Page 80 - Physical Chemistry

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Reversible Adiabatic Process in a Perfect Gas Section 2.8
For an adiabatic process, dq 0. For a reversible process in a system with only P-V Perfect Gases and the First Law
work, dw PdV. For a perfect gas, dU C dT [Eq. (2.68)]. Therefore, for a re-
V
versible adiabatic process in a perfect gas, the first law dU dq dw becomes
C dT P dV 1nRT>V2 dV
V
C V,m dT 1RT>V2 dV
where PV nRT and C C /n were used. To integrate this equation, we separate
V,m V
the variables, putting all functions of T on one side and all functions of V on the other
side. We get (C /T)dT (R/V)dV. Integration gives
V,m
2 C V,m dT 2 R dV R1ln V ln V 2 R ln V 1 (2.75)
1
2
1 T 1 V V 2
For a perfect gas, C V,m is a function of T [Eq. (2.69)]. If the temperature change in the
process is small, C V,m will not change greatly and can be taken as approximately constant.
Another case where C V,m is nearly constant is for monatomic gases, where C V,m is essen-
tially independent of T over a very wide temperature range (Sec. 2.11 and Fig. 2.15).
2
2
The approximation that C V,m is constant gives (C V,m /T) dT C V,m T 1 dT
1
1
C V,m ln (T /T ), and Eq. (2.75) becomes C V,m ln (T /T ) R ln (V /V ) or
2
2
1
1
2
1
ln 1T >T 2 ln 1V >V 2 R>C V,m
1
2
2
1
k
where k ln x ln x [Eq. (1.70)] was used. If ln a ln b, then a b. Therefore
T 2 V 1 R>C V,m
a b perf. gas, rev. adiabatic proc., C const. (2.76)
V
T 1 V 2
Since C is always positive [Eq. (2.56)], Eq. (2.76) says that, when V V , we
2
V
1
will have T T . A perfect gas is cooled by a reversible adiabatic expansion. In ex-
2
1
panding adiabatically, the gas does work on its surroundings, and since q is zero, U
must decrease; therefore T decreases. A near-reversible, near-adiabatic expansion is
one method used in refrigeration.
An alternative equation is obtained by using P V /T P V /T . Equation (2.76)
1
2 2
2
1 1
becomes
P V >P V 1V >V 2 R>C V,m and P V 1 R>C V,m P V 1 R>C V,m
2
2
1
1
2
1 1
1
2 2
The exponent is 1 R/C V,m (C V,m R)/C V,m C P,m /C V,m , since C P,m C V,m R
for a perfect gas [Eq. (2.72)]. Defining the heat-capacity ratio g (gamma) as
g C >C V
P
we have
g g
P V P V perf. gas, rev. ad. proc., C const. (2.77)
V
1 1
2
2
For an adiabatic process, U q w w. For a perfect gas, dU C dT. With
V
the approximation of constant C , we have
V
¢U C 1T T 2 w perf. gas, ad. proc., C const. (2.78)
V
2
1
V
To carry out a reversible adiabatic process in a gas, the surrounding constant-
temperature bath in Fig. 2.11 is replaced by adiabatic walls, and the external pressure
is slowly changed.
We might compare a reversible isothermal expansion of a perfect gas with a
reversible adiabatic expansion of the gas. Let the gas start from the same initial P and Figure 2.12
1
V and go to the same V . For the isothermal process, T T . For the adiabatic Ideal-gas reversible isothermal and
2
2
1
1
expansion, we showed that T T . Hence the final pressure P for the adiabatic adiabatic expansions that start
2
1
2
expansion must be less than P for the isothermal expansion (Fig. 2.12). from the same state.
2