Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.4 HEATING ATCONSTANTVOLUME ORPRESSURE 174

N 2 (s)

N 2 (l)

N 2 (g)

H 2 O(s)

H 2 O(l)

H 2 O(g)

C(s)

0

20

40

60

80

0 100 200 300 400 500 600

T=K

Cp

;m

=J K

^1

mol

^1

Figure 7.5 Temperature dependence of molar heat capacity at constant pressure (pD

1 bar) of H 2 O, N 2 , and C(graphite).

7.4 Heating at Constant Volume or Pressure

Consider the process of changing the temperature of a phase at constant volume.^6 The rate
of change of internal energy withTunder these conditions is the heat capacity at constant
volume:CV D.@U=@T /V(Eq.7.3.1). Accordingly, an infinitesimal change ofUis given
by

dUDCVdT (7.4.1)

(closed system,

CD 1 ,PD 1 , constantV)

and the finite change ofUbetween temperaturesT 1 andT 2 is

ÅUD

ZT 2

T 1

CVdT (7.4.2)

(closed system,

CD 1 ,PD 1 , constantV)

Three comments, relevant to these and other equations in this chapter, are in order:

1.Equation7.4.2allows us to calculate the finite change of a state function,U, by inte-

gratingCV overT. The equation was derived under the condition thatV is constant

during the process, and the use of the integration variableTimplies that the system

has a single, uniform temperature at each instant during the process. The integrand

(^6) Keeping the volume exactly constant while increasing the temperature is not as simple as it may sound. Most
solids expand when heated, unless we arrange to increase the external pressure at the same time. If we use solid
walls to contain a fluid phase, the container volume will change with temperature. For practical purposes, these
volume changes are usually negligible.