ENTROPY AXD HEAT-CAPACITY
BY J, E. TREVOR
I . Adiabatic paths. - The question may be asked : What is the relation between the entropy of a system of bodies and the heat-capacity of the system with respect to any reversible thermodynamic path ? For the particular case of adiabatic paths, the entropy 9 may be regarded as a sort of heat-capacity, in the following way. From the equation
d(8rl) = 0d77
+ rld0
for the differential d(8q) of the store of heat -of ‘ bound energy ’ - of a system, we find
(1)
Helmholtz’s
which equation interprets the entropy as \he increase of the store of heat, per degree) during adiabatic change,-as the heatcapacity for heat produced from adiabatic work. This conception resembles that of Helmholtz,’ that ( ( T h e entropy appears as the heat-capacity for heat produced at the expense of the free energy during adiabatic change.” Helmholtz’s idea is deduced from his formulation of the energy as the sum of the free and bound energies of the system :
EzF-87 dF=d’W--77d8
{ d(8rl)
=
+W ,
from which last equation the term 8 d ~disappears for adiabatic changes. But the matter is unnecessarily complicated by thus -4bh.
2,
Helmholtz. Ueber die Thermodynamik der chemischen Prozessen. Ges. 958 (1882).
530 treating the term 7de as supplied by the free energy ; it is far more rational to regard it as the work that is expended in increasing the store of heat 87 of the system, while the remainder (d’W - 7d9) of the work is the contribution from without to the store of free energy. 2 . The general case. - To formulate generally the relation between the entropy and the heat-capacity of a thermodynamic system, we proceed essentially as before. I n (I) we have
d(e7) = ed7
+ qde,
(3)
=c+rl,
setting C for the heat-capacity d’Q/de. Equation (3) is, of course, subject to the conditions which determine the path. This equation (3) states simply that, on any path, the heat a?Q/de added per degree, and the work 7deld9 stored per degree as heat, together give the total increase per degree of the store of heat.I Otherwise worded : a system’s entropy, plus its heatcapacity with reference’ to any reversible path, is equal to the change per degree, on the same path, of the system’s store of heat 97.
3. Further illustration. - Further to illustrate the general relation
let the line ab in the accompanying figure represent any reversible path between two states, of any thermodynamic system of bodies, having a temperature-difference of one degree. T h e ordinates of the diagram represent the thermal potentials, the absolute temperatures ),-while the abscissas give a general representation of the remaining state-variables. Drawing adiabatic lines through a and b, and an adiabatic
-
I. e., the heat-capacity C is the heat added per degree, and the entropy 7 is the work stored, per degree, as heat.
53'
line through the zero state of reference, we see that the entropy 71a at the state a is the work-value of the horizontally shaded one-degree Carnot cycle. For the changes of the entropy of a thermodynamic system are defined as the total work-values of the one-degree Carnot cycles that correspond to the heat-additions of any reversible path between the end-states of the system. Further, the heat-capacity C of the system is the work-value of the vertically shaded Carnot cycle, regarded as extending to 8 = 0. For
= e(% - 7,), l
and (qb - q a ) is the work-value of a one-degree Carnot cycle between the adiabatics 7 = 76 and 7 = qa. Again, the quantity d0
is the sum of the work-values of the two shaded Carnot cycles. For
and (8 T 1)7~is the work-value of the entire series of one-degree Carnot cycles between the adiabatics 7 = q6and 7 = 0, and from ~~~
1 Very approximately. We assume that d?l,'de= const., for thegiven path and through the temperature interval of one degree.
532
Ext ~ 0 j - vand Heat-cn$acity
8 = o to 9
+
=8 I ; while 8qa is that of the series between 7 = qa and 7 = 0, and from 8 = o to 0 = 8.
Thus the quantity
4017) d0
appears as the sum of the quantities qn and C, which is the relation whose physical significance was to be made clear. Cornell Universiiy, Febmary, 1900.
