COR R ES PO ND ENCE
Comment and Replies on Kircho$’s
Laws
SIR : Y
There appears in your September 1966 issue a paper entitled “The Independence of Temperature and Heat of Reaction.” I would like to make the following comments : 1. The meaning of Professor Giacalone’s discussion of “unjustified assumptions,” in reference to Equations 2 and 3, is not clear to me. T h e crux of the argument seems to be that the assignment of the words heat and work to the energy transports across the boundaries of the system during either a constant volume or constant pressure process is incorrect and, therefore, Equations 2 and 3 of the paper are wrong. It is true that the definitions of heat and work vary among thermodynamicists (2, 3) and, therefore, there can be disagreement about what is meant by work and heat. However, in the case of Equations 2 and 3 (Kirchoff’s equations), the derivation can be done independently of whether the transports of energy into or out of the system are called heat or work. For example, Equation 3 may be derived merely by requiring that the gases be perfect and noting the definition of the heat of reaction (I) :
( b A U / b T ) , = T(bAS/bT), = AC, Now, if AU is independent of temperature, then AS is independent of temperature and ACv is identically zero. The conclusion that AS is independent of temperature is a t variance with the third law, “ . . .for a chemical reaction, the reactants and products in quasistable states must have the entropy at zero temperature” (2). One concludes, then, that AS is zero at all temperatures, and this is not a satisfactory result. Also, there is no apparent reason why ACv should be zero. Therefore, it is doubtful that the conclusion that AU is independent of temperature is valid. literature Cited (1) Denbigh, K., “ T h e Principles of Chemical Equilibrium,” p. 142, Cambridge Univ. Press, New York, 1964. (2) Hatsopoulos, G. N., Keenan, J. H., “Principles of General Thermodynamics,” Wiley, New York;‘ 1765. (3) Obert, E. F., Concepts of Thermodynamics,” p. 268, McGraw-Hill, New York, 1960.
John H. Linehan AMV-ANL Fellow Argonne National Laboratory Argonne, Ill. 60440
AH = ~ v , h , t
where v t are the stoichiometric coefficients and h, are perfect gas molar enthalpies. Because the enthalpy of a perfect gas is a function of temperature only,
h , = Cz, T the derivation of Equation 3 is straightforward. Of course, because Kirchoff’s equations presume perfect gases, it should not be expected that these equations have universal validity. 2 . In the third paragraph, Professor Giacalone is led to the conclusion that AU will remain constant a t all temperatures. I t is interesting to examine the ramifications of the reasoning that led to this conclusion. First, it is pointed out that
( b A F / b T ) , = -AS Now, if Equation 4 is differentiated with respect to temperature a t constant volume, the result is
(~Au/AT). = (~AF/AT).
+ T ( ~ A S / A T ) .+ AS
SIR: The recent criticism (2) of Kirchoff’s relations is in part a criticism of the ignoration of constraints imposed in their derivation-a fault arising in practice rather than in the conventional thermodynamic texts (3-7, IO,
?I). These remarks, however, are pertinent: 1. The first law can be variously stated: energy, U , is conserved in the universe; U is a function of state; dU is exact; dU = dQ - dW. Similarly, from its p V , enthalpy is also a function of definition, H = U state. This means that AU and AH are each independent of the path and depend only on the initial and final states for a process. I t is this statement which is the basis for the Kirchoff relations.
