V the appropriate correction factor is 0.11 ± 0.04 V, where the estimated error is a guess based partly on a fear that the meter's inaccuracy may not be as smooth as the four data points indicate. Thus, the corrected Philips reading can be calculated.

(You may wish to know that all the numbers in this example are real data and that when the Philips meter read 6.50 V, the Fluke meter measured the voltage to be 6.63 ± 0.02 V.)

Finally, a further subtlety: Ohm's law states that the resistance is related to the voltage and the current across the resistor according to the following equation.

V = IR

Imagine that we are trying to determine an unknown resistance using this law and are using the Philips meter to measure the voltage. Essentially the resistance is the slope of a graph of voltage versus current.

If the Philips meter is systematically measuring all voltages too big by, say, 2%, that systematic error of accuracy will have no effect on the slope and therefore will have no effect on the determination of the resistance . So in this case and for this measurement, we may be quite justified in ignoring the inaccuracy of the voltmeter entirely and using the reading error to determine the uncertainty in the determination of .

3.5 Summary of the Error Propagation Routines

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How do you find the correct literature value easily?

For my practical write ups, I’ve often been asked to find literature values so that I can compare them with the experimental values. Does anyone have any tips on how I can find correct literature values easily?

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Comparison for Experimental vs Literature values for alumnium

Thread starter uzman1243
Start date Apr 2, 2014
Tags Comparison Experimental Literature
Apr 2, 2014

A PF Molecule

Homework statement.

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Aluminum comes in a variety of different alloys and heat treatments. I hope you have compared like with like, in terms of the material used in the experiment versus the material properties obtained from the literature search.

SteamKing said: Aluminum comes in a variety of different alloys and heat treatments. I hope you have compared like with like, in terms of the material used in the experiment versus the material properties obtained from the literature search.

Related to Comparison for Experimental vs Literature values for alumnium

The purpose of this comparison is to determine the accuracy and reliability of experimental data by comparing it to previously established values found in literature. This can help identify any discrepancies or errors in the experimental procedure or equipment.

There are several factors that can contribute to the difference between experimental and literature values for aluminum. These include variations in experimental technique, sample purity, and environmental conditions such as temperature and pressure.

The experimental and literature values for aluminum can be compared by calculating the percent error. This is done by taking the absolute value of the difference between the two values, dividing it by the literature value, and multiplying by 100%. A lower percent error indicates a closer agreement between the two values.

Having accurate experimental values for aluminum is important for several reasons. Firstly, it ensures the validity and credibility of scientific research. Additionally, accurate values are necessary for the development of new technologies and materials that use aluminum. It also aids in the understanding and prediction of the properties and behavior of aluminum in different environments.

Some potential sources of error in experimental values for aluminum include human error, instrument limitations, and systematic errors in the experimental procedure. It is important for scientists to carefully control and account for these factors to minimize their impact on the accuracy of their results.

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On the influence of h 2 addition on nh 3 laminar flame speed under engine-like conditions.

1. Introduction

2. the model, 2.1. governing equations, 2.2. numerical method, 3. results and discussion, 3.1. model validation, 3.2. a parametric analysis, 3.3. structure of the nh 3 /h 2 flame, 4. conclusions.

LFS exponentially increases with H 2 mole fraction, at 300 K and 800 K and for all pressures and equivalence ratios considered in this work;
In a semi-logarithmic scale, a second-degree polynomial regression accurately predicts the numerical results, with R β 2 varying in the range of 0.99748–0.99997. However, even a linear regression provides a good accuracy with H 2 mole fractions in the range of 0.0–0.6, and such a regression may be employed for less time-consuming computations. Both regressions may be used for further studies, specifically for CFD simulations of engines fueled with synthetic fuels;
α values show that the enhancement of LFS as the H 2 mole fraction increases in the range of 0.0–0.6 is lower as pressure and temperature increase, and as ϕ increases from 0.8 to 1.2;
As expected, LFS decreases as pressure increases. However, the more pressure increases, the less it influences the LFS. Furthermore, the increase in LFS with temperature is higher as pressure increases and H 2 mole fraction decreases;
Under thermodynamic engine-relevant conditions, i.e., 800 K and 4.0 MPa, the flame thickness for a stoichiometric mixture of N H 3 / H 2 / Air with 40% by mole of H 2 in the fuel blend is halved compared to the case of pure ammonia;
The analysis of the flame structure shows the kinetics that leads to LFS enhancement by H 2 : the presence of hydrogen in the fuel mixture enhances the formation of radicals, such as H, O and OH, thus increasing the reactivity of the mixture and, subsequently, the laminar flame speed.

Author Contributions

Data availability statement, conflicts of interest.

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Click here to enlarge figure

Pressure [MPa]	Temperature [K]	Equivalence Ratio	H Mole Fraction in the Fuel Mixture
0.1, 0.3, 0.5, 1.0, 2.0, 3.0, 4.0, 8.0	300, 373, 473, 800	[0.7–1.5]	[0.0–1.0]

Reference	Label	Number of Species	Number of Reactions
Zhang et al. [ ]	M1	38	263
Otomo et al. [ ]	M2	32	231
Gotama et al. [ ]	M3	32	165
Stagni et al. [ ]	M4	41	203
Singh et al. [ ]	M5	32	259

Grid Refinement	Gradient Parameter	Curvature Parameter
0	0.1	0.5
1	0.08	0.3
2	0.06	0.1
3	0.03	0.07
4	0.01/0.015	0.05

Reference	p [MPa]	T [K]	X(H )	ϕ
Ichikawa et al. [ ]	0.1, 0.3, 0.5	298	[0–1]	[0.8–1.2]
Han et al. [ ]	0.1	298	[0–0.45]	[0.8–1.4]
Shrestha et al. [ ]	0.1	473	[0–0.3]	[0.8–1.4]
Smallbone et al. [ ]	0.1	298	1	1
Lee et al. [ ]	0.1	298	0.1, 0.3, 0.5	1
Li et al. [ ]	0.1	293	[0.4–0.6]	1, 1.3, 1.4
Kumar and Meyer [ ]	0.1	298	0.25, 0.55, 0.8, 1	1
Zitouni et al. [ ]	0.1	298	[0–0.8]	[0.8–1.4]
Wang et al. [ ]	0.1, 0.3, 0.5	298	0.4, 0.6	[0.7–1.5]
Gotama et al. [ ]	0.1, 0.5	298	0.4	[0.8–1.4]
Lhuillier et al. [ ]	0.1	298, 373, 473	[0–0.6]	[0.8–1.4]
Lee et al. [ ]	0.1	298	0.5	0.8, 1
Mei et al. [ ]	0.1	300	0	[0.9–1.3]
Zakaznov et al. [ ]	0.1	293	0	[0.8–1.3]

