FREE ENERGY AND H E A T CAPACITY BY J. M. BELL
Introduction T h e question of the existence of a fundamental connection between changing heat capacity, change of free energy, and heat of reaction was raised by Richards‘ in 1902. I n his paper are given experimental data, from which the inference is drawn that the change of total energy with the temperature stands in a simple ratio to the change of free energy with the temperature. Equivalent to this conclusion of Richards is that of van’t H ~ f f , ~ namely, that the change of heat capacity stands in a simple ratio to the temperature coefficient of electromotive force. Quite recently, Haber and Tolloczko3 have declared that the conclusions arrived at by Richards and van’t Hoff are of the utmost importance, an importance which will perhaps direct physicochemical research along somewhat new channels. T h e objects of this paper are, first, to make a critical examination of the data from which the above conclusions are drawn, with special reference to the extent of the probable error in the observations, and to the degree in which errors of such magnitude would affect the calculations ; second, to examine critically the assumptions which are made in the calculations ; and, third, to discover whether the concltisions are in accord with the assumptions and in accord with the available experimental data.
Notation T h e notation used in this paper is that employed by T r e v ~ rviz., ,~ E is the total energy of the system before the isothermal change, and A E is the change of energy during the reaction. Proc. Am. Acad., 38, 293 (1902); Zeit. phys. Chem., 42, 129 (1902).
* Boltzniann Festschrift, p. 233 (1904). Zeit. anorg. Chem., 41, 436 (1904). Jour, Phys. Chem., 9, 299 (1905).
/. M. Bell
382
This latter corresponds to U used by Richards, and to Q used by van’t Hoff. F is the free energy of the system before the isothermal change, and AF is the change of free energy during the reaction. AI? corresponds to A used by Richards, and to E used by van’t Hoff. 8 is the absolute temperature; Richards and van’t Hoff use
T. c is the heat capacity of the system at constant normal’ parameters, the state variables of the system being the normal’ variables x,, . . . xv, 6 ; and Ac is the change of the heat capacity in the isothermal. reaction. (
Diseussion of the data T h e possible experimental errors in the data, upon which Richards bases his conclusions, will be ascertained, in order to determine how errors of such magnitude would affect the results. T h e calculation of the change of heat capacities, upon which the coiiclusions are based, have been made mainly from the data of Marignacl. Regarding the degree of accuracy of his results, J’ai portk les chaleurs specifiques ‘avec Marignac says : quatre decimales, telles qu’elles m’Ctaient donnkes par la moyenne des cinq ou six determinations, faites sur chaque solution. On peut sans inconvknient supprimer la q u a t r i h e dkcimale, puisque la troisiPme peut &re deja affectke d’une erreur de I ou 2, rarement de 3 uiiitks.” Taking account of the errors, we shall repeat Richards’s calculation’ for the case of the Daniel1 cell in which the initial solution is of the composition CuSO,. 200H,O. The equation is, of course, Zn CnSO,.zooH,O = Cu ZnS04.200H,0. For the specific heat of a solution of the composition CuSO,. 200H,O, Richards takes the mean of two figures given by Marignac, (a)0.9504, the specific heat between 18’ and 25’, and (6) 0.9527, the specific heat between 22’ and 53’, thereby assuming that for this temperature range, the specific heat does not perceptibly change. This assumption will be discussed ((
+
Ann. Chim. Phys., (5) 8, 418 (1876).
Loc. cit., p. 296.
+-
Fvee Eizevgy and Heat Cajacity
383
more at length later. Supposing for the moment that this assumption is justifiable, it is evident that there is an error of at least 2 in the third place of decimals. However, as the calculation has been made taking the mean as the true value, let us suppose that there is an error of I in the third place. The calculation, taking account of the errors is : Mayers
Factors :
CuS04.200H,0, (0.9516 Zn
_t
o.ooro) X (3762.7) X (4.181) = 14970 ‘5 . 26 Total factors, 14996 t 15
*
Products :
ZnS0,.2ooH,O, (0.9523 t 0.0010) X (3764.5) X (4.181) = 14989 F- I5
.
