CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 73
Example
A heat ofQ= 100 J was supplied to a system for isochoric heating of a substance fromT 1 =
300 K toT 2 = 305 K. Estimate the value ofCV. Do the given data also allow us to estimateCp
if we know that the substance is an ideal gas?
Solution
We can approximate the derivative of the function by the ratio of differences. From the definition
(3.17) we thus obtain
CV =
(
∂U
∂T
)
V
≈
∆U
∆T
.
It follows from the specification and from equation (3.5) that the supplied heat equals the change
in internal energy∆U. Hence
CV ≈
100
305 − 300
= 20 J K−^1.
For an ideal gas
H≡U+pV =U+nRT −→ ∆H= ∆U+nR∆T.
From the definition ofCpwe have
Cp=
(
∂H
∂T
)
p
≈
∆U+nR∆T
∆T
=
100 +n× 8. 314 ×(305−300)
305 − 300
= 20 + 8. 314 ×n.
In this case it does not matter thatCpis a derivative with respect to temperature at constant
pressure, or that the studied process is not isobaric (in an ideal gas, the heat capacities depend
on temperature only [see3.5.1]). However, we cannot calculate the value ofCpbecause we do
not know the amount of substancen.
