Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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Rumford’s Experimental Challenge to Caloric Theory: “Big Science” 18th-Century Style with Important Results for Chemistry and Physics Frederic E. Schubert* Friends Seminary, 222 East 16th Street, New York, New York 10003, United States
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S Supporting Information *
ABSTRACT: The cannon boring experiment of Count Rumford, where eight kilograms of water were boiled by metal on metal friction, is investigated. Consideration of this dramatic demonstration can enrich classroom discussions of calorimetry, units of measure, elements, and thermodynamics. A section pertaining to use of the article in the classroom appears after the experiment is discussed. The cannon work was one of the Count’s many efforts to understand heat and discredit the then solidly entrenched belief that heat was an element, caloric. Joule lauded Rumford’s work and mined his data decades later for comparison with his own on the mechanical equivalent of heat. The discussion is laid out in the pattern familiar to students of a laboratory experiment and adds in commentary and context when pertinent. Further support for Joule’s work is found in the Count’s data. Also mentioned are a number of other discoveries by the Count attacking the caloric theory and thereby supporting Joule in his efforts to establish that a mechanical equivalent of heat even existed. The development of the kinetic theory of gases in the mid-19th century gave a firm grounding to the early controversial idea of the Count that heat is particle motion. Further, it allowed a formal derivation of another early idea, Avogadro’s hypothesis. The Count’s earnest question “What is heat?” echoes across the centuries and still gives rise to interesting and worthwhile classroom discussions. KEYWORDS: General Public, High School/Introductory Chemistry, History/Philosophy, Interdisciplinary/Multidisciplinary, First-Year Undergraduate/General, Second-Year Undergraduate, Physical Chemistry, Calorimetry/Thermochemistry, Kinetic-Molecular Theory, Thermodynamics
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Joule remarked that this was an “ingenious (set of) experiments” by this “justly celebrated philosopher” that were the first to come down “decidedly in the favor of that view” (ref 4, pp 298 and 299) that motion could generate heat.
GENERATING MEASURABLE QUANTITIES OF HEAT FROM MECHANICAL MOTION: BACKGROUND
The Experimenter
Contemporary Thoughts (≈1800) on the Nature of Heat
Benjamin Thompson, Count Rumford (1753−1814), was a remarkable character. “Many-sided men have always attracted me”1 wrote President Franklin Delano Roosevelt, as he listed Jefferson, Franklin, Thompson, and Napoleon as the standout figures of their time. The record of the Count’s formidable accomplishments in many fields is unique, and he would likely be better known but for his questionable character.2 His spying, opportunism, and general abrasiveness led to a downplaying of his scientific contributions in life and afterward.3 The Supporting Information gives a brief synopsis of the espionage work of the Count for several countries, often playing one against the other. In one clandestine endeavor, he was outed and came quite close to sharing the fate of his coconspirator, who was drawn and quartered. He performed a number of experiments to understand the nature of heat. The concept of convection currents, as well as his revolutionary fireplace and kitchen designs, drip coffee pot, and clothing with air trapping weaves for warmth, all arose from his carefully documented endeavors. The cannon experiments examined here were part of his heat investigations. © XXXX American Chemical Society and Division of Chemical Education, Inc.
When the Count worked over two centuries ago, the caloric theory of heat was the dominant view. Caloric was one of Lavoisier’s elements or “simple substances”. Lavoisier pulled together into his understanding of caloric the following ideas: “heat, principle or element of heat, fire, igneous fluid and matter of fire and heat.”5 Importantly, he noted, “We are not obliged to suppose this to be a real substance, it being sufficient...that it be considered as the repulsive cause...which separates particles of matter from each other.” There was experimental support for caloric as a substance. Calorimetry experiments, where hot and cold materials are mixed, assuming Qlost + Qgained = 0, where Q is heat, proved fruitful. This equality can be seen as involving a flow of “fluid” from a hot object to a cold object in a closed system. This conservation of heat idea was employed in the 18th century to Received: January 15, 2019 Revised: June 24, 2019
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DOI: 10.1021/acs.jchemed.9b00039 J. Chem. Educ. XXXX, XXX, XXX−XXX
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gather useful information on heats of fusion, heats of vaporization, and specific heats of materials. Heat as a substance gained importance because it gave useful results. It appeared to be conserved. The Count’s experiment would indicate otherwise. Rumford appreciated Lavoisier and his colleagues maintaining a circumspect posture concerning caloric, calling them (ref 6, volume 3, p 168): “[M]en of superior excellence. They proposed the word to avoid circumlocutions and to render the language of science more concise, rather than to introduce a new opinion.” Unfortunately, the inclusion of caloric in Lavoisier’s list of elements reinforced its reality in the eyes of most, including, influentially, John Dalton. Figure 1 shows models of blocks he used in lecture, examples of his stationary atoms surrounded by a radiating atmosphere of caloric.