p [MPa]	α		β	β
ϕ = 0.8
0.1	4.01535	0.99912	4.46199	−0.85382	0.99928
0.3	3.78417	0.99910	3.66447	0.13347	0.99965
0.5	3.65400	0.99816	3.28495	0.57963	0.99982
1.0	3.44229	0.99674	2.76635	1.13354	0.99993
2.0	3.16464	0.99565	2.24763	1.56685	0.99978
3.0	2.97080	0.99551	1.98115	1.69605	0.99954
ϕ = 1.0
0.1	3.74508	0.99900	4.01731	−0.54984	0.99900
0.3	3.51541	0.99596	3.21302	0.45000	0.99937
0.5	3.35729	0.99387	2.77596	0.94952	0.99971
1.0	3.10613	0.99217	2.18010	1.57654	0.99997
2.0	2.84256	0.99228	1.65541	2.04374	0.99953
3.0	2.69021	0.99296	1.41644	2.19024	0.99885
ϕ = 1.2
0.1	3.51514	0.99274	3.38537	0.14472	0.99748
0.3	3.31057	0.99204	2.73452	0.96245	0.99939
0.5	3.20450	0.99201	2.41179	1.35772	0.99985
1.0	3.03999	0.99156	1.94449	1.89475	0.99993
2.0	2.82580	0.99130	1.44550	2.38350	0.99916
3.0	2.67350	0.99202	1.16852	2.58980	0.99804

p [MPa]	α		β	β
ϕ = 1.0
0.5	3.00838	0.99207	2.33794	1.07392	0.99938
1	2.79248	0.98931	1.85683	1.54954	0.99986
2	2.53740	0.98838	1.45236	1.82900	0.99989
4	2.27164	0.98941	1.19617	1.82809	0.99944
8	2.01777	0.99128	1.08260	1.59253	0.99907

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Bochicchio F, D’Amato M, Magi V, Viggiano A. On the Influence of H 2 Addition on NH 3 Laminar Flame Speed under Engine-like Conditions. Energies . 2024; 17(16):4181. https://doi.org/10.3390/en17164181

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Nanoporous carbons from hydrothermally pre-treated avocado waste: experimental design, hydrogen storage behavior, and energy distribution analysis

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Published: 20 August 2024

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Lehlohonolo Mohale 1 ,
Jibril Abdulsalam ORCID: orcid.org/0000-0001-5072-4996 1 &
Jean Mulopo 1

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Avocado waste, which includes the peel and seed, is a promising biomass source for highly porous materials crucial to establishing an economical and efficient hydrogen economy. Hydrothermally pre-treated avocado waste was explored as a precursor for biocarbon optimized for hydrogen storage, employing the design of experiment to vary activation temperature and impregnation ratio. The resulting optimized biocarbon, synthesized at 800 °C with a 1:3 impregnation ratio, exhibited appreciable hydrogen uptake at 77 K and 1 bar, surpassing some reported literature values. Notably, the optimized biocarbon (AC 200 ), hydrothermally pretreated at 200 °C, demonstrated a remarkable 3.07 wt% hydrogen uptake, attributed to its narrower micropores facilitating extensive adsorption. The study employed a modified Langmuir model incorporating homotattic patch approximation for a universal isotherm model, providing insights into the surface characteristics of the optimized biocarbon in terms of adsorption site availability and energy distribution. The modeling offers insight into the heterogeneous surface characteristics, specifically regarding the availability of adsorption sites, elucidating the distinct behavior exhibited by each optimized biocarbon.

Influence of Pyro-Gasification and Activation Conditions on the Porosity of Activated Biochars: A Literature Review

How the activation process modifies the hydrogen storage behavior of biomass-derived activated carbons

Preparation and Application of Hierarchical Porous Carbon Materials from Waste and Biomass: A Review

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1 Introduction

Sustainable energy development driven through shrewd energy policies can fortify the economy and improve social welfare. At the same time, the population’s safety and security are strengthened, and the environment is conserved through pollution reduction [ 1 ]. Energy is crucial for the growth and long-term viability of any economy. It drives manufacturing, business, transportation of goods, and the delivery of services in the country. The energy industry plays a crucial role in driving both the economic and social growth of South Africa [ 2 ].

The global energy landscape is marked by a significant reliance on fossil fuels [ 3 ]. This dependence has resulted in significant progress in economic and industrial endeavors, albeit at the expense of significant environmental deterioration. Fulfilling the current energy needs entails guaranteeing a consistent and dependable energy supply and tackling the environmental consequences of energy generation and usage. This necessitates developing and adopting cleaner, renewable energy sources that provide sustainable solutions.

Renewable energy sources, including solar, wind, hydro, geothermal, and biomass, have notable benefits compared to fossil fuels. They are abundant, environmentally friendly, and have minimal greenhouse gas emissions during operation [ 4 ]. Incorporating renewable energy sources into the worldwide energy portfolio is essential for diminishing reliance on non-renewable resources, decreasing greenhouse gas release, and alleviating climate change’s impact.

Among the various renewable energy sources, biomass stands out due to its unique attributes and potential for widespread application [ 5 ]. Biomass energy is obtained from organic materials such as plant and animal waste, agricultural wastes, and forestry by-products [ 6 ]. Porous carbon materials derived from biomass have demonstrated significant potential. These materials, derived from biomass waste, possess large specific surface areas, tunable pore structures, and exceptional adsorption capabilities, making them suitable for hydrogen storage applications [ 5 ].

Hydrogen, as a clean energy carrier, has the potential to play a significant role in the transition to a sustainable energy future. Hydrogen is a cleaner and safer option with enormous energy potential. Hydrogen can be generated from renewable energy sources (such as solar, wind, and biomass) and converted to power in fuel cells, emitting just water and no greenhouse gases or other pollutants [ 3 ]. However, the absence of affordable storage options hinders the widespread adoption of hydrogen. This has a lot to do with the low density of hydrogen. Several other solutions for using hydrogen as a fuel are currently being explored. Hydrogen storage in adsorbent materials is a promising option. Adsorbents for this purpose must have extremely specific attributes depending on the temperature and physical state of storage in which they will be used.

Activated carbons are viable adsorbents for hydrogen storage due to their low cost, large surface area, and ease of synthesis [ 7 ]. Pyrolysis of a carbonaceous feedstock produces char with a basic pore structure, which is then activated chemically or physically to increase porosity. Over the past decade, porous carbon materials derived from biomass waste have seen diverse applications. Activated carbons from biomass waste have been widely studied as hydrogen storage material [ 8 , 9 , 10 , 11 , 12 ]. Ariharan et al. [ 13 ] demonstrated that biomass-derived porous carbon, through simple carbonization, is a superior material suitable for hydrogen storage and utilization in supercapacitors. Carbon products from biomass pine nutshells showed high surface area, large pore volume, and excellent electrochemical performance as supercapacitors [ 14 ]. These materials’ highly developed porous structure makes them attractive for storage applications.