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2. From the statements dU = dQ - dW = -pdV d W ’ , where dW’ is the infinitesimal net work-Le., all dW‘. nonexpansion work-then dU = dQ - pdV When dV = dW’ = 0 , dU = d Q v , w i . If heat transfer is attended by a temperature change, dU = CV,,/dT. Similarly, dH = d Q Vdp dW’ and dH = dQ,,,, = C,>,tdT. More often than not the subscript W’ is omitted; this can be a source of error in application. 3. Although the first law is not stated in terms of a temperature scale, it is not indifferent to this property. The ideal gas temperature scale is used to define the state of the system. The Carnot cycle permits an identification of this scale with a purely thermodynamic one. 4. Rather that being in a position to show that U = F TS, one commonly uses this equation to define F, the Helmholtz function. Also the Gibbs-Helmholtz equation includes the constraint that dW’ = 0. From
+
+
+
+
dF = dU - d ( T S )
dU = d Q - pdV
+ dW’
and dQ,,, = TdS
I t is not entirely clear what was meant by “the invariance of AS with temperature is required by the second law.” For an isobaric reversible boiling process
AS, = Q,/T
AH,,/T = (All, +pAV,)/T
and
(bAS,/dT), = (AC, - AS,)/T where:
A c , = (bAff,/bT), = C,(vapor) - C,(liquid) Only if AC, = AS, will AS, be temperature-independent. If, for example, Trouton’s rule is used, the derivative of the entropy of boiling is negative. This means that the isobaric heat of evaporation does not increase as fast as does the temperature; the entropy of boiling decreases as the temperature rises. I t is probably not safe to extend the invariance of the Eotvos constant to T,; the Ramsay-Shields equation is an improvement over Eotvos’ for just this restriction ( 9 ) . However, there is no disagreement over the observation that the total molar surface energy is essentially independent of temperature (8). This arises from the Gibbs-Helmholtz equation for surfaces (7)
LT8= y - Tdy/dT
then
dF = -SdT - p d V
+ dW’
The derivation of the Gibbs-Helmholtz equation involves the term
(bF/aT),,w/ = --S
+
5. The equation AU = AF TAS is only valid for isothermal processes; for other processes dU = dF TdS SdT must be employed. The conclusion that A t r is independent of temperature is based on an erroneous premise. 6. Although it is indeed difficult to verify Kirchoff’s relations for reactions at very high temperatures, the derivation is based on the assumption that changes in energy and in enthalpy are independent of path. One can, for example, go from a set of reactants at one temperature to a set of products at another temperature by two different isobaric paths and obtain the Kirchoff relation concerning enthalpy changes and heat capacities at constant pressure and with the condition that W’ = 0. T o expand on my earlier remarks, for any cyclic process AU = AH = AV = 0 . The concept of enthalpy is introduced exactly to have a convenient thermodynamic function of state for isobaric processes. Because for isobaric processes
+
+
AH = AU
+ p A V = ( Q - pAV f
W’)
+ PAV
=Q,+W then simply AH = Q , when W‘ = 0 ; no work of expansion has been ignored. If a cyclic process is isobaric throughout and if W‘ = 0, then AH = Q , = 0 . For the same process the total work of expansion ispAV(tota1) = 0. This is, however, not the case for the nonisobaric Carnot cycle. 78
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
and the empirical linear variation of y with T . The constancy of U8 is as good as the temperature-linearity of y. It doesn’t seem necessary to destroy the Kirchoff equations to show this. LITERATURE CITED (1) Adamson, A. I“., “Physical Chemistry of Surfaces,” Interscience-Wiley, New York, 1960. (2) Giacalone, A,, IND. ENG.CHEY.5 8 , 5 4 (1966). (3) Glasstone, S.: “Thermodynamics for Chemists,” Van Kostrand, New York, 1947. (4) Guggenheim, E. A , , “Thermodynamics,” Iiorrh Holland Publishing Co., Amsterdam, 1959. (5) Kirkwood, J. G., Oppenheim, I., “Chemical Thermodynamics,” McGrawHill, New York;c 1961. ( 6 ) Klotz, I. hi., Chemical Thermodynamics,” Benjamin, New York, 1964. (7) Lewis, G . h-.,Randall, M.,Pitzer, K. S., Brewer, L., “ThermodynaInics,” IvicGraw-Hill, Yew York, 19:;. ( 8 ) Moelwyn-Hughes, E. A , Physical Chemistry,” 2nd ed., Pergamon Press, Oxfnrrl. 1961.
(9yP&trAgton,. J. R., “An Advanced Treatise on Physical Chemiswy,” Vol. 11, Longmans Green, London, 1051. (IO) Prigogine, I., Defay, R., “Chemical Thermodynamics,’’ Wiley, New York, 10?6
(1 i j ’ z i m a n s k y , M., “Heat and Thermodynamics,“ McGraw-Hill, K c w York, 1943.
Avrom A. Blumberg Department of Chemistry De Paul University Chicago, Ill.
Professor Giacalone’s Reply SIR: I have read with due attention the remarks of Dr. Blumberg and Dr. Linehan on my recent paper (‘The Independence of Temperature and Heat of Reaction.” The remarks seem to indicate a great desire to defend an established position and little desire to evaluate a new and perhaps better one.
