CORRESPONDENCE
The Independence of Temperature and Heat of Reaction Antonino Giacalone
s a consequence of the increasing use of high tem-
A perature processing, our means for estimating reaction heats a t extreme conditions have assumed critical importance. Experimental difficulties in measuring reaction heats have made it necessary to estimate over progressively greater temperature ranges with means devised in a n era before high temperatures were extensively used. It is appropriate, therefore, critically to review our means of estimation in order to assess their reliability. T h e means for estimating reaction heats a t elevated temperatures are based on the work of Person (1847-51) and Kirchhoff (1858). Kirchhoff's work depends upon the First Law of Thermodynamics in the form
AU= Q -
W
(1)
where AU = internal energy change in the system Q = heat absorbed from the surroundings W = work done on the surroundings For an isochoric process, where W = 0, Kirchhoff derived the relation
d ( A U ) / d T = ZC,
(2)
where ZC, = the sum of the specific heats of the reaction products minus the sum of the specific heats of reactants For a n isobaric process he obtained the corresponding relation
d ( A H ) / d T = ZC,
(3)
It can be shown that Equations 2 and 3 are based upon unjustified assumptions. I n Equation 2 it is generally assumed that W represents only the work of expansion. Consequently, the change in internal energy of the reacting system ( A U ) is attributed entirely to the gain or loss of heat by the system. I n general, this is incorrect. T o properly characterize Q and W , it is necessary to consider some of the peculiar properties of heat which are a consequence of the Second Law of Thermodynamics. 54
INDUSTRIAL AND ENGINEERING CHEMISTRY
I n principle, all the various forms of energy (chemical, surface, electrical, atomic, heat, etc.) are interconvertible. Further, the various forms are convertible to mechanical work and to heat, via the familiar equivalence involving the conversion factor, J. The conversion of heat to work is subject to restrictions imposed by the Second Law. Equation 1 implies only two kinds of energy-viz., heat and work. Included in the work term, W , is not only the work resulting from the conversion of heat, but also that resulting from any other conversion. Equation 1 makes no distinction other than that Q is different from W. Neither does Equation 1 place any limitation on the values Q and W may assume, in particular no temperature dependence. The concept of temperature is completely extraneous to Equation 1. Equation 3 is even more difficult to accept because it assumes that Q, = AH without accounting for the expansion work which is already included in AH. Rather, Equation 3 was obtained by placing ZQ, = 0 for a system undergoing a cyclic process-Le., where the initial and final states are identical. However, from the First Law, AU = 0, and ZQ = ZW, for a cyclic process. The Second Law allows us to define quantities such as the entropy, S, and the free energy, F. I t can be shown that the internal energy, U, of a system is the sum of two essentially different kinds of energy ; the free energy, F, is' completely convertible, under proper conditions to work; another energy, T S , manifests itself as heat, also in an isothermal and reversible change (Qr,,). Therefore, in any given isothermal process, we may measure AF and TAS for the process in addition to the total energy change AU. These quantities are related by the Gibbs-Helmholtz equation
AU = AF
+-
T A S = AF - Td(At;)/dT
(4)
I n Equation 4, AS represents the entropy change of the system relative to a reference state, such that AF = 0 ; AF is the free energy change of the system at a
temperature T. For a reversible change, where A F = 0, AU = T’AS = Qr,, where T’ = the temperature of the reference state. For an irreversible change, at T < T’, the heat in excess of - Qr,, = - TAS arises from the failure to completely convert - A F to work. Close examination of Equation 4 discloses that conversion between heat and work is unidirectional and specific-Le., only unutilized work is spontaneously transformed into heat. Equation 4, unlike Equation 1 which treats only the external manifestations of change Q and W , is concerned with variables of state-Le., all the variables have exact differentials. Consequently Equation 4 ignores the form in which energy manifests itself externally. T o determine the change in internal energy with temperature, it is necessary to know the change in A P (free energy) and in TAS (“bound” energy) with temperature. The invariance of A S with temperature is required by the Second Law. Obviously this is true only for ideal systems. However, in all real cases, deviations from ideality must be accounted for. Thermodynamics tells us that the change of free energy with temperature equals the entropy change and that the free energy varies in the opposite sense from the bound energy. If we consider the definition of bound energy to be TAS, for every degree increase in temperature, the bound energy will increase by an amount A S and the free energy will decrease by the same amount. I t follows, therefore, that AU will remain constant iit all temperatures. The foregoing discussion leads to conclusions different from those of Kirchhoff with respect to the variation of AU with temperature. We note that Kirchhoff’s conclusions have never received experimental verification, but have been generally accepted and used nonetheless. Experimental verification of the above was obtained (2) by the study of the total molar surface energy (AllsM) of normal liquids which was calculated from the relation