cu
Total products,
26
15013
15 Difference, I 7 & .ZO. I t will be seen that the error is out of all proportion .to the small difference which the figures seem to show. Further, in case the possible error is 2 or 3, which it may be, the error in the data indicating the change of heat capacities will be approximately 60 or 90 mayers respectively. I n the ten cases cited by Richards, the greatest change of heat capacity is calculated as 124 mayers, while in most of the cases this change of heat capacity is given as numerically less than 70 mayers. Hence it follows that not only is the actual change of heat capacity uncertain within rather wide limits, but also that the sign of this difference in a great majority of the cases actually cited is just as uncertain. Thus it can be asserted that Richards’s conclusions, both of which involve statements regarding the sign and magnitude of the change of heat capacity, are based on insufficient experimental evidence.
Richards’s eonelusion is at variance with an assumption One of the inferences which Richards has sought to establish aAF aAE is that __ and ae stand in a simple ratio, and it is clainied a0
~
that this inference is supported by experimental data.
a A F is ~
ae
the temperature coefficient of the free energy change, or in this
J. M. Bell
384
case the temperature coefficient of the electromotive force, and
where Ac is directly calculated from the data, the accuracy of which has been discussed above. However, even if the data were approximately correct, the figures do not warrant the inference of a linear relation between a--A F a A E
ae-. __ ae '
is given as varying from
aAE ~
ae
and
I :0.13
to
aAF
--.a0
T he ratio
I : I. 70
giving an
average of I : 0.53, an average" which can scarcely have any significance. From this average" it is deduced that ((
a__ AF -
ae
--
aAE
-
M --,
ae
in which the value of 31 "averages" about 2. We shall assume temporarily that this equation holds, and examine whether it is at variance with the assumption, tacitly made by Richards, that Ac is independent of the temperature. It has been shown by Trevor' that this hypothesis of Richards, of the proportionality aAF
of - and
ae
aAE
-,ae
is tantamount to the hypothesis that Ac is
proportional to the IM-/th power of :he temperature. Assuming the average" of Richards to be correct, it follows that ((
where a is independent of 8; in other words, the isothermal change of heat capacity is dependent upon the temperature. However, in calculating the specific heat of CuSO,. zooH,O, Richards takes the mean of two measurements, the one having the end temperatures 18" and 25", and the other the end temperatures 22' and 53". I n taking such an average the tacit assumption is made that the specific heat of CuS0,.200H20 is independent of the temperature. Further, Richards has taken the only figure given for ZnSO,.zooH,O, the end temperatures being zoo and 52'. These specific heats are assumed to be LOC.cit., p. 299.
Free Eizevgy and Heat Cajacity
385
equivalent to the Specific heats at 18" C, an assumption which is not in accord with the result which the data are adduced to prove. A linear relation between a4E/ae and ahF/a&does n o t exist I t has been shown that the data used by Richards involve errors of observation which vitiate the conclusions drawn from them, and that, even though the data were correct, an assumption has been made in the calculation which is out of harmony with the inferences drawn from the data. Further, the existence of such a relation as
a4F
aAE
-
-a7
XT - -
cannot be proved or disproved from purely theoretical considerations alone, wherefore we must appeal to experiment. By differentiation of the free energy equation AF=AE+0-
a4F
ae
we obtain agAF'
-0,,,---
-
a4E
ao
= Ac.
From this equation it follows that the sign of A c is different from that of the change of the temperature coefficient of AF a4Fwith.the temperature. Further, it follows that if ~is con-
ae
stant over any range of temperature, then A c o over that range. Several cases will be considered. I t has been shown by Streintz' that for the lead storage cell in which the sulphtiric acid concentration is 0.0005 gram-mol. per liter, the electromotive force is a linear function of the temperature, between 20' and 65". This cell has even been suggested AF by Dolezalek' as a thermo-element. Streintz found that a-3
ae
1 2
Wied. Ann. 46, 499 (1892). pie Theorie des Bleiaccuniulators, p. 54 (1901).
