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From Joule to Caratheodory and Born: A Conceptual Evolution of the First Law of Thermodynamics Robert M. Rosenberg Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113
[email protected] Since the first law of thermodynamics was first formulated in the 19th century, it has had a substantial evolution, mainly in the definition of heat and in the number of terms requiring independent definition. This essay describes the course of this evolution. The 19th Century Development The first law arose after almost two centuries of controversy about the nature of force, heat, motion, energy, kinetic energy, and potential energy. This era is engagingly described by Rogers in “An Informal History of the First Law of Thermodynamics” (1). Our discussion will start with James Prescott Joule's experiment on “the mechanical equivalent of heat” (2). James Prescott Joule (1818-1889) was a brewer and physicist who did his experiments in his private laboratory. Joule carried out his experiments starting in 1843 at a time when the caloric theory of heat was being challenged, and heat was beginning to be considered as a form of motion rather than the “substance” caloric. He measured the change in temperature of a water bath when work was performed against friction, by a falling weight driving a paddle wheel, or by passage of an electric current through a resistor immersed in the bath. His major finding is that the change of temperature for a given quantity of work is the same no matter what kind of work was done. Joule clearly used the change in temperature as a measure of the quantity of heat. He concluded that there is a fixed relationship between the quantity of heat and the quantity of work, hence, the mechanical equivalent of heat. Unknown to Joule, the concept had already been propounded by Clapeyron (3) in 1834 and J. R. Mayer (4) in 1842, but these ideas were ignored. Later, Mayer applied to the biological world what would later become the first law of thermodynamics. When Hermann von Helmholtz (5) and Rudolph Clausius (6) extended the work of Joule to include the concept of energy of a system, they expressed the first law of thermodynamics as (in modern form), dU ¼ DQ þ DW
the form of heat or work, dU is equal to zero. All these discussions of the first law take for granted that the experimental quantities to be defined are work and heat, with work defined mechanically, and heat defined in terms of temperature change and heat capacity. J. Willard Gibbs, in his classic treatise (7) in 1875, like many other authors, takes for granted that ΔU is the sum of the heat exchanged and the work done. But Gibbs gives little consideration to operational definitions of these terms. When, in 1897, Max Planck (8) reinvented modern thermodynamics without knowledge of Gibbs' work,1 he also used heat and work as the primary experimental quantities, with the statement of the law of conservation of energy as ΔU ¼ Q þ W
G. N. Lewis and Merle Randall, in their classic work (9), also take the conservation of energy as a postulate, with the conclusion that U is a state function and that eq 2 is a statement of the first law. An alternative version for the earlier system has been proposed by Henry Bent (10). He designates the mutually interacting parts of the universe as (i) the system, (ii) the thermal surroundings, and (iii) the mechanical surroundings. These three entities are considered to be isolated from the rest of the universe. He then expresses the total energy change as the sum of the energy changes in the three entities, ΔUtotal ¼ ΔUσ þ ΔUθ þ ΔUwt
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ð3Þ
where σ represents the system, θ represents the thermal surroundings, and wt represents the mechanical surroundings, essentially a statement of the conservation of energy. In this convention, ΔUwt is determined by measurement of mechanical quantities such as the change in height of a weight, and ΔUθ is determined by a separate experiment such as Joule's of the mechanical equivalent of heat that involves only θ and wt. Thus, Joule's experiment does not involve a “system” according to this point of view. Bent defines Q as -ΔUθ and W as -ΔUwt, so that ΔUσ ¼ Q þ W
ð1Þ
which introduces the concept of the energy of the system, U, in terms of work, W, and heat, Q. They make the additional statement that for a given change of state, dU is a constant, although DQ and DW may vary (the capital D for heat and work, in contrast with the lower case d indicates this distinction symbolically). In this distinction, they introduced the concept of a state function, U. The conservation of energy follows from eq 1, since in the absence of interaction with the environment in
ð2Þ
ð4Þ
essentially the conventional form of the first law of thermodynamics, although he prefers not to use the symbols Q and W. In essence, this convention takes the conservation of energy as a postulate, rather than using Joule's experiment as evidence that U is a state function. Modern Developments Constantin Caratheodory (1873-1950) was Professor of Mathematics at the University of Munich from 1924 to 1950.
