Physical Chemistry , 1st ed.

variables. (Other derivatives have also been expressed as partial derivatives.)
Since dHdqp, we can substitute on the left side of the equation to get



H

T



p

CV+ 

U

V



T

• p

V

T



p

The term ( H/ T)phas already been defined as the heat capacity at constant
pressure,Cp. We now have a relationship between CVand Cp:

CpCV+ 

U

V



T

• p

V

T



p

(2.37)

If the system is composed of an ideal gas, this is straightforward to evaluate.
The change in internal energy at constant temperature is exactly zero (that’s
one of the defining features of an ideal gas). We can also use the ideal gas law
to determine the derivative ( V/ T)p:



V

T



p



n

p

R



Substituting into equation 2.37:

CpCV+ (0 + p)

n

p

R



CpCV+ nR

or, for molar quantities:

CpCV+ R (2.38)

for an ideal gas. This is an extremely simple and useful result.
The kinetic theory of gases (to be considered in a future chapter) leads to
the result that, for a monatomic ideal gas,

CV^3

2

R12.471

mo

J

lK

 (2.39)

Therefore, by equation 2.38,

Cp

5

2

R20.785

mo

J

lK

 (2.40)

Gases like Ar and Ne and He have constant-pressure heat capacities around
20.8 J/molK, which is not surprising. The lighter inert gases are good approx-
imations of ideal gases.*
Ideal gases have a temperature-invariant heat capacity; real gases do not.
Most attempts to express the heat capacity of real gases use a power series, in
either of the two following forms:

Cpa+ bT+ cT^2

Cpa+ bT+ 

T

c

 2

where a,b, and care experimentally determined constants. Example 2.10, along
with equation 2.31, illustrates the proper way to determine changes in state
functions using heat capacities of this form.

2.8 More on Heat Capacities 47

*Kinetic theory of gases also predicts that for ideal diatomic or linear molecules,CV
^52 R; for ideal nonlinear molecules,CV 2 ^7 R.Cpis thus ^72 Rand ^92 R, respectively. (We include
this to illustrate that thermodynamics isn’t just applicable to monatomic gases!)