Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.5 PARTIALDERIVATIVES WITHRESPECT TOT,p,ANDV 176

(More general versions of the two preceding equations have already been given in Sec.
4.6.1.)
SinceCV is positive, we see from Eqs.7.4.2and7.4.7that heating a phase at constant
volume causes bothUandSto increase.
We may derive relations for a temperature change at constantpressureby the same
methods. FromCpD.@H=@T /p(Eq.7.3.2), we obtain

ÅHD

ZT 2

T 1

CpdT (7.4.9)

(closed system,

CD 1 ,PD 1 , constantp)

IfCpis treated as constant, Eq.7.4.9becomes

ÅHDCp.T 2 T 1 / (7.4.10)

(closed system,CD 1 ,

PD 1 , constantpandCp)

From dSD∂q=Tand Eq.7.3.2we obtain for the entropy change at constant pressure

dSD

Cp

T

dT (7.4.11)

(closed system,

CD 1 ,PD 1 , constantp)

Integration gives

ÅSD

ZT 2

T 1

Cp

T

dT (7.4.12)

(closed system,

CD 1 ,PD 1 , constantp)

or, withCptreated as constant,

ÅSDCpln

T 2

T 1

(7.4.13)

(closed system,CD 1 ,

PD 1 , constantpandCp)

Cpis positive, so heating a phase at constant pressure causesHandSto increase.
The Gibbs energy changes according to.@G=@T /pD S(Eq.5.4.11), so heating at
constant pressure causesGto decrease.

7.5 Partial Derivatives with Respect toT,p, andV

7.5.1 Tables of partial derivatives

The tables in this section collect useful expressions for partial derivatives of the eight state
functionsT,p,V,U,H,A,G, andSin a closed, single-phase system. Each derivative is
taken with respect to one of the three easily-controlled variablesT,p, orVwhile another