Studies on hydrogen storage utilizing activated carbon derived from biomass have demonstrated encouraging outcomes in terms of increased surface area, porosity, and storage capabilities. Research has concentrated on using several types of biomasses to synthesize activated carbons with remarkable hydrogen storage capacities. Activated carbons from pinecones, synthesized through dehydration at 453 K followed by KOH activation in varying ratios, exhibit significant hydrogen storage properties. The resulting activated dehydrated pinecone powders achieved a remarkable gravimetric capacity of 5.5 wt% at cryogenic temperature and 80 bar [ 12 ]. Porous carbon produced from food waste by hydrothermal carbonization at 220 °C and KOH activation at 800 °C resulted in a surface area of 2885 m 2 /g and a pore volume of 1.93 cm 3 /g. The hydrogen storage capacity increased from 4.53 to 6.15 wt% when the KOH/hydrochar ratio was increased from 2:1 to 4:1 [ 15 ].

Using KOH as an activating agent in synthesizing porous carbons from onion peel waste results in activated carbons with diverse morphologies and high surface areas. These activated carbons possess a sheet-like structure. The sample with the highest surface area of 3150 m 2 /g and a pore volume of 1.64 cm 3 /g also exhibited the highest hydrogen storage capacity, at 3.67 wt% at 77 K and 1 bar [ 16 ]. Porous-activated carbon from crushed Palimera sprout, carbonized and activated at 900 °C, showed a specific surface area of 2090 m 2 g⁻ 1 and a pore volume of 1.44 cm 3 g⁻ 1 . These properties resulted in a gravimetric hydrogen storage capacity of 1.06 wt% at 298 K and 15 bar [ 17 ]. Activated porous carbons derived from bark and camellia shell using KOH and a mixed chemical pre-treatment showed significantly enhanced properties, with surface areas of 2849 m 2 g⁻ 1 and pore volumes of 1.08 cm 3 g⁻ 1 . These carbons achieved notable hydrogen storage capacities of up to 3.01 wt% at 77 K and 1 bar and 0.85 wt% at room temperature [ 11 ]. Overall, the research status on biomass-based activated carbon for hydrogen storage shows promise for future practical use.

In 2019, global avocado production exceeded 7.18 million metric tonnes, increasing from 6.77 million in 2018. South Africa ranks as the world’s 12th-largest producer of avocados [ 18 ]. Avocado’s global output is predicted to increase between 2010 and 2030 as people become more health-conscious because of its health benefits [ 19 ]. As production output and demand increase, so does the waste generated during manufacturing and processing. Due to rigorous government requirements and a scarcity of landfill space, disposing of industrial waste has become increasingly challenging. In South Africa, regulations such as landfill levies are enforced to reduce the quantity of solid waste deposited in landfills. The increased expense of conveying a bulky and moist substance drives up the cost of landfilling even more. This gives an opportunity, especially considering the increased interest in the bio-economy, to develop value-added products such as activated carbons from avocado waste to become adsorbents for hydrogen storage applications.

Our literature analysis indicates no studies have been published on using avocado waste as a feedstock for porous carbon for hydrogen storage. Previous studies have demonstrated that activated carbons derived from avocado waste, specifically the seed and peel, may effectively eliminate various emergent organic compounds [ 20 ], facilitate microextraction [ 21 ], remove phenol from water [ 22 ] and aqueous solutions [ 23 ], adsorb ammonia [ 24 ], eliminate dyes [ 25 ], and extract fluoride from groundwater [ 26 ].

The design of experiment (DoE) approach optimizes carbonization and activation processes by evaluating the interaction between multiple parameters [ 27 ], unlike the traditional one-variable-at-a-time method, which is time-consuming. Previous studies [ 27 , 28 , 29 , 30 ] have used DoE for optimizing activated carbon.

This study explores the potential of avocado waste as a feedstock for producing highly porous carbons for hydrogen storage, addressing the growing need for sustainable energy solutions and waste management. The study uses the design of experiment (DoE) method to optimize the activation process, evaluating multiple parameters simultaneously. By incorporating hydrothermal pre-treatment, the study seeks to enhance the efficiency and effectiveness of the activation processes. This study represents the first application of the DoE method to avocado waste for hydrogen storage, highlighting its innovative approach and contribution to sustainable energy development.

2 Materials and methods

2.1 sample collection and preparation.

Avocado waste, comprising peel and seed, was collected in an airtight sample bag from an avocado processing plant in Limpopo, South Africa, and was appropriately labeled for accurate identification. The avocado waste was dried under laboratory conditions for 48 h. After drying, the sample was ground and sieved to an average 100–200 µm particle size. The resulting powder was stored for characterization and hydrothermal carbonization.

2.2 Hydrothermal pre-treatment

Avocado waste underwent hydrothermal pre-treatment at varying temperatures to enhance its suitability as a precursor for activated carbon. A Berghof stirred pressure reactor, whose contents were stirred at 250 rpm, was used for hydrothermal pre-treatment. The biomass-to-water ratio was kept at 0.125, and the reaction temperature varied between 180 ℃, 200 ℃, and 220 ℃. Retention time was kept constant at 6 h (Masoumi et al., 2021). Following the reaction, the reactor was allowed to cool before the content was emptied onto a filter paper and allowed to separate the liquid and solid contents. The solid content was dried and stored for analysis. The hydrochars obtained were labeled HC 180 , HC 200 , and HC 220 .

2.3 Activated biocarbon preparation

Activation was carried out using a suitable activating agent—potassium hydroxide (KOH)—and applied at varying impregnation ratios. The hydrochar (HC 200 ) with the highest surface area was selected as the precursor for the experiments in line with the experimental design. The activation temperatures varied from 700 to 900 °C, and the KOH-carbon precursor weight ratio varied from 1:1 to 3:1, respectively, to examine how they affected the properties of the resulting activated carbon. The material was cooled to room temperature under nitrogen gas flow after activation. The samples were washed using 1 M hydrochloric acid (HCl) and distilled water. The washed samples were dried at 105 °C in an oven overnight to obtain activated biocarbons.

2.4 Design of experiments (DoE)

The response surface methodology (RSM) has received considerable scholarly interest as a highly effective optimization technique. By utilizing the RSM method and a central composite design, the system was constructed to reduce the number of experiments [ 31 , 32 ]. This statistical method facilitates the development and examination of empirical data, providing an efficient way to understand the correlation between analyzed parameters and system response, especially when used in conjunction with central composite design (CCD). RSM enables the examination of parameter interactions, resulting in cost and time savings [ 33 ]. The central composite was designed to fit a quadratic polynomial model with the fewest number of experiments, allowing for the evaluation of the interaction between process parameters and identifying the major component for response optimization. The chosen parameters, including activation temperature and impregnation ratio, as shown in Table 1 , were systematically varied to evaluate their impact on the final activated carbon’s characteristics.

