Duncan M a c R a e
Route 3 Be1 Air, Maryland
The Fundamental Assumptions of Chemical Thermodynamics
"Ah, take one consideration with another" -the student's lot-like that of Gilbert and Sullivan's policeman-"is not a happy one!" At least, so it seems a5 far as thermodvnamics is concerned. Not nnlv are there many differeht verbal statements of the firit and second laws ( I ) , but a number of sets of fundamental assumptions have been proposed to take their place (2-6). There are four kinds of approaches to the subject (7) and the situation is such that teachers of parts of thermodynamics in different courses for students of chemistry and chemical engineering, instead of using the student's previously acquired knowledge (S), begin each year by saying, ''Well now look boys, just forget about what you learned last year, this is the way we're going to treat these problems" (9). It may, therefore, be worthwhile for teachersespecially those who plan to write textbooks-to go hack to the beginning and review the fundamental assumptions of chemical thermodynamics as they were presented in three papers by the founder of that science (10). He had an uncommon knowledge of mechanics, both classical (11) and statistical, as well as of thermo~
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dynamics, and clear ideas as to their relations to each other (12). His assumptions must be good ones, because the many theorems derived rigorously from them have stood the test of time and experiment. A closer look at them might help eliminate some of the present confusion without further complicating the subject by introducing yet more assumptions or laws, which may not rest on so firm an experimental basis. It would, no doubt, be useful to identify a11 laws, postulates, principles, axioms, definitions, and other fundamental assumptions, stated or implied, in the three papers, in much the same way as has been done recently for Euclid's geometry ( I S ) , and still more recently for the whole of mathematics as now taught in schools and colleges (14). This might also be done for other formulations of chemical thermodynamics and the results compared. For, in thermodynamics as in mechanics, "the greater the number of points of view from which we examine a subject, the deeper should be our understanding of it" (15). A less ambitious program is to call attention to a postulate, two principles, and seven equations, two of which "may be considered as," and four of vhich are
called, definitions. The postulate is evidently such from the way it is stated and its use in the first two papers, but becomes a theorem when it is derived from other assumptions in the third paper. It is stated in the first paper as follows: It is taken for granted that the body has a uniform temper* tore throughout, and that the pressure (or expansive force) has a uniform value both for all points in the body and for all directions. This, it will be observed, will exclude irreversible processes. but will not entirelv exclude solids. althoueh the condition
If certain of the original symbols are replaced by others now more commonly used, three of the equations are given in the first paper as follows: We have to consider the following quantities: V, the volume, 1 P, the pressure, of a given body T, the (absolute) temperature, in state, E. the enerw. s,' the entropy, w, the work done, by the body in passing slso and q, the heat received. from one state to another.
I
I1
These are subject to the relations expressed by the following differential equations: dw = a P d V (a) d E = bdq-dw (b) (c) d S = dq/T where a and b are constants depending upon the units by which V, P, w and q sre measured. We may suppose our units so chosen t h a t a = 1and b = 1. .(17).
.
The following is found in the second paper:
...the principle that the volume, entropy, and energy of the whole body are equal to the sums of the volumes, entropies, and energies respectively of the parts, while the pressure and temperature of the whole are the same as those of each of the parts (18). I n the third paper, the following principle and equations are added: I t is an inference naturally suggested by the general increase of entropy which accompanies the changes acourring in any isolated system that when the entropy of the system has reached a maximum, the system will he in a. state of equilibrium. Although this principle has by no means escaped the attention of physicists, its importance does not appear to have been duly appreciated. Little has been done to develop the principle as a foundation for the general ththry of thermodynamic equilibrium (19). d E = TdS - P d V #,dm, ndm p.drn, (12) (80) A=E-TS (105)($1) G=E-TS+PV (116) ($1) H = E + P V (118) ($1)
+
+
+ ... +
in which some of the original symbols have been replaced by others so that they have the following meanings:
and
m, the masses of the components, n, the number of components, A, the Helmholtz free energy, G, the Gibbs free energy, and the p, the chemical potentials, H, the enthalpy.