J. M. Bell
386
is negative and constant over the temperature range considered. It follows that in this case Ac = 0, and hence that there exists no linear relation between
aAF __
a0
aAE
and a0
Another case of reversible cell where
aAF
is constant is that
of the amalgam cells. Richards and Lewis' have shown that for the cadmium amalgam cell, the temperature coefficient is constant and positive over the range o to 2 4 O , and for the zinc amalgam cell, the temperature coefficient is constant and positive over the range oo to 30°. aAF
I n the above cases was found to be constant over a coma0 paratiaely small temperature range. Weber,Zhowever, has stated that within temperature ranges of over 400°,there is no appreciable change in the heat capacity of many systems of the type Pb I PbCl, I C1, and'of the type Pb I PbCl, I PbBr, 1 Pb, and consequently i3AE - = 0.
a0
aAF
Yet in all such cases he found that the value of --,
ao
the tem-
perature coefficient of the electromotive force, is of considerable magnitude. From these results also it follows that Richards's concliision of a fundamental linear relation between
*a ae
is at variance with the experimental data. I n all the above cases
with
--e
aAF ae and
the
temperature,
aevanishes. Zeit. phys. Chem., Zeit. anorg Chem.,
aAF
has been found to vary linearly azAF
hence -ae2
am
Consequently --, a0 28, I
(1899). (1899).
21, 305
vanishes, and
hence
at least Over consider-
B e e Enevgy aizd Heat Capacity able ranges of temperature, is zero, and, as
aAF ~
ae
387
in the cases con-
sidered is of considerable magnitude, it follows that there is no aAF
aAE
-.ae
algebraic relation between -- and
ae
Richards' has more recently said: ((Recent study has made it appear highly probable that a change in heat capacity during a reaction is the chief, if not the only, reason why the total energy change (or the heat of the reaction) is not equal to the electrical work which the reaction performs in a galvanic cell." This is eqnivalent to the statement that the term aAF O x of
the Helmholtz equation probably vanishes when
Ac = 0. That this is more probably incorrect has been indicated by the experiments of Streintz, Richards and Lewis, and Weber. I n these cases Ac was found to vanish 'over conA F had a definite value. siderable temperature ranges and ?__
ae
There are certain cases where
aAF
ae
does not vary linearly
~-
with the temperature, i. e., in a temperature-electromotive force diagram the curve is not a straight line. Such cases are given by Ochs' for oxidation and reduction cells. Of the large number of such cells studied by Ochs, the three cited below most probably conform to the necessary condition of reversibility, and only in so far as they are completely reversible do the conclusions drawn from the data hold good. At all points on the curve for HMnO,
+ H,SO,, a A F is positive.
a point of inflexion, and therefore
~
ae
The curve shows
has both positive and
negative values, depending on the temperature. The same is true of the curve for CrO, H,SO,. Hence for these cases there can be no linear relation beaAF a'AF aAF aAE tween - and -, and ---. In the case of a0 ae i. e., between __ a0 a0
+
Trans. Am. Electrochem. spc., 6, 11, 12 (1904). Inaugural Dissertation. Uber Oxydations uiid Basel, 1895.
Reduktionsketten
/. M. Bell
388
Mn,(SO,),
+ H,SO,,
is positive and
is ae
3°F
negative, and
UF aAE Consequently -- and - a0 ae have the same sign, whereas Richards’s equation a-aT A F =-M-, a A E
hence -6’
__ = __
ae
ae
is positive.
ae
aAF
where M is positive, would necessarily make ae and
aAE
ae
--
carry
opposite signs.
Van’t Hoff’s calculation T h e subject of a possible relationship between the change of heat capacity and the temperature coefficient of electromotive force has been taken up quite recently byvan’t Hoff.’ I n the Helmholtz’ equation AF=AE
aAF + 63g-
+
van t Hoff replaces A E by A E , Ac.8 where AE, is the heat of reaction at the absolute zero of temperature, It is assumed in this substitution that Ac is independent of the temperature, an assumption which in general is inadmissible. Assuming, however, that Ac is constant over a temperature range 8, to 8, we obtain, on replacing A E by A E , (6 - e,)&,
+
or, rearranging, AF F -eI -a a0 --=A -8 2> - - A 0E2
he
Ac
e -I-@ 80,
and integrating between temperature limits 8, and 8 AF = (AE, -Ace,)
(a)
-t
which is the same in form as van’t Hoff’s equation (46). Differentiation yields
__
.___-.--
I
Boltzmann Festschrift, p. 233 (1904).
