Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.4 HEATING ATCONSTANTVOLUME ORPRESSURE 175

CVmay depend on bothV andT, and we should integrate withVheld constant and

CVtreated as a function only ofT.

2.Suppose we want to evaluateÅUfor a process in which the volume is the same in the

initial and final states (V 2 DV 1 ) but is different in some intermediate states, and the

temperature isnotuniform in some of the intermediate states. We know the change

of a state function depends only on the initial and final states, so we can still use Eq.

7.4.2to evaluateÅUfor this process. We integrate withVheld constant, althoughV

was not constant during the actual process.

In general: A finite changeÅXof a state function, evaluated under the condition

that another state functionYis constant, is the same asÅXunder the less stringent

conditionY 2 DY 1. (Another application of this principle was mentioned in Sec.

4.6.2.)

3.For a pure substance, we may convert an expression for an infinitesimal or finite

change of an extensive property to an expression for the change of the corresponding

molarproperty by dividing byn. For instance, Eq.7.4.1becomes

dUmDCV;mdT (7.4.3)

and Eq.7.4.2becomes

ÅUmD

ZT 2

T 1

CV;mdT (7.4.4)

If, at a fixed volume and over the temperature rangeT 1 toT 2 , the value ofCVis essen-
tially constant (i.e., independent ofT), Eq.7.4.2becomes

ÅUDCV.T 2 T 1 / (7.4.5)

(closed system,CD 1 ,

PD 1 , constantVandCV)

An infinitesimal entropy change during a reversible process in a closed system is given
according to the second law by dSD∂q=T. At constant volume,∂qis equal to dUwhich
in turn equalsCVdT. Therefore, the entropy change is

dSD

CV

T

dT (7.4.6)

(closed system,

CD 1 ,PD 1 , constantV)

Integration yields the finite change

ÅSD

ZT 2

T 1

CV

T

dT (7.4.7)

(closed system,

CD 1 ,PD 1 , constantV)

IfCV is treated as constant, Eq.7.4.7becomes

ÅSDCVln

T 2

T 1

(7.4.8)

(closed system,CD 1 ,

PD 1 , constantVandCV)