These experimental parameters were combined, and the effect of their combination was evaluated using three response parameters: the BET surface area (S BET ), pore size, and hydrogen uptake at 77 K and 1 bar. The selected responses enabled the screening of materials in an efficient, high-throughput manner. The BET surface area is a dependable indicator of hydrogen absorption patterns among materials of the same group, providing a practical estimation of the number of adsorption sites [ 34 , 35 ]. Prior studies have also shown a good correlation between gas sorption and the average pore size [ 27 , 36 , 37 , 38 ]. As a result, the experimental design used BET surface, pore size, and hydrogen uptake as screening criteria to evaluate the performance of the synthesized carbon materials.

Design Expert Version 11.1.2.0 (Stat-Ease Inc., Minneapolis, USA) was used to evaluate the data. The quadratic polynomial model is often used to fit experimental data in RSM [ 31 , 39 ]. The effect of factors was determined by fitting a quadratic polynomial model according to the equation ( 1 ):

where y represents the response, x is the process factors, β o denotes a constant term, β terms represent the model parameters, and ε represents the residual response variation [ 27 ]. By ascertaining numerical values for model parameters (β), the influence of each factor on the measured response was determined, as well as how these factors interacted to affect the response as a whole to ascertain the optimal process conditions [ 40 ].

2.5 Characterization methods

The proximate analysis of the avocado waste and hydrochars was conducted in accordance with the ISO 11722, 1171, and 562 standards [ 41 ], using a Leco TGA 701. The ultimate analysis of the sample (C, H, N, S, and O) was evaluated using LECO TruSpec CHN equipment and LECO Sulphur Analyzer, following ISO 12902 and ISO 19579 standards [ 42 ], respectively.

SEM/EDS analysis of the hydrochars and synthesized activated biocarbon was carried out using a Carl Zeiss Sigma Field Emission Scanning Electron Microscope (SEM) equipped with an Oxford X-act EDS detector. X-ray diffraction (XRD) was conducted using a Bruker D2 phaser equipped with Cu-Kα radiation (λ = 1.5405 Å), operating at 30 kV and 10 mA. Fourier transform infrared (FTIR) spectra were obtained using a PerkinElmer Spectrum 2 FTIR spectrometer. The surface area, pore volume, and pore size distribution were measured by analyzing nitrogen adsorption–desorption isotherms at 77 K using the Anton Paar Autosorb iQ instrument (Quantachrome Instruments, USA).

2.6 Hydrogen adsorption studies

The synthesized activated biocarbons produced were tested for hydrogen sorption at 77 K and 1 bar using Anton Paar Autosorb iQ. Adsorption tests were performed after degassing 0.1 g of each sample for 10 h at 200 °C [ 43 ]. Multiple samples synthesized under different conditions were tested to identify optimal parameters for enhanced hydrogen storage performance.

2.7 Modeling

The universal adsorption isotherm model developed by Ng et al. [ 44 ] was used to predict the adsorption behavior of the activated biocarbons. The model, a unified approach for the adsorption isotherm, is an improvement of previous adsorption models that could only accurately represent a particular region of adsorption but failed to predict other regions universally. The model combines three components: (i) homotattic patch approximation (HPA), (ii) updated Langmuir model, and (iii) fractional probability factor for the distribution of site energy sets. This universal model aims to simplify the process of adsorption on a complex and heterogeneous porous surface [ 44 ].

The heterogeneous surface of the adsorbent is represented by smaller homogeneous surfaces that are assumed to have the same energy level within each site. Each homogeneous surface was defined through the adsorption site energy distribution function (EDF). The probability of an adsorption site indicated the localized adsorption uptake within the monolayer. This depends on the type of pores in the layer. A fractional probability component was included to account for changes in the multi-layer formation. This factor indicates the proportion of the total surface covering or energy distribution of each pore size or adsorbate layer in the formation [ 44 ].

The assumption was that there would be just one molecule per adsorption site, and a homotattic patch approximation would be used. The adsorption capacity at a uniform patch/surface is presented in equation ( 2 )

where ${n}_{j}$ represents the number of molecules and ${s}_{j}$ the local adsorption sites available. As a consequence, the total adsorption uptake available at an adsorbent surface is given in equation ( 3 ):

where ${N}_{a}$ represents the total number of adsorbed molecules and ${S}_{o}$ the total number of available sites.

According to Ng et al. (2017), Type-VI isotherm energy distributions have four peaks, so four probability factors are used to express the energy distribution functions, ${\alpha }_{1}+{\alpha }_{2}+{\alpha }_{3}+{\alpha }_{4}=1$ . However, two terms are sufficient for a Type-IV isotherm as shown in equation ( 4 ).

where ${\theta }_{t}$ is the total adsorption uptake, $\alpha$ the probability factor, and ${p}_{s}$ is the saturation pressure at the maximum possible uptake by an adsorbent,

Equation 4 is an expanded form of Eq. 3 , without the intermediate steps, to reflect the number of sets of adsorption sites available and their distributions in an isotherm. Two terms for Type-I to Type-V isotherms and four terms for Type-VI isotherms but less or more than four terms can be used for Type-VI isotherms considering the complexity of the multi-layer behavior. [ 44 ].

3 Results and discussion

3.1 avocado waste characterization.

Table 2 shows the results of the avocado waste proximate, ultimate, and total sulfur analysis. The results of proximate analysis for avocado waste show crucial properties that make them potential precursors of activated carbon. It has an 8.68% moisture and 12.95% fixed carbon. Low moisture content is generally preferred to enhance the efficiency of the carbonization process and improve the yield of activated carbon [ 45 ]. A high volatile matter content of 76.15% is beneficial as it promotes the formation of a more porous structure in the resulting char [ 46 ]. The low ash content of 2.22% is favorable as high ash content can impede the development of porous structures by blocking pores and reducing the available surface area for adsorption [ 45 ]. From the proximate analysis, it can be seen that avocado waste is a viable feedstock for synthesizing high-porous carbon materials. This is consistent with results from other biomass sources, strengthening the argument for using agricultural waste for sustainable carbon materials [ 47 ].

The ultimate analysis of avocado waste material provides valuable insights into its elemental composition, which is crucial for understanding its potential applications. The results obtained from the ultimate analysis are shown in Table 2 . According to the results, the avocado waste has a high percentage of carbon and oxygen and a low amount of hydrogen and nitrogen. This composition agrees with the high starch content in avocado seeds reported by Bressani et al. [ 48 ].

The carbon content of 43.69% observed in the avocado waste sample suggests its potential as a carbon-rich feedstock. The high carbon content shows the presence of organic carbonaceous materials in avocado waste, which can serve as precursors for carbonization processes [ 49 ]. Hydrogen storage in porous carbon materials relies on physisorption, where hydrogen molecules are attracted to and held onto the surface and pores of the carbon matrix [ 50 ]. The presence of nitrogen (1.1%) and oxygen (49.04%) in the avocado waste sample can enhance hydrogen adsorption by providing more surface sites and promoting polar interactions with hydrogen molecules [ 51 ]. The absence of sulfur in avocado waste is advantageous, as sulfur can lead to undesirable emissions or corrosion issues during thermochemical conversion processes. With zero sulfur content, avocado waste poses minimal environmental risks and can be processed without concerns about sulfur-related complications.