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THE EXPERIMENT
The Design
In 1797, overseeing the production of artillery for the army of Bavaria, Count Rumford noticed the large amounts of heat released by the process of drilling the bore hole into a cannon blank. His curiosity aroused, he set out to study this unwanted byproduct in a purely scientific investigation. Science can advance in unusual ways. Here, Rumford saw a humdrum annoyance, observed for centuries, associated with the important work of cannon boring, as an opportunity for a unique investigation. He modified his cannon boring apparatus, not to bore a hole but simply to generate heat by rubbing two pieces of dull metal together, as shown in Figures 2 and 3. Part of the apparatus, shown in the Supporting Information, maintained a force of 10,000 pounds at the frictional interface (see ref 6, volume 2, pp 472−493). One piece of metal, an iron bar, was held stationary and rubbed against the other that was part of the end of a cannon blank which had been formed into a hollow cylinder while still being connected to the main body of the blank by a narrow neck. This blank and its cylindrical extension was rotated 32 times per minute by horse power. Metal−Metal Friction Boils 18 Pounds of Water
The combination was placed in a stationary wooden box with two gasketed openings and filled with 18 pounds of water. Figure 3 is a close up of the box, drawn from the Count’s point of view, with the front removed. The front of the cylinder is partially cut out. The rotating hollow cylinder, the stationary iron bar and the boiling water are all visible. A gasket at the left end of the cylinder, that keeps water out, is not shown. The box had a cover and the whole assembly was blanketed with insulating material.
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THE DATA During a run, the rotation of the system was occasionally paused and the temperature measured. At the 2.5 hour mark, “the WATER BOILED!” (Capital letters and exclamation mark are the Count’s.) Bystanders expressed surprise and astonishment. The result afforded the Count a “childish pleasure” he could not suppress. The resulting data are plotted in Figure 4. On the y-axis, the initial temperature, 60 °F, is shown as zero. All but the point at 30 min are from the Count’s experiment 3. That point is derived from the related experiment 1. The Count himself did not present his data graphically. Considering the complexity of the setup, horses, gearing, cannon blank, metal bar, box, and water, the consistency is remarkable. The linearity proves to be of further significance.
Figure 1. Models of Dalton’s lecture block atoms of three different elements with their caloric atmospheres radiating outward. (Blocks painted and photographed by Judy Schubert, created with information in ref 7; photograph used with permission.)
One of his students, when asked about atoms, responded (ref 7, p 780): “Atoms are blocks of wood, painted in various colors and invented by Dr. Dalton.” The prevailing view was that raising the temperature of an object increased the amount of caloric “atmosphere” between its atoms, and cooling it would decrease the same. Caloric was as real to Dalton as his indestructible spherical atoms. Dalton felt supported in his stationary atom view by a hypothetical calculation of Newton that derived Boyle’s law assuming stationary atoms repelling one another.8 This was a mathematical aside for Newton but taken as proof by Dalton.9 He required his stationary atoms surrounded by a caloric atmosphere in his rather convoluted explanation of the law of partial pressures. The Count’s work would suggest an alternate view: that atoms were in motion and caloric was not a viable concept.
