Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.2 TOTALDIFFERENTIAL OF THEINTERNALENERGY 136

As explained in AppendixF, we may identify the coefficient of each term in an expres-
sion for the total differential of a state function as a partial derivative of the function. We
identify the coefficients on the right side of Eq.5.2.2as follows:

TD



@U

@S



V

pD



@U

@V



S

(5.2.3)

Now let us consider some of the ways a system might have more than two independent
variables. Suppose the system has one phase and one substance, with expansion work
only, and isopenso that the amountnof the substance can vary. Such a system has three
independent variables. Let us write the formal expression for the total differential ofUwith
S,V, andnas the three independent variables:

dUD



@U

@S



V;n

dSC



@U

@V



S;n

dVC



@U

@n



S;V

dn (5.2.4)

(pure substance,

PD 1 ,∂w^0 D 0 )

We have seen above that if the system isclosed, the partial derivatives are.@U=@S/V DT
and.@U=@V /S D p. Since both of these partial derivatives are for a closed system in
whichnis constant, they are the same as the first two partial derivatives on the right side of
Eq.5.2.4.
The quantity given by the third partial derivative,.@U=@n/S;V, is represented by the
symbol(mu). This quantity is an intensive state function called thechemical potential.
With these substitutions, Eq.5.2.4becomes

dUDTdSpdV Cdn (5.2.5)

(pure substance,

PD 1 ,∂w^0 D 0 )

and this is a valid expression for the total differential ofUunder the given conditions.
If a system contains a mixture ofsdifferent substances in a single phase, and the sys-
tem is open so that the amount of each substance can vary independently, there are 2 Cs
independent variables and the total differential ofUcan be written

dUDTdSpdVC

Xs

iD 1

idni (5.2.6)

(open system,

PD 1 ,∂w^0 D 0 )

The coefficientiis the chemical potential of substancei. We identify it as the partial
derivative.@U=@ni/S;V;nj§i.

The termpdVon the right side of Eq.5.2.6is the reversible work. However, the

termTdSdoes not equal the reversible heat as it would if the system were closed. This

is because the entropy change dSis partly due to the entropy of the matter transferred

across the boundary. It follows that the remaining term,

P

iidni(sometimes called

the “chemical work”), should not be interpreted as the energy brought into the system

by the transfer of matter.^2

(^2) Ref. [ 93 ].