64
PART I
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 c is defined as being equal to C/nR and the expression can be revised as:
(db3.10)
a ln x = ln(x
a )
−ln(x/y) = ln(y/x)
ln
ln
T
T
V
V
nal
initial
c
nal
initial
3
3
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=
⎛
⎝ ⎝
⎜
⎜
⎞
⎠
⎟
⎟
C
T
T
nR
V
V
T
T
V
V
initial
nal
initial
nal
d
d
3
3
∫
∫
=
−
C
T
T
nR
V
V
= −
d
d
C T
P V
nRT
V
V
d
d
d
= −
= −
C
T
T
nR
V
V
nal
initial
nal
initi
ln
ln
3
3
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
= −
a al
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
(
)
(
)
(
constant
d
constant
d
constant
x
x
x
x
∫
∫
=
=
) )ln( )
x
9781405124362_4_003.qxd 4/30/08 19:05 Page 64