Chemists usually take the unit mass of a component to be equal to its molecular weight so that its mass is equal to n, the number of moles of it. Some other
letter must then he used for the number of components. The ahove quotations must he interpreted strictly in the light of their original context-not all 434 pages of it; but, at least, the pages cited in the references. Of come, all teachers of students of chemistry and chemical engineering might profitably take an occasional glance at the biographical sketch (22) and the synopsis and abstract of the third paper (28). Yurther, since the general theory of thermodynamic equilibrium is patterned after Lagrange's theory of mechanical equilibrium, some acquaintance with configurations, displacements, constraints, generalized coordinates, and the principle of virtual work (24) is highly desirable because these have their counterpart8 in the thermodynamic theory. All ten quotations come from the first 144 pages of the book containing the three papers, and do not, therefore, include the additional assumptions required for the treatment of systems influenced by gravity, electricity, distortion of solids, and capillary tension. Not all of these assumptions are mentioned as such even in the later pages. It appears that, in general, the assumptions of mechanics are employed, as required, without comment. For example, equation (a), which comes from mechanics, is not called a definition or a principle; mass is not included with volume, entropy, and energy in the additivity principle, although a similar assumption as to mass seems to be employed in the derivation of "the conditions characteristic of chemical equilihrium" (25) ; and definitions of gravitational and electric potentials are not mentioned, when these potentials are introduced (26, 27). These quotations do not even include all the principles and definitions in the first 144 pages. They do not include "the principle of continuity" (28) or the "principle that, for a path which begins and ends on the same isodynamic, the work and heat are equal" (29); nor the definitions of an ideal gas (SO), fundamental equations @I), phases (Sf?), coexistent phases (52), critical phases (SS), and several verbal definitions of potentials (54); nor the criteria of stability "readily deduced from that of equilibrium," but, nevertheless called principles (55). Yet, incomplete as it may be, the set of fundamental assumptions included in the ten quotations given ahove ought to be something on which all teachers of students of chemistry and chemical engineering could agree. The six equations are found in every textbook of chemical thermodynamics. The postulate is implied whenever the temperature and the pressure of a body are mentioned. The two principles are, at least tacitly, accepted by all teachers of that subject. Indeed, four of the equations and the postulate are taught in a first course in physics for students of science and engineering in universities (36); and the two principles might easily be added alougwith the additivity ofweight and electric charge, and the principle of the increase of entropy from which the principle of thermodynamic equilibrium is inferred. Perhaps, however, the most immediate advantages of this set of fundamental assumptions lie in their method of presentation: 1. All the many diierent verbal statements of the first and second laws are disposed of with the statement: Volume 43, Number 1 1 , November 1966
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That functions of the state of the body exist, the differentials of which satisfy these equations, may easily he deduced from the first and second laws of thermodynamics (57).
