CHAPTER 3 THE FIRST LAW
3.5 APPLICATIONS OFEXPANSIONWORK 76
In a reversible adiabatic expansion with expansion work only, the heat is zero and the
first law becomes
dUD∂wD pdV (3.5.4)
We equate these two expressions for dUto obtain
CVdTD pdV (3.5.5)
and substitutepDnRT=Vfrom the ideal gas equation:
CVdTD
nRT
V
dV (3.5.6)
It is convenient to make the approximation that over a small temperature range,CV is
constant. When we divide both sides of the preceding equation byT in order to separate
the variablesTandV, and then integrate between the initial and final states, we obtain
CV
ZT 2
T 1
dT
T
D nR
ZV 2
V 1
dV
V
(3.5.7)
CVln
T 2
T 1
D nRln
V 2
V 1
(3.5.8)
We can rearrange this result into the form
ln
T 2
T 1
D
nR
CV
ln
V 2
V 1
Dln
V 1
V 2
nR=CV
(3.5.9)
and take the exponential of both sides:
T 2
T 1
D
V 1
V 2
nR=CV
(3.5.10)
The finaltemperatureis then given as a function of the initial and final volumes by
T 2 DT 1
V 1
V 2
nR=CV
(3.5.11)
(reversible adiabatic
expansion, ideal gas)
This relation shows that the temperature decreases during an adiabatic expansion and in-
creases during an adiabatic compression, as expected from expansion work on the internal
energy.
To find theworkduring the adiabatic volume change, we can use the relation
wDÅUD
Z
dUDCV
ZT 2
T 1
dT
DCV.T 2 T 1 / (3.5.12)
(reversible adiabatic
expansion, ideal gas)