Y
The application of the Gibbs-Helmholtz equation to calculate the total molar surface energy, U S M ,in Equation 5 shows that U s , is independent of temperature because the Eotvos constant, -d(VM2J3y)/dT= 2.12 = AS, is independent of temperature. This fact, which evidently contradicts the assumptions of the Kirchoff Equation 2, removes any doubts. The Kirchoff equations are based on experimental data and the application of familiar formulas. On the basis of Equation 5, with the deductions of Equations 6, 7, and 8, I have shown that -dAF/dT = ACv. The calculated value of AC, = CVnL- Cv,G = 12.82 cal. is in agreement with experimental values for normal liquids. None of these arguments was discussed by my correspondents. The matter of the distinction between heat and work, which is more than a matter of confusion of words as suggested by one correspondent, is essential to my argument. The two kinds of energy are essentially different according to the second law. Ignorance of the distinction affected the validity of the Kirchoff equations. Because of the distinction between heat and work made by the second law, we can determine (with the Gibbs-Helmholtz or the Clausius-Clapeyron equations) U or H at each temperature from the maximum work of reaction and its change with the temperature. I feel that the answers to the criticism of my article are adequately answered by the above reply and a closer reading of the article by my correspondents. I thank them for their interest and comments in this matter. Antonio Giacalone Palermo, Italy
SIR: I have read Professor Giacalone’s article in the September 1966 issue and offer the following comments. He states that Equations 2 and 3 are based on unjustified assumptions. O n the contrary, they are based on nothing but the first law of thermodynamics and the definition of specific heat. Does this mean that he questions the first law? He further states that it is incorrect to attribute the AU of a reacting system entirely to gain or loss of heat. Surely this is not incorrect if the only external force is a fluid pressure; and when one is considering a heat of reaction, all other forces are ruled out. This is basic to the definition of the heat of reaction. Perhaps Professor Giacalone is using a unique definition of heat of reaction. If so, there is no argument, but it would help matters to have his definition. He states that “Equation 3 is even more difficult to accept because it assumes that Q , = AH,” etc. He doesn’t define Q p , but if he intended to represent the heat at constant pressure, then it does equal AH if one
takes, as one should, the case of pressure being the only external force. This is the only assumption involved. He then introduces a discussion of the second law which seems wholly irrelevant because the change of AU and AH with temperature depends entirely on the first law. Later he states, “The invariance of AS with temperature is required by the second law.” This is certainly not the case. There is nothing in the second law which requires this and, in fact, the third law requires just the opposite, namely that AS decreases as T is lowered and becomes zero at T = 0. The relation of AS to T is given by the equations
and
(%),
ACP =
T
For A S to be independent of T , either ACv or ACp must be zero. This might be true in a few special cases, but in general such is not the case. He also states that it follows from Equation 4 that AU is independent of T and we have just seen that such is not the case. Professor Giacalone notes that Kirchoff’s conclusions (presumably Equations 2 and 3) have never received experimental verification. This is equivalent to saying that the first law has never received experimental verification. The fact that not one exception (excluding nuclear phenomena) has ever been found to the first law in over 100 years seems sufficient verification to me. The treatment of surface energy seemed irrelevant to me. Professor Giacalone states, “ I n chemically reacting systems, we observe that free energy varies directly with temperature.” Please don’t include me in this “we.” Examination of any text on physical chemistry or thermodynamics will verify the fact that AF is not, in general, a linear function of T and, therefore, AS is not a constant independent of temperature. In yet another place Professor Giacalone says, “The above consideration shows beyond doubt that summation of specific heats. . . has the dimensions of entropy.” I t doesn’t require the “above consideration” to show this; both specific heat and entropy involve a heat quantity divided by a temperature and it seems obvious to me that they have the same dimensions. I disagree completely with Professor Giacalone’s conclusions. Many thermodynamics problems have been solved with the first law and I can’t believe that we have been wrong all these years. The Kirchoff equations have a good foundation in thermodynamics and their use is not at all questionable or unjustified. I may have misunderstood what Professor Giacalone was driving at. If so, I hope that he will show me where I have erred. Barnett F. Dodge Department of Chemical Engineering Yale University New Haven, Conn. VOL. 5 9
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