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In the Classroom
In 1909, he (11) tried to unify and simplify the treatment of the first law of thermodynamics by reducing thermal concepts to mechanical terms. From this point of view, Joule's experiment is assumed to be carried out adiabatically because Joule made very careful corrections for the small quantities of heat transferred between system and surroundings.2 The system is defined as the water in the tank, the device that transfers the work to the system, and the wall of the tank. The surroundings include the devices that do work on the system; in this case, the pulley and weights or the electrical circuit that does electrical work. His results demonstrate that adiabatic work of a given quantity produces the same change in temperature no matter how the work is produced, whether by friction, by turbulent motion of water, by compression of gas, or electrically. Because the adiabatic work is independent of the kind of work that is done, it is equal to the difference between two values of a state function U, which is named the energy, so that the energy change is defined in differential form as ð5Þ
dU ¼ DWadiabatic
where D is used with W because it is not in general a state function. If a change of state is not carried out adiabatically, the work is no longer equal to dU. The difference between dU and DW is attributed to the transfer of thermal energy to or from the surroundings as a result of a difference of temperature across a thermally conductive boundary, and we call this energy transfer, heat, DQ.3 This concept was first put forth by Max Born (12). Max Born (1873-1970) won the Nobel Prize in Physics in 1954. He defined heat in terms of energy and work as DQ ¼ dU - DW
ð6Þ
dU ¼ DWelectrical - PdV
ð9Þ
where -PdV is the pressure volume work. Because P is constant, we can add -VdP to the right side of eq 9 without its value changing, and obtain dU ¼ DWelectrical - PdV - V dP ¼ DWelectrical - dðPV Þ ð10Þ or dU þ dðPV Þ ¼ dðU þ PV Þ ¼ DWelectrical
ð11Þ
The parenthesized sum, (U þ PV) is a state function, the familiar enthalpy, H, so that ð12Þ DWelectrical ¼ dH for a constant pressure calorimeter. Thus, although the name calorimeter implies the measurement of heat, as a remnant of the caloric theory of heat, the calorimeter actually measures Z T2 DWadiabatic dT Wadiabatic ¼ T1
which is equal to ΔU or ΔH. An important question is: what is the change of state to which the value of ΔU or ΔH applies? Consider that we are carrying out the reaction XþY f Z The change of state that occurs in the calorimeter is XðT1 Þ þ YðT1 Þ f ZðT2 Þ ðreaction AÞ
or DQ ¼ DWadiabatic - DW
ð7Þ
The statement that U is a state function together with eqs 5 and 6 constitute the first law of thermodynamics according to this point of view. In principle, then, one can map the values of U, relative to some chosen reference state, by carrying out changes of state that can be reached by an adiabatic process. It has been pointed out that it is not possible to go from one adiabat to another reversibly without violating the second law of thermodynamics, so that eq 1 only describes values of U along a single adiabat (13). However, it is possible to go from one adiabat to another irreversibly. If one carries out that change of state in the spontaneous direction quasi-statically, with infinite slowness, one can determine Wadiabatic and hence ΔU in going from one adiabat to another. Thus, all values of U in the coordinate space P, V, T, can be mapped (12-17). The most common method for measuring “heat” is with a calorimeter. Most calorimeters are adiabatic, with the system defined as the calorimeter and the surrounding water bath. The value of dU is obtained by measuring the quantity of electrical work that is required to produce the same temperature change in the system as occurred when the state changed. The measurement is actually one of the adiabatic work, DWadiabatic, where the work is equal to dU or dH, depending on whether the change is carried out at constant volume or constant pressure, dU ¼ DWelectrical ðconstant volumeÞ
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where DWelectrical is the infinitesimal electrical work required to produce a given infinitesimal change in temperature and therefore a given change of state. If the calorimeter is held at constant pressure,
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ð8Þ
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But we wish to know ΔU for the reaction XðT1 Þ þ YðT1 Þ f ZðT1 Þ
ðreaction BÞ
We can obtain Reaction B by adding reaction C to reaction A, where reaction C is ZðT2 Þ f ZðT1 Þ
ðreaction CÞ
Therefore, ΔU ðreaction BÞ ¼ ΔU ðreaction CÞ þ ΔU ðreaction AÞ because U is a state function. But Z ΔU ðreaction CÞ ¼
T1
T2
DUZ DT
dT
ð13Þ
V
where (∂UZ/∂T)V is the conventional isochoric heat capacity CV. “Capacity” is a reference to the caloric theory of heat, implying that heat is contained in a system. The integral in eq 13 is the electrical work used to change the temperature of Z from T2 to T1 in the calorimeter in which the reaction was carried out. If T2 is greater than T1, then no addition of electrical work will change the temperature from T2 to T1, and we need to cool the system from T2 to T1 and measure the work required to heat the system from T1 to T2. Thus, the measurement of heat capacity is also a measurement of Wadiabatic. Some later authors follow Caratheodory in using eq 5 as their definition of dU, beginning with Born (12) and including
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In the Classroom
Callen (14); Denbigh (15); Zemansky and Dittman (16); and Berry, Rice, and Ross (17). Both Callen and Berry, Rice, and Ross followed Born in calling attention to the use of irreversible processes to extend the applicability of eq 1 to the whole P, V, T coordinate space of a system. As heat transfer engineers, Keenan and Shapiro (18) present an alternative statement of the first law based on a one variable definition that uses a no-work rigid wall instead of an adiabatic wall and uses heat instead of work as the defining variable. That is
Notes 1. Because Gibbs published his work in the Transactions of the Connecticut Academy, a journal not read by most European scientists, he had a mailing list of those to whom he sent reprints. At the time he published his thermodynamics papers, Planck was not on the list, because he was still a graduate student. When Gibbs published Statistical Mechanics in 1901, Planck received a reprint. It is interesting that Planck cited both earlier and later thermodynamics papers of Gibbs, but not his primary work on the equilibrium of heterogeneous substances. 2. Most of us infer from reading textbook descriptions of Joule's experiment that Joule's water bath was insulated. According to Joule, it was not. 3. This word has the unfortunate connotation that it is a material substance, a historical legacy of the caloric theory of heat. Some have suggested that we should not use the word heat at all, but its usage is too entrenched in the literature to be ignored.