Free Enevgy and Heal Cajaciky
389
or
which is identical in form with van? Hoff’s equation make
(44 if
we
T h e above equation ( a ) as has been pointed out by Trevor’ was deduced by Helmholtz in his famous free-energy paper. T h e physical interpretation given by van’t Hoff to this constant A is that A = 2 log
CB
-
CA
where C, and CB are the end concentrations of the solution in the cell, and the term AT of his equation (46) corresponds in some way to the energy derivable from concentration changes. Haber and Tolloczkoz have put the same interpretation on this term, although no proof is given. T h e constant A is made to vanish on the ground that C, = C,. By the vanishing of A
and the condition for the vanishing of A is
which it is desired to prove. T h e above equation really indicates that if, at any temperature eo of a temperature range oyer which Ac is constant, equation (44 holds, then equation (44 holds at every temperature of that temperature range. It is possible, however, to show that sufch an equation as (44 cannot hold under the conditions which van’t Hoff has imposed. For it has been demonstrated by Trevor‘ that the ex1. c. p. 299. 1. c . p. 438.
J. &I. Bell
390
istence at any definite temperature of a proportionality between
2% ae
and Ac is the necessary and sufficient condition that
nc = a0P where a and p are independent of 0. This is at variance with the original assumption that Ac is independent of the temperature. It may be definitely stated, therefore, that under the conditions postulated, A does 7zot vanish, and consequently any deductions made on the assumption that A = o are incorrect. aAF \JTherefore, the correct form of the relation between __ and Ac a0
is __ = A --&(I
ae
+ log e),
and as A depends on the initial temperature and does not aAF vanish, it follows that -and Ac) cannot stand in a simple 80. ratio to each other. I n conclusion it should be stated explicitly that the most general formulation of the relation between the heat of reaction and the free energy is the Helmholtz equation. I n his book recently published Traube’ says : “ Die Gleichung von Helmholtz bedarf nach diesen fundamentalen Festellungen von Richards eines Korrektionsgliedes, dessen Grosse von dem Unterschiede der spezifischen Warmen abhangt.” This statement is incorrect, for the assumption of Richards of a linear re-
aAE
lation between -ae and
E ae merely imposes one
arbitrary spe-
cific limitation, namely that the change of the heat capacity is a particular function, in this case a power, of the temperature. The assumption of van’t Hoff, that the change of heat capacity is independent of the temperature, merely imposes another arbitrary specific limitation. Each of these formulations is therefore less general than that of Helmholtz. Both assumptions involve attempts to formulate some integral expression for
Eo
~ c d and 0 thus to divide the term 1
Grundriss der physikalische11 Chemie, p.
A E of the Helmholtz 321
(1904).
h
e Energy and Heat Cajacity
39 1
equation into the sum of two terms, AB, the heat of reaction at the temperature
e,,
and the integral expression for
1;A d o . 0
I n this paper it has been shown that I. T h e inferences of Richards and van’t Hoff regarding the sign and magnitude of the change of heat capacity are based on insufficient experimental data. 2. Richards’s conclusion of a proportionality between
aAE
-~
ae
aAF and __ is at variance with the assumption tacitly made in the a0 calculation that a c is independent of the temperature. 3. T h e existence of a general linear relation between
aAE ~
ae
and
aAF ~
a0
does not accord with the experimental data avail-
able.
4. T h e dependence of the constant A upon concentration changes was not proved by van’t Hoff or by Haber and Tolloczko, and in the case of no change of concentration of the cell-solution, the assumption that A vanishes leads to a result which is out of harmony with the assumption that Ac is independent of the temperature. Cormdl University,Jnnuary, ‘905.