The supplemental material includes a detailed characterization of the hydrochars obtained from pre-treating the avocado waste at different temperatures (Table S1 , Figs S1 , and S2 ).

3.2 Impact of factors on the properties of avocado waste-derived biocarbon

RSM was used to study the effect of process factors on the adsorptive characteristics of avocado waste biocarbons. Table 3 presents the finalized experimental design and the corresponding measured results. The parameters investigated were activation temperature (A Temp ) and KOH weight ratio (K wr ). The measured responses included the BET surface area (S BET ), average pore size (PS), and hydrogen uptake at a temperature of 77 K and pressure up to 1 bar. The impact of temperature and chemical reagent weight ratio on the surface area, pore size, and hydrogen adsorption of synthesized activated biocarbons is explored in this section.

3.2.1 Impact of activation temperature on adsorptive properties of synthesized biocarbon

The adsorptive properties of the synthesized activated biocarbons as obtained from nitrogen and hydrogen adsorption at 77 K are summarized in Table 3 . The influence of temperature on the properties of the activated samples can be observed from the results shown in Table 3 and Fig. 1 . The results demonstrate that as the temperature increases, the surface area also increases. The largest surface area recorded was 2488.98 m 2 /g at a temperature of 800 °C, corresponding to a larger hydrogen uptake. Multiple studies have shown that the temperature at which activation occurs substantially impacts the formation of the porous structure and is the most influential factor in the synthesis of activated carbon [ 52 , 53 ]. As shown in Table 3 and Fig. 1 A, the activation temperature caused an increase in surface area from 700 to 800 °C; however, it declined once the temperature surpassed 800 °C. Higher temperatures enhance the carbonization and gasification processes, leading to developed pore formations, which help increase the total surface area with a corresponding improvement in adsorption capacity [ 32 , 52 ]. The study by Wang et al. [ 54 ] highlights the temperature-dependent evolution of surface area, providing insights into the nuanced mechanisms governing this phenomenon. While higher activation temperatures generally lead to increased surface area, maintaining balance is crucial. Excessive temperatures can induce pore collapse, diminishing the beneficial effects on surface area. The results demonstrate this phenomenon, as shown in Table 3 , where an observed reduction in surface area occurs at an activation temperature of 900 °C. Excessive heat during activation can disrupt pore structures, which is undesirable when preparing porous materials [ 55 ]. Linares-Solano et al. [ 56 ] also reported that temperatures above 800 °C negatively impact reaction yield and narrow pore size distribution. Consistent with the findings of this study, a rise in temperature leads to a drop in yield, mostly because of the increased burn-off of carbon material. A sharp decrease in yield of 16% was obtained at 900 °C. As shown in Table S3 of the supplemental material, yields decreased as temperature increased. This drop was attributed to the increasing carbon burn-off and the volatilization of non-carbon components [ 57 ]. The optimal activation temperature, therefore, lies at the intersection of enhanced pore development and potential structural integrity concerns.

A Radar chart showing the maximum nitrogen adsorbed (Max Q ads ), total pore volume (V tot ), micropore pore volume (V micro ), BET surface area (S BET ), micropore surface area (S micro ), and pore size (W avg ) of activated carbons synthesized at 700 °C (Exp. 2), 800 °C (Exp. 10), and 900 °C (Exp. 9). B Hydrogen isotherms show hydrogen uptake at 77 K up to 100 kPa

3.2.2 Impact of KOH weight ratio on adsorptive properties of synthesized biocarbon

The proportion of precursor material to KOH by weight significantly influences the adsorptive properties of activated carbons. Several studies have investigated using alkaline hydroxides to prepare activated carbons [ 32 , 41 , 42 , 58 , 59 ]. Activated carbon synthesized using hydroxides stands out for its unique qualities, such as low ash content, narrow pore size distribution, and high adsorption capacity [ 2 , 56 ]. Table 3 and Fig. 2 present the findings of the conducted analysis on the synthesized biocarbon’s textural properties and hydrogen uptake, considering different weight ratios of KOH to precursor ratio at 800 °C.

A Radar chart showing the maximum nitrogen adsorbed (Max Q ads ), total pore volume (V tot ), micropore pore volume (V micro ), BET surface area (S BET ), micropore surface area (S micro ), and pore size (W avg ) of activated carbons synthesized at 800 °C and KOH weight ratio of 1:1 (Exp. 1), 1:2 (Exp. 3), and 1:3 (Exp. 10). B Hydrogen isotherms show hydrogen uptake at 77 K up to 100 kPa

Table 3 shows that the surface area of the activated biocarbon samples increases when the chemical weight ratio of KOH increases. An increase in the chemical weight ratio leads to a greater extent of the reaction, resulting in an increased surface area. This is mostly owing to the etching effect caused by KOH on the carbon precursor. This results in more adsorption sites and improved accessibility for adsorbates [ 60 ]. An increase in the weight ratio from 2 to 3 corresponds to a 68% increase in surface area. Figure 2 shows that a larger weight ratio of chemical reagents results in a greater increase in hydrogen uptake. In the studies conducted by Kwiatkowski et al. [ 60 ] and Hassen [ 61 ], a similar finding was observed, indicating that an increase in the ratio of KOH impacted both the increase of surface area and the enhancement of adsorption capacity.

Table S3 of the supplementary material shows that an increased KOH weight ratio decreases the activated carbons’ yield. As stated, higher KOH ratios typically enhance chemical activation, leading to greater pore development and higher surface areas. However, this also tends to decrease activated carbon yield because more carbon material is consumed in the activation process [ 62 ].

3.3 Process of optimization

3.3.1 statistical analysis.

The assessment of model validity was conducted by examining the R 2 parameter, which is measured on a scale of 0 to 1. Higher values are considered more desirable, ideally with a maximum difference of 0.3 [ 63 ]. The F -value and P value are the parameters used to assess the model’s significance and fitness. Figure 3 shows the alignment between the predictions made by the model and the experimental outcomes for all responses. Additionally, Table 4 presents the specifics of the ANOVA analysis. For each response, experimental data points exhibit a significant alignment with the model predictions, leading to a high value of R 2 .

Experimentally observed points vs model-predicted points (denoted by dotted line) for the three responses: A surface area, B pore size, and C hydrogen uptake at 77 K. Each data point is labeled with its respective experiment number

The R 2 values for the surface area, pore size, and H 2 uptake are 0.990, 0.997, and 0.995, respectively. These values are close to 1 and are considered ideal. The adjusted R 2 values of 0.975, 0.991, and 0.988 demonstrate satisfactory agreement with the predicted R 2 values of 0.887, 0.896, and 0.955 for surface area, pore size, and H 2 uptake, respectively. In order for the Adj. R 2 and Pred. R 2 to be considered quite agreeable, their difference should be below 0.20 [ 64 ]. The Adj. R 2 and Pred. R 2 differ by 0.088, 0.095, and 0.033 for surface area, pore size, and H 2 uptake, respectively. Adequate precision is determined by comparing the anticipated values at the design point to the average prediction error. A model is considered acceptable and has adequate precision if its value exceeds 4 [ 64 ]. The surface area, pore size, and H 2 uptake values that meet the required precision are 22.11, 33.92, and 32.58, respectively. According to Hou et al. (2013), the F -value and P value are important indicators of a model’s statistical significance and fitness. The F -values of 66.85, 151.84, and 139.14 for surface area, pore size, and H 2 uptake, respectively, together with the P values of all responses being less than 0.05, indicate that the models effectively align with the experimental data and give a satisfactory fit.