2. Since the six equations are regarded as definitions, they require no jmtificatinn other than the well-known utility of the formulation of chemical thermodynamics in which they occur. 3. Heat and the absolute temperature are left undefined and without "intuitive descriptions" (38) so they require no more discussion than to call attention to rules for their measurement (39)in elementary textbooks of physics or the extensive literature of calorimetry and thermometry (40). Literature Cited (1) HALLIWELL,H. F. AND NYBURG,S. C., Scienm Progress, 46, 451 (1958). "Statements of the F i n t and Second Laws of Thermodynamics.'' (2) CAR~THE~DORY, C., Mathematische Annalen, 67, 355 (1909). "Investigations of the Foundations of Thermodynamics." (3) LE CHATELIER, H., Rev. Quesl. Sci., 94, 363 (1928). "Principesfondamentam del'energetique." (4) LAMER,V. K., FOSS, O., AND REISS,H., Acta Chem. Scand., 3, 1238 (1949). A clear exposition of Br$nsted's two principles. ( 5 ) CALLEN,H. B., "Thcrmodynamics," John Wiley & Sons, Inc., New York, 1960, p. 192. Four postulates. ( 6 ) H ~ ~ s o ~ o uG. ~ oN., s , AND KEENAN,J . H., "Prineiple~of General Thermodynamics," John Wiley & Sons, Inc., New York, 1965. Contains a "new set of postulates and definitions." ( 7 ) DIXON,J. R., A N D EMERY,A. H. JR., Am. Scientist, 53, 428 (1965). "Semantics, Operationalism, and the Molecnlar-St,at,istiee*lmodel in Thermodynamics." ( 8 ) M A C ~ ~ AD., E , J. CHEM.EDUC.,32, 172 (1955). "The Int,roduction to Thermodynamics 11." ( 9 ) IBELE,W., ROSSINI,F., AND TISZA,L., Int. Sci. and Tech., No. 42, p. 56 (1965). "What Is Thermodynamics?" (10) Glass, J. W., "Scientific Papers," Dover Publications, New York, 1961, Val. I, pp. 1 4 3 4 . "Thermodynamics." (11) ibid, Vol2, pp. 1-15, "On the Fundamental Formulae of Dynamics." (12) G I B B ~J., W., "Collected Works," Longmans, Green, and Co., New York, 1931, Vol. 2, Part 1, Elementary Prin-
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ciples in Statistical Mechanics. Chap. 14. Discwsion of Thermodynamic Analogies, pp. 164-167. 1131 D'ABRO.A,. "The Rise of the New Phvsics." Dover PuhlicatLns,'~ewYork, 1951, Vol. I, pp. i93-i98. AI~I~ENDOERFER, C. B., AND OAKLEY,C. O., "Pl'in~iple~ of Mathematics," McGraw-Hill Book Co., New York, 2nd Ed., 1963, pp. 1 4 7 , Logic and Sets. LINDSAY,R. B., A N D MARGENAU, H., "Foundations of Physics," Dover Publications, New York, 1957, p. 158. GIBBS,J. W., "Scientific Papers," Ilover Puhlirations, New York. 1961.Vol. I. D. I Note. (17) ibid, pp. 1-2. i1U ) ihid. n. Re.
(23) ibid, pp. 35+371. (24) SYNGE,J. L., AND GRIFFITH,B. A,, "Principles of M e chanics." McGraw-Hill Book Co.. New Yark. 2nd ed.. 1949, pp. 285-302. (25) GIBRS,J. W., lac. Cit., p. 65. (26) ibid, p. 148. (27) ibid, p. 332. (28) ibid, p. 5. (29) ibid, p. 9 Note. (30) ibid, p. 12. (31) ibid, p. 86. 132) ibid. o.96. ibid, p. 129. ibid, pp. 92-96. ibid, p. 61. SEARS,F. W., AND ZEMANSKY, M. W., "Univemity Physics," Addison-Wsley Publishing Co., Reading, Mass., 3rd Ed., 1963. GIBBS,J. W., lac. cit., Val. 1, p. 2 Note. C. B.. AND OAKLEY.C. 0.. lac. el.. DD. , , ALLENDOERFER. 6 Para. 1.2 Definitions and para.'1.3 ~rabositions: ' ' (39) SEARS,F. W., AND ZEMANSKY, M. W., 10~.cit., p. 1. para. 1-1. The fundamental indefinahles of mechanics. "The rule for measuring an indefinable takes the place of a definition." (40) DONNAN, F . G., A N D HAAS, A,, "Commentary on the Scientific Writ,inga of J. Willard Gihhs, Vol. 1, Thermodvnarnics." Yale Univenitv Press. New Haven. 1936. 16. According to E. B. ~ i l s o nGibbs , began his Lketures an Thermodynamics on October 3, 1899 with a s t a t e ment to the effect that "The measurements in our subject fall into two sets, thermometry and calorimetry."
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