Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.5 PARTIALDERIVATIVES WITHRESPECT TOT,p,ANDV 177

Table 7.1 Constant temperature: expressions for partial derivatives of state

functions with respect to pressure and volume in a closed, single-phase system

Partial General Ideal Partial General Ideal

derivative expression gas derivative expression gas



@p

@V



T

1

TV



p

V



@A

@p



T

TpV V



@V

@p



T

TV 

V

p



@A

@V



T

p p



@U

@p



T

.TCTp/V 0



@G

@p



T

V V



@U

@V



T

T

T

p 0



@G

@V



T

1

T

p



@H

@p



T

.1T /V 0



@S

@p



T

V 

V

 T

@H

@V



T

T 1

T

0



@S

@V



T

T

p

T

of these variables is held constant. We have already seen some of these expressions, and the
derivations of the others are indicated below.
We can use these partial derivatives (1) for writing an expression for the total differen-
tial of any of the eight quantities, and (2) for expressing the finite change in one of these
quantities as an integral under conditions of constantT,p, orV. For instance, given the
expressions 
@S
@T



p

D

Cp

T

and



@S

@p



T

DV (7.5.1)

we may write the total differential ofS, takingTandpas the independent variables, as

dSD

Cp

T

dTVdp (7.5.2)

Furthermore, the first expression is equivalent to the differential form

dSD

Cp

T

dT (7.5.3)

providedpis constant; we can integrate this equation to obtain the finite changeÅSunder
isobaric conditions as shown in Eq.7.4.12.
Both general expressions and expressions valid for an ideal gas are given in Tables7.1,
7.2, and7.3.

We may derive the general expressions as follows. We are considering differentia-

tion with respect only toT,p, andV. Expressions for.@V=@T /p,.@V=@p/T, and

.@p=@T /Vcome from Eqs.7.1.1,7.1.2, and7.1.7and are shown as functions of

andT. The reciprocal of each of these three expressions provides the expression for

another partial derivative from the general relation

.@y=@x/zD

1

.@x=@y/z

(7.5.4)