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DISCUSSION Taking into account the temperature increase of the water and metals in his box, the Count calculated that the heat released would bring the equivalent of 27.0 pounds of water10 from freezing to boiling. The heat to take a pound of water from 32 to 212 °F was his reference unit. The box contained only 18 pounds of actual water. The Supporting Information contains details of the Count’s calculations. He noted one horse could comfortably do the job. For the Count, the experiment showed that caloric can apparently be generated in nearly unlimited amounts. It is clearly not conserved. In one of many scrupulous checks on his interpretation of his results, he verified, from their appearance and calorimetric analysis, that the shavings B
DOI: 10.1021/acs.jchemed.9b00039 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Figure 2. Experimental setup: The gunmetal cannon blank rotates by horsepower. Its hollow cylindrical end rubs against the end of the fixed iron bar with a flat hardened steel end inside the water-filled box. See close-up in Figure 3. The Count takes notes. (Photograph by the author of an exhibit at the Rumford Birthplace Museum, Woburn, MA; reproduced with permission.)
Figure 3. Close-up (with front and top of the box removed) of Figure 2 from the Count’s point of view. The rectangular window at center shows a cutout view into the partially hollowed out cylinder. It is connected to the cannon blank at right by a narrow neck. Blank and cylinder rotate at 32 rpm. The box and the dark iron bar do not rotate. Force is applied against the bar from the left, creating a high-pressure frictional interface where the narrowed bar, with its blunt hardened steel end, meets the cylindrical end of the cannon blank. A gasket at the left end of the cylinder prevents water from entering the hollowed-out space. The box is 29 cm long. The water is boiling. (Painting by Judy Schubert from Count Rumford’s description in ref 6, volume 2, pp 472−493; reproduced with permission.).
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DOI: 10.1021/acs.jchemed.9b00039 J. Chem. Educ. XXXX, XXX, XXX−XXX
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constant rate and heat energy was being added to the water at a constant rate, there is a mechanical equivalent of heat (MEH) implied in the linearity of the plotted data in Figure 4. This was Joule’s view of the Count’s work as his analysis in a subsequent section shows, where he mines the Count’s data. Joule: The Count Was ahead of the Scientific World
Joule, writing in 1843, indicated (ref 4, p 157): “We shall be obliged after all, to admit that Count Rumford was right in attributing the heat evolved by boring cannon to friction, and not...to any change in the capacity of the metal.” This late acknowledgment was because (ref 4, p 302) “[T]the scientific world, preoccupied with the hypothesis that heat is a substance...almost unanimously denied the possibility of generating heat in this way.” Joule’s experimental work, carried out over four decades after Rumford’s, was informed by the later period’s clearer understanding of energy. By 1843, after calculating several values for the MEH in varied and ingenious experiments,11 he finally announced (ref 4, p 204): “I may therefore conclude that the existence of an equivalent relation between heat and the ordinary forms of mechanical power is proved.”
Figure 4. Plot of Rumford’s data as contents of the box in the setup in Figure 3 was heated from 60−210 °F. r2 = 0.99. Box, water, iron bar, and cannon blank in Figure 3 were all initially at 60 °F, the y-axis origin.
Looking Back: Mining The Count’s Data
Joule looked back to “one of the most important parts” (ref 4, p 299) of Rumford’s cannon experimental work to calculate a value for the MEH. Rumford’s comment that one horse could comfortably have driven his experimental setup and using the accepted value of 33,000 ftlbs/min = 1 horsepower, unknown when Rumford worked, indicated that his data gave the work needed to heat a pound of water one degree Fahrenheit as 1034 ftlbs.12 Joule goes on (ref 4, p 299), “This result is not very widely different from that which I have deduced from my own experiments...772 ftlbs”. He points out the high value would be expected as the Count acknowledged heat lost to the box and surroundings was not accounted for. Precision here is an issue because the Count was talking about an actual horse at work rather than a defined unit. Clearly, Joule looked closely at Rumford’s data. He would have noticed the uniformity of the temperature increase with time shown in Figure 4. Did he find this evidence for the MEH supportive of his endeavors? The importance of the MEH to Joule is shown by the fact that the 772 ftlbs value, which he judged to be his most accurate, is etched on his gravestone. Following in Joule’s footsteps, a bit more can be gleaned from the Count’s data. Assume his box, which he gives as weighing 15 and 1/4 pounds, was heated to the boiling point. The Count had covered it with insulation. Taking the heat absorbed by the wooden box into account, with a specific heat of 0.41 relative to water, drops the Rumford value for the MEH to about 900 ftlbs, allowing one significant figure for the limits of precision of the horsepower value. Converting to more familiar units, Joule’s best value of 772 ftlbs corresponds to 4.17 Joules = 1 calorie, close to the actual 4.18 Joules = 1 calorie. Using the Count’s 900 ftlbs, yields 5 Joules = 1 calorie. Details on the calculations for these values are presented in the Supporting Information. At the conclusion of a paper in 1849 summing up years of experiments, Joule states as a principle (ref 4, p 328; italics are Joule’s): “1st, That the quantity of heat produced by the f riction of bodies, whether solid or liquid, is always proportional to the quantity of force expended.” He is using the term force to stand for energy here. The linearity of Rumford’s data in Figure 4, from 1797, was supportive of this statement.