3.3.2 Optimization of preparation parameters

This section examines the optimal parameters for preparing activated carbon to achieve maximum surface area, hydrogen uptake, and smaller pore size necessary for effective hydrogen storage. The optimal preparation conditions were established using the desirability function of Design Expert software to yield the desired outcomes. The surface plot in Fig. 4 depicts desirability while maintaining activation temperature and KOH weight ratio within the studied range.

Three-dimensional surface plots of desirability for numerical optimization of the goal of maximized surface area, hydrogen uptake, and smaller pore size

The optimal conditions derived from numerical optimization for achieving the desired responses of maximizing surface area, hydrogen uptake, and smaller pore size are presented in Table 5 . The proposed optimal conditions outlined in Table 5 have a desirability value of 0.957. Validation experiments conducted using these optimal processing conditions resulted in a material with a surface area of 2529.8 m 2 /g, a pore size of 2.31 nm, and a hydrogen uptake of 3.07 wt% at 77 K and 1 bar, with predicted and experimental results showing good agreement within an error range of 2–4%. These results validate the predictions of the ANOVA model for the responses under experimental conditions.

3.4 Analysis of optimal activated biocarbons

The optimal process parameters were then employed to synthesize activated biocarbons with hydrochars H 180 , H 200 , and H 220 as precursors. These materials’ (AC 180 , AC 200 , and AC 220 ) physiochemical and adsorptive characteristics were evaluated to assess the impact of hydrothermal pre-treatment temperatures.

3.4.1 Characterizations

The surface morphology of the hydrochars (H 180 , H 200 , and H 220 ) and activated biocarbons (AC 180 , AC 200 , and AC 220 ) are shown in Fig. 5 . The SEM analysis of the hydrochars (depicted in Fig. 5 A–C) revealed a surface featuring macro pore formation, indicating the potential of avocado waste hydrothermal pre-treatment to yield a porous carbon material. Subsequent activation of the hydrochars with KOH at high temperatures led to the formation of smaller pores on the material surface, as illustrated in Fig. 5 D–F. These images indicate that while the activated materials maintained their original morphology, they underwent size reduction and surface roughening, likely attributed to material decomposition and shrinkage during activation, thus generating additional surface pore structures.

SEM images of A HC 180 , B HC 200 , C HC 220 , D AC 180 , E AC 200 , and F AC 220

The X-ray diffraction (XRD) spectrum of the optimized activated biocarbon, as shown in Fig. 6 A, reveals an identified peak at around 2θ = 10°, associated closely with (002) [ 65 ]. This peak indicates the existence of an amorphous or disordered carbon structure, typical of biomass-derived activated carbons where the disordered carbonaceous materials contribute to peaks at lower angles [ 42 ]. The second identified peak at 2θ = 35° is associated with (100) graphitic planes [ 66 , 67 ]. This peak indicates some degree of graphitic structure within the optimized activated biocarbons. The analysis of the XRD spectrum indicates that the optimized activated biocarbon derived from avocado waste exhibits a blend of amorphous and graphitic carbon structures. Prior studies have also documented that the carbonization and activation process substantially influences the formation of amorphous carbon and porous architecture [ 42 , 68 ].

Characterizations of the optimized activated carbons: A XRD pattern; B FTIR spectra; C N 2 isotherms at 77 K; D pore size distribution (PSD) using the NLDFT method; and E radar chart showing the maximum quantity of nitrogen adsorbed (Q ads ), total pore volume (V tot ), micropore volume (V micro ), BET surface area (S BET ), micropore area (S micro ), and average pore diameter

The Fourier transform infrared (FTIR) spectrum, as shown in Fig. 6 B and Fig. S3 in supplementary material, exhibits many distinct peaks, indicating the existence of diverse functional groups. The aliphatic hydrocarbons, most likely derived from the lipid content in the avocado waste, are indicated by the aliphatic C-H stretch at 2882 cm −1 . The identification of a C = C stretching vibration at 1631 cm −1 indicates the existence of aromatic structures, maybe originating from lignin or other phenolic chemicals present in the waste material. Furthermore, the presence of aromatic structures is further supported by the peak observed at 1492 cm −1 , which can be attributed to the stretching vibrations of aromatic C = C bonds. The observation of a peak at 1293 cm −1 indicates the C-O stretching vibration, suggesting the existence of oxygen-containing functional groups such as alcohols, ethers, or esters [ 69 ]. These functional groups may originate from cellulose or hemicellulose constituents in the avocado waste.

Additionally, the observation of a peak at 1075 cm −1 implies the presence of a C-O bond, which may imply the presence of carboxylic acids or esters. These compounds might originate from pectin or other organic acids found in the avocado waste. The FTIR spectrum exhibits a combination of aliphatic and aromatic hydrocarbons and oxygen-containing functional groups commonly found in carbon compounds generated from biomass [ 42 , 69 ].

Nitrogen (N 2 ) sorption experiments were conducted to evaluate the optimized biocarbons’ specific surface area and porous structure, with pore size distributions (PSDs) analyzed using the non-local density functional theory (NLDFT) model. The N 2 adsorption–desorption isotherm is shown in Fig. 6 C, exhibiting a type IV isotherm according to the IUPAC classification [ 70 ]. The significant increase before reaching relative pressure P/P o at 0.4 indicates the presence of micropores in the materials, facilitating increased adsorption at low pressures [ 71 ]. When the relative pressure P/P o exceeds 0.4, the desorption and adsorption isotherms do not overlap, resulting in a hysteresis loop. This loop indicates the presence of mesopores [ 72 ]. This clearly illustrates the coexistence of micro- and mesoporous structures within the avocado waste-derived biocarbons. The combination of micropores and mesopores is often most effective for enhanced hydrogen storage in activated carbons. Micropores maximize the surface area available for hydrogen adsorption. At the same time, mesopores facilitate the accessibility of hydrogen molecules to the interior regions of the material. This synergistic effect improves both the adsorption capacity and the kinetics of hydrogen uptake and release [ 73 , 74 ].

