1.3.6 Work Done by an Ideal Gas During an Adiabatic
Expansion
Having established the relationships between the thermodynamic coordinates during
an adiabatic process, let us now determine the work done during an adiabatic
expansion. Writing again the mathematical expression for the first law of thermodynamics as
dq ¼ c V dT þ dW:
ð1:27Þ
Since the process is adiabatic, dq ¼ 0, hence,
dW ¼ Àc V dT
ð1:28Þ
and integrating gives
W ¼ c V T i À T f
À
Á
,
ð1:29Þ
where T i and T f are the initial and final temperatures, respectively. Since the
process in going from T i to T f passes through equilibrium states, we can use the
ideal gas equation and write the latter equation as
W ¼
c V
R
p i υ i À p f υ f
À
Á
ð1:30Þ
and using the fact that R ¼ c P À c V in conjunction with γ ¼ c P /c V Eq. (1.30) becomes
W ¼
p i υ i À p f υ f
γ À 1
:
ð1:31Þ
Alternatively, by using the relationship between p and υ for an adiabatic process,
one can show that the work done by an ideal gas during an adiabatic expansion is
W ¼
p i υ i
γ À 1
1 À
p f
p i
γÀ1
γ
"
#
:
ð1:32Þ
1.3 Some Elements of Thermodynamics
9