from the grinding metal had not undergone any change in properties that might have explained the rise in temperature. Rumford addressed the question of whether the powder generated by grinding in the experiment could have given up caloric to the cylinder and produced the increase in temperature observed. In a trial run of his apparatus without the box and water, he raised the temperature 70 °F to 130 °F. He found the powder generated by grinding at the interface was about 1/1000 part of the cylinder weight. To raise the cylinder 70 °F, the powder would have to have given up, in his terminology, 70,000 “degrees of heat”. The Count found this implausible (ref 6, volume 2, pp 478,479); capital letters are the Count’s): He mused after the experiment, “What is heat?Is there any such thing as an igneous fluid?Is there anything that can with propriety be called caloric?” Further, he said, “It appears to me extremely difficult, if not quite impossible to form any distinct idea of anything capable of being excited, and communicated in the manner the heat was excited and communicated in these experiments, except it be MOTION.” He saw the particles or atoms of a material as being in constant motion. This was more than speculation from this experiment. He had seen it as the explanation of his experiments on the mixing, without stirringdiffusion in modern termsof layered liquids of different densities. His view of particle motion working its way through an object as heat was qualitative. It was too limited to be seen as modern kinetic theory, as some later scientists (e.g., Tyndall, see ref 2, p 198) suggested. Looking Back: The Data Suggest There Is a Mechanical Equivalent of Heat
Rumford indicated that his system produced heat “equably or in a continuous stream”. This was a consequence of the constant 32 rpm rotation rate of the horses and the frictional pressure being held constant as well throughout the 2.5 hour run. Looking back at the work from a contemporary perspective, because mechanical energy was exerted by the horses at a D
DOI: 10.1021/acs.jchemed.9b00039 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Further Challenges by the Count to the Caloric Theory
caloric, the essence of heat, and its relative phlogiston, the essence of fire, can be seen as the last incarnation of Aristotle’s element fire in science. A brief outline of phlogiston theory is given in the Supporting Information. Acceptance of the importance of conservation of mass had discredited the phlogiston idea only shortly before Rumford’s cannon experiment.
The Count carried out other experiments of a less sensational nature that supported Joule by raising further questions about the caloric idea. He showed, while a vacuum was a good insulator, caloric traveled through it in a regular measurable way. What could be its nature? How could his findings that radiative cooling favored coarse black surfaces over smooth white ones be explained by caloric theory ideas? Further, his tests indicated that changes in temperature of an object, which should add or subtract caloric from it, produced no measurable mass change. The results of these and other studies attempted to chip away at the entrenched caloric idea.
Units of Measure
A problem the Count faced, working in the early days of science, was what units to employ in his work. His choice, the amount of heat to raise a pound of water from freezing to boiling, involved the most common of materials and phase changes set its upper and lower bounds. It could be reproduced in another lab. However, he had to contend with the effect of atmospheric pressure on boiling point, a concern that students can understand. In Munich, altitude 550 m, the boiling point of water is 210 °F. This discussion easily moves into consideration of how international scientific units are defined today. A recent case of interest involves the change in the kilogram’s definition.17 A more detailed discussion of the SI system of units is also pertinent.18
The Kinetic Theory of Heat Leads to a Derivation of Avogadro’s Hypothesis
Dalton’s view, in Figure 1, positing stationary atoms surrounded by the element caloric, was influential and a dead end. As Joule indicated later, the preoccupation with the belief that heat was a real substance was a problem for science. Acceptance of the Count’s view of his experimental results, a nascent kinetic idea, that heat and particle motion were intimately related, was a key to moving forward. By 1860, Clausius and at least 14 others had derived proof of Avogadro’s hypothesis assuming a gas to be composed of fast-moving molecules in collision, modern kinetic theory.13 Cannizzaro was on firm physical ground in bringing Avogadro’s idea into the mainstream of chemistry by 1858 (ref 7, p 612). As a consequence of the fact that equal volumes of gas contain the same number of molecules, vapor densities could provide a direct method of finding relative molecular masses.