Figure 6 D shows the PSDs estimated using the NLDFT model. All three materials showed a bimodal PSD, with the first peak at 0.7–0.8 nm and the second at 2.3 nm in the micropore and mesopore regions. In addition, the three materials exhibited a broad peak at 0.7–0.8 nm and a shoulder around 1–2 nm. The existence of supermicropores (0.7–2 nm) [ 75 ] was observed. When the pores are filled with nitrogen, the pore volume diminishes and becomes insignificant beyond the mesopore domain at a pore diameter of around 4–5 nm. The PSD curve revealed the presence of micropores and mesopores, aligning with the nitrogen isotherms’ observations. While all three materials have porous structures, including micropores and mesopores, their differences mostly lie in the micropore-to-mesopore volume ratio. This is primarily influenced by different hydrothermal carbonization (HTC) temperatures.

The radar chart in Fig. 6 E visually depicts the adsorptive properties of the optimized biocarbons. AC 200 exhibits the highest surface area (SA) and pore volume (PV) at 2529.8 m 2 /g and 1.45 cm 3 /g, respectively. In comparison, AC 180 and AC 220 show SA and PV of 2178.1 m 2 /g and 1.26 cm 3 /g and 1838.4 m 2 /g and 1.09 cm 3 /g, respectively. All biocarbons exhibit a significant degree of microporosity, wherein micropores account for 81.4% (AC 180 ), 80% (AC 200 ), and 76% (AC 220 ) of the total pore volume. The surface area and pore volume increase with rising hydrothermal carbonization (HTC) temperature up to 200 °C but decrease afterward due to intensified decomposition reactions. Higher HTC temperatures promote the breakdown of larger molecules in the precursor material, creating more void spaces and increasing specific surface area, leading to micropores and mesopores [ 76 ]. However, excessive heat can collapse the pore wall, reducing surface area and pore volume [ 77 ], as observed when HTC temperature surpasses 200 °C. Hence, identifying the optimal HTC temperature is crucial, as it varies depending on the precursor material. This study demonstrates that an HTC temperature of 200 °C for avocado waste yields porous carbon material with enhanced pore characteristics.

4 Hydrogen uptake of optimal biocarbons

Figure 7 A shows the hydrogen adsorption–desorption isotherms for the optimal biocarbons. The isotherms offer a glimpse into how the pore structure of the synthesized nanoporous materials influences hydrogen storage behavior. The isotherms closely resemble Type I, suggesting that saturation occurs due to forming a hydrogen monolayer, as commonly observed on microporous surfaces [ 9 ]. The hydrogen uptake for the three materials consistently rises with increasing pressure. However, it does not reach a distinct plateau because of the low pressure. All isotherms’ adsorption and desorption branches exhibit an insignificant presence of hysteresis, suggesting that adsorption is reversible [ 27 ]. The synthesized biocarbon demonstrates a hydrogen uptake ranging from 1 to 3% by weight, with biocarbon AC 200 , hydrothermally pretreated at 200 °C, exhibiting the highest uptake at 3.07 wt%. This is partly attributed to its well-developed micropores, as observed in the PSD. Following AC 200 , AC 180 shows an uptake of 2.44 wt% and AC 220 exhibits an uptake of 1.9 wt%. These values for the optimized biocarbons are comparable to or higher than those previously reported for bio-based carbons in the literature, including bean shells [ 78 ], walnut shells [ 79 ], Neolamarckia cadamba [ 80 ], sawdust [ 81 ], lignin [ 27 ], fungi [ 82 ], hemp [ 83 ], sucrose [ 84 ], corn grain [ 85 ], and rice husks [ 86 ].

A Hydrogen isotherms, showing uptake at 77 K up to 1 bar in the synthesized optimized biocarbon. B Hydrogen uptake plotted against the BET surface area for the optimized biocarbon and biomass-derived carbons reported in literature [ 4 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 ]

In comparison, commercial activated carbons like MAXSORB MSC-30 and MSP20X have shown hydrogen adsorption capacities of 5.8 wt% and 4.8 wt%, respectively, at 77 K and 40 bar [ 75 ], whereas AC 200 from this work achieved 3.07 wt% at 1 bar, indicating a promising performance at high pressure. Avocado waste-derived activated carbons for hydrogen storage application present a cost-effective and sustainable alternative, offering a novel avenue for biomass valorization and a sustainable and environmentally friendly approach to hydrogen storage. Table S2 in the supplementary material shows the hydrogen adsorption data at 77 K for commercial activated carbons and activated carbons from various biomass feedstocks in the literature to the ACs synthesized in this work.

The improved hydrogen uptake of the optimized biocarbon can be ascribed to its enhanced surface area and significant presence of micropores. This efficiency of narrow micropores is evident in Fig. 7 B, where the biocarbons, especially AC 200 and AC 180 , outperform materials in the literature with higher surface areas reported by Chen et al. (2838 m 2 /g, 2.63 wt%) [ 78 ] and Hu et al. (2743 m 2 /g, 2.85 wt%) [ 80 ]. This highlights the significant role of micropores in enhancing hydrogen uptake. While a higher surface area provides more sites for hydrogen adsorption, micropores facilitate strong van der Waals interactions between hydrogen molecules and the pore walls, enhancing adsorption capacity. The study demonstrates that avocado waste may be transformed into high-value activated biocarbon, resulting in a material with a substantial surface area and significant potential for hydrogen storage.

4.1 Modeling

The universal isotherm model, a type of Monte Carlo simulation developed for sorption modeling, was used to predict the experimental data. This model is also used to calculate the energy of adsorption for each site. The universal isotherm model for three sites, UNIV4, was used. This was also seen as appropriate based on other analyses carried out in terms of hydrogen sorption. The hydrogen heat of vaporization used in this model is 449.36 J/mol. Two terms were used for modeling all activated carbons [ 44 ].

Figure 8 A shows that AC 180 exhibits Type-IV isotherm. This isotherm is characterized by a rapid increase in the amount of hydrogen adsorbed between 0 and about 86 kPa as the pressure increases, with almost two saturation points or levels being reached. The first pseudo-saturation would be observed between 60 and 80 kPa. After that, we can see a rapid rise or shift in adsorption at pressures above 85 kPa. This shows the multi-layer formation during adsorption at this pressure for AC 180 . Due to the pressure used in this analysis, we cannot see the actual saturation points. This indicates a well-developed porosity if the pores can still adsorb so much hydrogen at such pressures. Only a fraction of the mesopores seem evident here, and most adsorption occurs in the micropores. The curve with rings shows the experimental data points. In contrast, the blue lines without rings show the predictions of the universal isotherm model.

Predicted and measured data for AC 180 A adsorption isotherm and B the corresponding adsorption site energy distribution. The curve with rings in A shows the experimental data points, while the blue lines without rings show the universal isotherm model predictions

The energy distribution function, Fig. 8 B, shows a non-symmetrical single peak, which is something only seen in type-IV adsorption [ 44 ]. Most surface coverage occurs at low pressures, with this region having the highest probability distribution of energy sites. It is also clear that these sites do not require much energy for hydrogen sorption, requiring less than 2000 J/mol for most of the gas sorption and slightly above 300–400 J/mol, which are the most dominant energy site requirements. The apparent mesoporous layer in the multi-layer formation does not require much adsorption energy. It does not have a significant energy site distribution. This could indicate that these are not true mesopores but micropores with higher pore widths, as seen in the PSD.

