Thermodynamics and Chemistry

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∂wDpdV (3.5.4)

We equate these two expressions for dUto obtain

CVdTDpdV (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

DnR

ZV 2

V 1

dV

V

(3.5.7)

CVln

T 2

T 1

DnRln

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)