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Experimental Reproducibility
The Count’s work, employing his involved apparatus, could not for all intents and purposes be easily reproduced. One simply had to believe him. This made it easier for critics to be skeptical of his methods and conclusions. In routine basic student experiments, inculcating the importance of consistency of results in multiple trials is a crucial goal. When Joule looked back and worked out calculations showing correspondence between his results and the Count’s, he was lending support to the validity of the earlier experiment.
RELATING THE COUNT’S EXPERIMENT TO CLASSROOM WORK
Thermodynamics
Calorimetry
For students with some exposure to thermodynamics, this curious experiment can be viewed as the workings of a complex apparatus whose purpose is to increase the entropy of the universe without producing useful work. There is a trace of what could be thought of as “useful” work done in that his dull steel tipped bar abraded away a small amount of metal from the cannon blank. This perspective was ahead of the Count’s time. Work and energy had yet to be defined. However, the combination of “thermo” and “ dynamics” characterizes the interrelationship issues he was spotlighting.
Early in the author’s introductory chemistry class, students have calculated the heat of combustion of a candle.14 They heat water in a soda can with a candle flame. A relevant worksheet for the experiment is given in the Supporting Information. They calculate the heat generated using Q = CmΔT, where C = 4.18 J/gC, m = g, and ΔT = change in temperature. Students then determine the mass lost by the candle and then the heat of combustion in kJ/g. Students can calculate the energy gained by the water in Rumford’s experiment. In their soda can case, 100 mL was heated to about 20 °C. He heated the equivalent of 12,300 mL by 100 °C. The question of heat not accounted for was discussed in the student’s work. Looking at the Count’s apparatus, they can suggest sources of loss there. Also, they can scale up their mass of candle wax lost to compare to a hypothetical value calculated by Rumford for his experiment.
Scientific Revolutions
The author discusses Rumford’s work in a history of science class. The cannon experiment fits into Kuhn’s19 ideas on how change comes to science. Rumford’s work was a Kuhnian “anomaly”, a direct challenge to what seemed settled science, that is, the conservation of heat and the existence of the substance caloric. He received the usual response given to anomalies. His results were challenged, denied, and then ignored. His continuing experimental attacks on caloric could not shake its hold on scientists. It took many years of work by Joule to convince his fellows of even the existence of the mechanical equivalent of heat. Joule’s work was accompanied by broadening theoretical ideas that culminated in the general principle of conservation of energy in all its forms. It was a crucial understanding of the 19th century that recast how all change in nature is viewed. It was a major paradigm shift. It had its beginnings in the troublesome work of the Count, who immersed himself in experimental studies addressing the question, “What is heat?”
