Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.6 EXPRESSIONS FORHEATCAPACITY 142

transform of the function. What this means for the thermodynamic potentials is that an
expression for any one of them, as a function of its natural variables, can be converted to
an expression for each of the other thermodynamic potentials as a function of its natural
variables.
Willard Gibbs, after whom the Gibbs energy is named, called Eqs.5.5.6–5.5.9thefun-
damental equationsof thermodynamics, because from any single one of them not only the
other thermodynamic potentials but also all thermal, mechanical, and chemical properties
of the system can be deduced.^3 Problem 5. 4 illustrates this useful application of the total
differential of a thermodynamic potential.
In Eqs.5.5.6–5.5.9, the coefficientiis the chemical potential of speciesi. The equa-
tions show thatican be equated to four different partial derivatives, similar to the equali-
ties shown in Eq.5.5.5for a pure substance:

iD



@U

@ni



S;V;nj§i

D



@H

@ni



S;p;nj§i

D



@A

@ni



T;V;nj§i

D



@G

@ni



T;p;nj§i

(5.5.10)

The partial derivative.@G=@ni/T;P;nj§iis called thepartial molar Gibbs energyof species
i, another name for the chemical potential as will be discussed in Sec.9.2.6.

5.6 Expressions for Heat Capacity

As explained in Sec.3.1.5, the heat capacity of a closed system is defined as the ratio of
an infinitesimal quantity of heat transferred across the boundary under specified conditions

and the resulting infinitesimal temperature change: heat capacity defD ∂q=dT. The heat
capacities of isochoric (constant volume) and isobaric (constant pressure) processes are of
particular interest.
Theheat capacity at constant volume,CV, is the ratio∂q=dTfor a process in a closed
constant-volume system with no nonexpansion work—that is, no work at all. The first law
shows that under these conditions the internal energy change equals the heat: dU D∂q
(Eq.5.3.9). We can replace∂qby dUand writeCV as a partial derivative:

CVD



@U

@T



V

(5.6.1)

(closed system)

If the closed system has more than two independent variables, additional conditions

are needed to defineCVunambiguously. For instance, if the system is a gas mixture

in which reaction can occur, we might specify that the system remains in reaction

equilibrium asTchanges at constantV.

Equation5.6.1does not require the condition∂w^0 D 0 , because all quantities ap-

pearing in the equation arestatefunctions whose relations to one another are fixed by

the nature of the system and not by the path. Thus, if heat transfer into the system

at constantVcausesUto increase at a certain rate with respect toT, and this rate is

defined asCV, the performance of electrical work on the system at constantV will

cause the same rate of increase ofUwith respect toTand can equally well be used to

evaluateCV.

(^3) Ref. [ 64 ], p. 86.