Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.2 INTERNALPRESSURE 165

benzene(l)

ethanol(l)

H 2 O(s)

H 2 O(l)

^0250255075100

5

10

15

t=ıC

T

=10

^5

bar

^1

Figure 7.2 The isothermal compressibility of several substances as a function of tem-

perature atpD 1 bar. (Based on data in Ref. [ 49 ]; Ref. [ 88 ]; and Ref. [ 167 ], p. 28.)

With the substitutions.@V=@T /pDV (from Eq.7.1.1) and.@V=@p/T D TV (from
Eq.7.1.2), the expression for the total differential ofVbecomes

dVDVdTTVdp (7.1.6)

(closed system,

CD 1 ,PD 1 )

To find howpvaries withTin a closed system kept at constant volume, we set dV equal
to zero in Eq.7.1.6: 0 D VdTTVdp, or dp=dT D=T. Since dp=dT under
the condition of constant volume is the partial derivative.@p=@T /V, we have the general
relation

@p
@T



V

D

T

(7.1.7)

(closed system,

CD 1 ,PD 1 )

7.2 Internal Pressure

The partial derivative.@U=@V /T applied to a fluid phase in a closed system is called the
internal pressure. (Note thatUandpV have dimensions of energy; therefore,U=V has
dimensions of pressure.)
To relate the internal pressure to other properties, we divide Eq.5.2.2by dV: dU=dV D
T .dS=dV /p. Then we impose a condition of constantT:.@U=@V /T DT .@S=@V /Tp.
When we make a substitution for.@S=@V /T from the Maxwell relation of Eq.5.4.17, we