Elements
The Count’s challenge to the existence of the element caloric in the cannon experiment arose from the fact that heat could be generated by motion in seemingly infinite amounts. Could it be a substance? Other experiments by Rumford indicated caloric was weightless, a further indictment against its element status. This historical case can serve as a starting point for consideration of contemporary elemental understandings and the difficulties new discoveries pose. The false element-118 discovery15 is a case that sparks interest. Aristotle saw the world as being constituted of four elements: earth, air, water, and fire. The “imponderables”,16 E
DOI: 10.1021/acs.jchemed.9b00039 J. Chem. Educ. XXXX, XXX, XXX−XXX
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(5) Lavoisier, A. L. Elements of Chemistry. In The Great Books; Encyclopedia Britannica Inc.: Chicago, IL, 1952, Vol 45; p 10. (6) Thompson, B. The Complete Works of Count Rumford; American Academy of Arts and Sciences, Macmillan and Company: London, 1876. (7) Partington, J. R. A History of Chemistry, Vol. 2; Martino Books: Mansfield Center, CT, by arrangement with St. Martin’s Press, 1963. (8) Newton, I. Newton’s Principia; Cohen, I. B., Whitman, A., translators; University of California Press: Berkeley, CA, 1999; Book II, p 697. (9) Holton, G.; Roller, D. H. D. Foundations of Modern Physical Science; Adison-Wesley Publishing Company: Reading, MA, 1958; p 378. (10) Value was adjusted for a calculation error by the Count. It is discussed in the Supporting Information. (11) Hamer, W. J. A Joule Centennial. J. Chem. Educ. 1968, 45 (2), 123−125. (12) Historical numbers and units are presented as published. The original documents do not address precision in presenting numerical results. The foot pound, ftlbs, is James Joule’s chosen unit of energy. 1.00 ftlbs = 1.36 Joules. (13) Rocke, A. J. Image and Reality; University of Chicago Press: Chicago, IL, 2010; 258−260. (14) Chemistry in the Community. Chem. Comm, 5th ed.; Heikkinen, H., Ed.; W.H. Freeman and Company: New York, 2006; pp 248−250. (15) Schwarzschild, B. Lawrence Berkeley Lab Concludes That Evidence of Element 118 Was a Fabrication. Phys. Today 2002, 55, 15. (16) The Oxford Companion to the History of Science; Heilbron, J. L., Ed.; Oxford University Press: Oxford, UK, 2003; pp 397−399. (17) Ghosh, P. Kilogram Gets a New Definition. https://www.bbc. com/news/science-environment-46143399 (accessed June 24, 2019). (18) Giunta, C. J. What Chemistry Teachers Should Know About the Revised International System of Units. J. Chem. Educ. 2019, 96 (4), 613−617. (19) Kuhn, T. S. The Structure of Scientific Revolutions; University of Chicago Press: Chicago, IL, 1996; pp 52−65.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.9b00039. Close-up views of model of Rumford’s apparatus; table of Count’s data used in Figure 4; class experiment (with worksheet) to determine the heat of combustion of a candle and relating the work to the Count’s experiment; comparison of Joule and Thompson data; brief discussion of phlogiston; description of aspects of the Count’s career-long espionage activities for several countries; character and the historical record; comparing the results of Joule and the Count to modern values for the conversion of Joules to Calories; details on the Count’s determination of the heat gained by his apparatus expressed in water equivalents; calculations of results in the manuscript presented in detail (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Frederic E. Schubert: 0000-0002-9503-9551 Notes
A DVD of a lecture about the Count, given by the author, a mixture of his life and work, is available from the author. It gives more of a picture of his extensive civic minded activities and brilliant ideas that changed day-to-day life of those in Munich and around the world. The author declares no competing financial interest.
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ACKNOWLEDGMENTS Thanks first to Judy Schubert for her artwork in the Graphical Abstract and Figures. Thanks also to colleagues John Garnevicus, Carla Moopenn, and Monica Witt for helpful comments on the manuscript. Thanks to Friends Seminary for supporting the author in offering his History of Science and Origin of Knowledge elective over many years. The support of The Portsmouth Athenaeum in sponsoring a talk on Count Rumford given by the author, that encouraged generation of this article, is greatly appreciated. Also, thanks to Len Harmon at the Rumford Birthplace Museum for his generous gift of time in discussing the exhibits there. Finally, The Stadtmuseum in Munich, Germany helpfully sent some useful materials relating to the Count’s activities there.
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REFERENCES
(1) Delbanco, N. The Strange Forgotten Life of America’s Other Ben Franklin by an Author So Fascinated He’s Writing a Novel About Him. American Heritage, 1993; Vol. 44; p 5. https://www. americanheritage.com/rumford (accessed June 24, 2019). (2) Brown, S. C. Benjamin Thompson, Count Rumford; The MIT Press: Cambridge, MA, 1979. (3) Linscott, R. B. The Contentious Count: Benjamin Thomson, Count Rumford; Talk to the Monday Evening Club: Pittsfield, MA, June 20, 2010. http://mondayeveningclub.blogspot.com/2010/06/ contentious-count-benjamin-thompson.html (accessed June 24, 2019). (4) Joule, J. P. The Scientific Papers of James Prescott Joule, Vol. 1; Scoresby, W., Ed.; Physical Society of London, Taylor and Francis: London, 1884. F
DOI: 10.1021/acs.jchemed.9b00039 J. Chem. Educ. XXXX, XXX, XXX−XXX