ð14Þ
dU ¼ DQðno workÞ In this treatment, work is defined as DW ¼ dU - DQ
ð15Þ
This approach shares the economy of the approach of Caratheodory, but Keenan and Shapiro do not specify an operational definition of heat, analogous to the operational definitions of various kinds of work provided by mechanics and by electricity and magnetism, and they ignore that work is defined clearly in mechanical or electrical terms. Heat is usually defined through heat capacity and temperature change, but that definition seems circular because heat capacity is measured calorimetrically by measuring Wadiabatic. Conclusion Three approaches to the first law of thermodynamics still survive, and their proponents still see the merits of each. First, the historical approach uses work and heat as primary experimental quantities and energy as a derived quantity, with work measured mechanically or electrically, and heat measured by temperature change, with the magnitude obtained from the heat capacity and temperature change. Energy changes are then calculated with eq 1, with the understanding that U is a state function and work and heat depend on the path taken in passing from state 1 to state 2. A major problem with this convention is that heat is defined as ð16Þ
DQ ¼ CV dT
and, as pointed out above, CV is obtained in an adiabatic calorimeter, hence, measured by Wadiabatic. The second approach, suggested by Keenan and Shapiro, uses only heat as a primary quantity with both work and energy change as derived quantities. But, in the heat-based convention, where heat is defined as Z T2 CV dT ð17Þ Q ¼ T1
the heat capacity is determined in a calorimeter and depends on measurement of Wadiabatic. Thus, calculating W from Q is circular in the second approach. The third approach based on Caratheodory's proposal uses only work as the primary quantity, with both energy change and heat as derived quantities. U is a state function and heat and work are path dependent. All three approaches provide a basis for deriving all the relationships usually associated with the first law of thermodynamics. Both modern approaches, based on Caratheodory's suggestion and that proposed by Keenan and Shapiro, provide a more economical set of assumptions than the historical method, but Caratheodory's is firmly based on operational definitions of work and does not use circular reasoning.
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Literature Cited 1. Rogers, D. W. Chemistry 1976, 49, 11–15. 2. Joule, J. P. Philos. Trans. R. Soc. 1850, 140, 61–82. 3. Clapeyron, E. J. de l'Ecole Polytechnique, 1834, 14, 153-190. English translation in Reflections on the Motive Power of Fire; Mendoza, E., Ed.; Dover Pub.: New York, 1960. 4. Mayer, J. R. Ann Chem. 1842, 42, 233–240. 5. Helmholtz, H. v. Sitzung Phys. Ges. zu Berlin 1847, July, 23. Reprinted in Helmholtz, H. Ueber die Erhaltung der Kraft, Wissenschafttliche Abhandlungen, Barth, Leipzig, 1882; pp 12-75. 6. Clausius, R. The Mechanical Theory of Heat; Macmillan & Co.: London, 1879: p 31. An English translation of his original article, On the Motive Power of Heat, appears in Mendoza, E., Ed., Reflections on the Motive Power of Fire, Dover, New York, 1960, and in Philos. Mag., 1851, Series 4, 2, 1-21, 102-119. 7. Gibbs, J. W. “On the Equilibrium of Heterogenous Substances”, Trans. Conn. Acad. 1875, 3, 103-246. Reprinted in The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, CT, 1948; pp 55-353. 8. Planck, M. Treatise on Thermodynamics; Longmans, Green, and Co.: NewYork, 1903. Based on the First German edition, 1897, pp 38-45. 9. Lewis, G. N.; Randall, M. Thermodynamics and the Free Energy of Chemical Substances; McGraw-Hill Book Co., Inc.: New York, 1923; pp 47-54. 10. Bent, H. A. J. Chem. Educ. 1972, 49, 44–46. 11. Caratheodory, C. Math. Ann. 1901, 67, 355–386. Redlich, O. Rev. Mod. Phys. 1968, 40, 556–563. 12. Born, M. Phys. Z. 1921, 22, 218. 13. Craig, N. C.; Gislason, E. A. J. Chem. Educ. 2002, 79, 193–200. 14. Callen, H. B. Thermodynamics; John Wiley & Sons, Inc.: New York, 1960; pp 15-21. 15. Denbigh, K. The Principles of Chemical Equilibrium; Cambridge University Press: Cambridge, 1971; pp 15-19. 16. Zemansky, M. W.; Dittman, R. H. Heat and Thermodynamics; The McGraw-Hill Book Companies: New York, 1997; pp 72-81. 17. Berry, R. S.; Rice, S. A., Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: Oxford, 2000; pp 379-382, pp 454-456. 18. Keenan, J. H.; Shapiro, A. H. Mech. Eng. 1947, 69, 915–921.
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