Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.3 THERMALPROPERTIES 167

7.3 Thermal Properties

For convenience in derivations to follow, expressions from Chap. 5 are repeated here that
apply to processes in a closed system in the absence of nonexpansion work (i.e.,∂w^0 D 0 ).
For a process atconstant volumewe have^3

dUD∂q CVD



@U

@T



V

(7.3.1)

and for a process atconstant pressurewe have^4

dHD∂q CpD



@H

@T



p

(7.3.2)

A closed system of one component in a single phase has only two independent variables.
In such a system, the partial derivatives above are complete and unambiguous definitions of
CV andCpbecause they are expressed with two independent variables—T andV forCV,
andT andpforCp. As mentioned on page 142 , additional conditions would have to be
specified to defineCV for a more complicated system; the same is true forCp.
For a closed system of anideal gaswe have^5

CVD

dU

dT

CpD

dH

dT

(7.3.3)

7.3.1 The relation betweenCV;mandCp;m.

The value ofCp;mfor a substance is greater thanCV;m. The derivation is simple in the case
of a fixed amount of anideal gas. Using substitutions from Eq.7.3.3, we write

CpCV D

dH

dT

dU

dT

D

d.HU /

dT

D

d.pV /

dT

DnR (7.3.4)

Division bynto obtain molar quantities and rearrangement then gives

Cp;mDCV;mCR (7.3.5)

(ideal gas, pure substance)

For any phase in general, we proceed as follows. First we write

CpD



@H

@T



p

D



@.UCpV /

@T



p

D



@U

@T



p

Cp



@V

@T



p

(7.3.6)

Then we write the total differential ofUwithTandVas independent variables and identify
one of the coefficients asCV:

dUD



@U

@T



V

dTC



@U

@V



T

dV DCVdTC



@U

@V



T

dV (7.3.7)

(^3) Eqs.5.3.9and5.6.1. (^4) Eqs.5.3.7and5.6.3. (^5) Eqs.5.6.2and5.6.4.