64 PARTI THERMODYNAMICS AND KINETICS
The ratio of volumes can be related to a ratio of temperatures for an ideal gas, since the change
in internal energy, dU, can be related to the work since the heat flow is zero (dq =0) for an
adiabatic process. In addition, the change in internal energy is proportional to the specific
heat and temperature change (eqn 2.23). Combining these two relationships yields:
dU=dw+dq=dw =−PdV (db3.5)
dU=CdT (db3.6)
CdT=−PdV (db3.7)
Using the ideal gas law, the pressure can be substituted, resulting in:
or (db3.8)
To determine the change for the entire process, these terms are integrated to yield:
(db3.9)
To simplify this relationship, the variable cis defined as being equal to C/nRand the expres-
sion can be revised as:
(db3.10)
alnx=ln(xa)
−ln(x/y) =ln(y/x)
ln ln
T
T
V
V
nal
initial
c
nal
initial
⎛ 33
⎝
⎜
⎜
⎞
⎠
⎟
⎟ =
⎛
⎝⎝
⎜
⎜
⎞
⎠
⎟
⎟
C
T
T
nR
V
T V
T
V
V
initial
nal
initial
nal
dd
3 3
∫∫=−
C
T
T
nR
V
V
=−
dd
CT PV
nRT
V
dd=− =− dV
C
T
T
nR
V
V
nal
initial
nal
initi
ln^33 ln
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟=− aal
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
()()(constant
d
constant
d
constant
x
x
x
∫∫x
==))ln( )x