Figure 9 A shows that AC 200 exhibits Type-V isotherm. This isotherm has a unique S-shape. Three different uptake rates characterize it. There is a smooth uptake at low pressures, between 0 and about 10 kPa, followed by a rapid vertical increase in the amount of hydrogen adsorbed at 10 kPa and a subsequent steep increase of adsorption with increasing pressure, though not as steep as at 10 kPa. This subsequent uptake then levels off at higher pressures, but we cannot see saturation due to the pressures used in this research. This also indicates a well-developed porosity if the pores can adsorb so much hydrogen at such pressures.

Predicted and measured data for AC 200 A adsorption isotherm and B the corresponding adsorption sites energy distribution. The curve with rings in A shows the experimental data points, while the blue lines without rings show the universal isotherm model predictions

Figure 9 B shows a Type-V characteristic energy distribution function with a higher probability peak of the energy distribution function at lower energy sites, indicating that most surface coverage occurs at lower concentrations [ 44 ]. It is also interesting to note that there are two apparent peaks, with the last peak showing a very small probability of energy sites, though an important amount of adsorption site energy of about 25 000 J/mol at higher pressures.

Figure 10 A shows that AC 220 exhibits Type-IV isotherm, similar to AC 180 . There is a rapid increase in the amount of hydrogen adsorbed between 0 and about 77 kPa as the pressure increases, with almost two saturation points or levels being reached. The first pseudo-saturation would be observed between 60 and 78 kPa. After that, we can see a rapid rise in adsorption at pressures above 82 kPa. This shows the multi-layer formation during adsorption at this pressure for AC 220 . This indicates a well-developed porosity with only a fraction of the mesoporous layer contributing to hydrogen adsorption; most adsorption occurs at the micropores. It could also indicate wider micropores and not a real mesoporous layer.

Predicted and measured data for AC 220 A adsorption isotherm and B the corresponding adsorption sites energy distribution. The curve with rings in A shows the experimental data points, while the blue lines without rings show the universal isotherm model predictions

The energy distribution function in Fig. 10 B shows a rather interesting non-symmetrical peak at the distribution’s lower concentration/pressure end. We can see lower adsorption energy sites exist for AC 220 , ranging between 0 and just over 1800 J/mol. Adsorption happens easily in the lower-pressure region for this material.

Per the discussion under Section 3.4.1 , there are a number of functional groups in the activated carbons. The aliphatic C-H is likely a result of the lipid composition in the avocado waste, while the aromatic C = C stretching vibration may be attributed to lignin or other phenolic chemicals. The presence of oxygen-containing functional groups, as indicated by the C-O stretching vibration, suggests the existence of activated carbons with van der Waals interactions with hydrogen and potentially weak hydrogen bonds. As alluded to in the discussions for Figs. 8 B, 9 B, and 10 B, there is a correlation between pore characteristics and energy distribution. Specifically, hydrogen adsorption requires less energy in well-developed micropores because the van der Waals forces/interactions between activated carbons and hydrogen are stronger.

On the other hand, mesopores require a greater amount of energy for the process of adsorption. Additionally, macropores or the bigger mesopores necessitate even more energy as the van der Waals forces weaken due to the increased contact distance. The micropores with well-developed structures exhibit the most prominent distribution of energy sites, primarily because of their substantial surface area to volume ratio. As a result, most adsorption occurs in the micropores, which need less energy for adsorption than the meso- and macropores. As a result, the micropores have a better capacity to adsorb hydrogen at lower pressures due to stronger connections. In contrast, the larger pores require higher pressures for hydrogen adsorption due to weaker contacts.

5 Conclusion

This study utilized the design of experiment methodology to prepare activated carbons from avocado waste, offering valuable insights into their potential for hydrogen storage. Regression analysis identified an activation temperature of 800 °C and a KOH weight ratio of 3 as optimal process conditions for maximizing hydrogen uptake. Validation experiments confirmed close agreement with predicted responses, yielding material with a surface area of 2529.8 m [ 2 ]/g, a pore size of 2.31 nm, and a hydrogen uptake of 3.07 wt%.

Additionally, investigation into hydrothermal carbonization temperatures (180 °C, 200 °C, and 220 °C) revealed superior performance of activated carbon derived from 200 °C pre-treatment, showing enhanced surface properties and higher hydrogen adsorption capacity. Notably, activated carbon from 200 °C pre-treatment exhibited the best pore volume at narrower pore widths, resulting in the highest hydrogen uptake of 3.07 wt%. Micropores predominantly characterized the resulting activated carbons, explaining their high hydrogen uptake. In addition, employing energy distribution functions and homotattic patch approximation, the study found that the two-term universal adsorption isotherm model accurately predicted experimental data, shedding light on the adsorption energy sites of synthesized activated carbons. This study demonstrates avocado waste’s potential as feedstock for producing high surface area, well-developed porosity-activated carbons for hydrogen storage, holding promise for advancing renewable energy technologies and addressing global energy challenges.

Data availability

All data generated or analyzed during this study are included in this published article.

Abbreviations

Analysis of variance

Brunauer-Emmett-Teller

Central composite design

Design of experiment

Energy distribution function

Energy dispersive X-ray analysis

Fourier transform infrared

Hydrochloric acid

Homotattic patch approximation

Hydrothermal carbonization

Potassium hydroxide

Non-local density functional theory

Pore size distributions

Pore volume

Coefficient of determination

Response surface methodology

Surface area

Scanning electron microscopy

X-ray diffraction

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Acknowledgements

The authors thank James Mehl from the South African Avocado Growers Association (SAAGA) and Andre Oosthuisen from Westfalia Fruits for supplying the avocado waste sample. Sincere gratitude to the Carnegie Corporation of New York for granting Dr. Jibril Abdulsalam the Carnegie research fellowship during this study.

Open access funding provided by University of the Witwatersrand. The corresponding author acknowledges the funding support from the Carnegie Corporation of New York.

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Lehlohonolo Mohale, Jibril Abdulsalam & Jean Mulopo

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LM: methodology, data curation, formal analysis, investigation, software, and writing—original draft. JA: conceptualization, formal analysis, software, writing—review and editing, visualization, and supervision. JM: resources, supervision, and project administration. All authors read and approved the final manuscript.

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Mohale, L., Abdulsalam, J. & Mulopo, J. Nanoporous carbons from hydrothermally pre-treated avocado waste: experimental design, hydrogen storage behavior, and energy distribution analysis. Biomass Conv. Bioref. (2024). https://doi.org/10.1007/s13399-024-06006-1

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Received : 29 May 2024

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Accepted : 28 July 2024

Published : 20 August 2024

DOI : https://doi.org/10.1007/s13399-024-06006-1

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