I t can be shown that the Eotvos constant is a number proportional to ZC, for the components of the system. Since AU,, is the energy required to vaporize a molecule of liquid to a molecule of ideal vapor
-
ZC, = COgg C O j z = -AC,
where C,,, and C,,z are the molar heat capacities of the vapor and liquid, respectively I n Equation 5, when T = T,, y = 0, and 3NA1l3V M 2 ’ 3 7 = 0. Thus,
A Us, = - 3NA1l3T, [d(V M 2 ’ 3 7 )/d T] = Qrev
- 0)
= AC,(T,
A Us,/
T,
=
= AC,Tc
(6)
- 3NA1/3[d(V M 2 ‘ 3 7 ) /d T]
- Qrey/Tc= AC, = -d(AF)/dT and
-d(AF)/dT
(7)
= -3N.41i3[d(V,2/3r)/dT] = 12.82 cal. = AC, =
AS
(8)
AC,, like the Eotvos constant, is a universal constant, independent of the temperature. Its value agrees with the experimental data for normal liquids (3). Integration of Equation 8 between T and T,yields
A F = 3N.41/3V,2/37= AC,(T, - T )
(9)
and since A S = AC,,
-
AUS, = AF $- TAS = AC,(T,
T)
+ TAC,
= AC,T, = constant
(10 )
The ACv value may be calculated also from the work of expansion by the following equation which relates the surface energy to the expansion energy ( 4 ) ) R T ln($,/p)
= 3NA1/3VM2’3y = AC,(Tc
- T)
(11)
The value of C, = R T ln@,/$)/( T, - T ) (and, therefore, of AU = the heat of vaporization at constant volume = A,) is not merely a constant, as it is when calculated from the surface energy (4). This can be explained as follows. For an ideal vapor, the ClausiusClapeyron equation gives the relation A, = RT2[d(lnp)/dT] =
where
NA = Avogadro’s number V , = the molar volume of the liquid = M/(d, - d,) y = the surface tension -d(VM2/3y)/dT = 2.12 ergs = the Eotvos constant for nonassociated liquids. I t is independent of T
Therefore, A Us, is also independent of temperature.
AUTHOR Antonino Giacolone is a professor in the School of
Engineering at the University of Palermo, Palermo, Italy. He is a well known contributor to the field of Thermodynamics.
-2.3R [d(log p ) /d(l / T) ]
(12)
and differentiation of
+B
log$ = - A / T
(13)
with respect to temperature gives = A
-d(log$)/d(l/T)
(14)
Substitution of Equation 1 4 into Equation 12 yields A, = 2.3 RA = constant
(15)
Note that AC, = A J T , = 2.3 RA/T,. Since a plot of log@ us. ( I / T ) is a curved line ( I ) , A in Equation 15 is not a constant. However, the familiar Antoine equation logp = - A ’ / ( t
+ C) + B’
does involve a constant A’, in the temperature range from the triple point to T, = 0.75 (7). VOL. 5 8
NO. 9
SEPTEMBER 1966
55
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I n chemically reacting systems, we observe that free energy varies directly with the temperature, indicating the constancy of AS and, hence, of AU, with respect to temperature. The above consideration shows beyond doubt that summation of specific heats of the substances participating in a reaction has the dimension of entropy (6) and corresponds to d ( A F ) / d T , and not to d ( A U ) / d T as was supposed by Kirchhoff. I n the case of the total surface energy, equivalent to the heat of vaporization (when the volume of the liquid is neglected), we note that the sum of the specific heats corresponds to the entropy change at the limiting temperature T,, where AF = 0, and where AU = T,AS. Such a limiting temperature must exist for every system in equilibrium (5). If a reference temperature lower than T , is chosen, the entropy change will result from the summation of specific heats of the components plus the additional entropy change corresponding to the isothermal work necessary to bring the system from the limiting state to the chosen state. The heat of reaction at constant pressure may be obtained by adding to the heat a t constant volume, the heat corresponding to the expansion work, usually BnR T . Conclusion
T h e above exposition demonstrates that thermodynamics problems may not be resolved with the First Law alone, which limits itself to an energy equivalence independent of restrictions imposed by the Second Law. The Kirchhoff equations, which ignore the Second Law, therefore, are without thermodynamic foundation, and their use is questionable and unjustified. REFER ENCES (1) Fishtine, S. H., IND.ENC.&EM. 5 5 (4), 20 (1963). ( 2 ) Giacalone, A , , Gam. Chirn. R a l . 77, 73, 82, 444 (1947). (3) Zbid., p. 8 8 . (4) Zbzd., p. 448. ( 5 ) I b i d . , 81, 185 (1951). (6) Giacalone, A , , Ric. Sn’. 19, 706 (1949). (7) Thornson, G. W., Chem. Reu. 3 8 , 1 (1946).
C0R R ECTION
In the article, “Hot Applied Coal Tar Coatings,” by J. J. McManus, W. L. Pennie, and A. Davies [IND.END. CHEM. 58 (4),43 (1966)], Table I1 was inadvertently printed with the dots reversed. T h e correct pattern is shown below.
I
TABLE I I . COATING SYSTEMS
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I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Circle No. 1 on Readers’ Service Card-
