Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.5 PARTIALDERIVATIVES WITHRESPECT TOT,p,ANDV 178

Table 7.2 Constant pressure: expressions for partial derivatives of state func-

tions with respect to temperature and volume in a closed, single-phase system

Partial General Ideal Partial General Ideal

derivative expression gas derivative expression gas



@T

@V



p

1

V

T

V



@A

@T



p

pVS 

pV

T

S



@V

@T



p

V

V

T



@A

@V



p

p

S

V

p

TS

 V

@U

@T



p

CppV CV



@G

@T



p

S S



@U

@V



p

Cp

V

p

CVT

V



@G

@V



p

S

V



TS

 V

@H

@T



p

Cp Cp



@S

@T



p

Cp

T

Cp

 T

@H

@V



p

Cp

V

CpT

V



@S

@V



p

Cp

T V

Cp

V

Table 7.3 Constant volume: expressions for partial derivatives of state functions with

respect to temperature and pressure in a closed, single-phase system

Partial General Ideal Partial General Ideal

derivative expression gas derivative expression gas



@T

@p



V

T

T

p



@A

@T



V

S S



@p

@T



V

T

p

T



@A

@p



V

TS



TS

 p

@U

@T



V

CV CV



@G

@T



V

V

T

S

pV

T

S



@U

@p



V

TCp

T V

T CV

p



@G

@p



V

V

TS

V

TS

 p

@H

@T



V

CpC

V

T

.1T / Cp



@S

@T



V

CV

T

CV

 T

@H

@p



V

TCp

CV.1T /

CpT

p



@S

@p



V

TCp

T

V

CV

p

This procedure gives us expressions for the six partial derivatives ofT,p, andV.

The remaining expressions are for partial derivatives ofU,H,A,G, andS. We

obtain the expression for.@U=@T /Vfrom Eq.7.3.1, for.@U=@V /Tfrom Eq.7.2.4,

for.@H=@T /pfrom Eq.7.3.2, for.@A=@T /Vfrom Eq.5.4.9, for.@A=@V /Tfrom Eq.

5.4.10, for.@G=@p/Tfrom Eq.5.4.12, for.@G=@T /pfrom Eq.5.4.11, for.@S=@T /V

from Eq.7.4.6, for.@S=@T /pfrom Eq.7.4.11, and for.@S=@p/Tfrom Eq.5.4.18.

We can transform each of these partial derivatives, and others derived in later

steps, to two other partial derivatives with the same variable held constant and the

variable of differentiation changed. The transformation involves multiplying by an