Page 82 - Modern physical chemistry

4.8 Reversible Adiabatic Change in an Ideal Gas 71

whether P is fixed or not.

Example 4.4

Determine how Cp is related to Cv when the system is an ideal gas.
The equation of state for n moles of ideal gas is

PV=nRT.
By definition, the enthalpy is given by
H=E+PV.
Combine these two equations,
H=E+nRT,
differentiate
dH=dE+nRdT,
and introduce relations (4.43) and (4.38):

Cp dT=Cv dT+nR dT =(Cv +nR) dT.
Thus, we find that
Cp =Cv +nR
for the ideal gas.

4.8 Reversible Adiabatic Change in an Ideal Gas
We now have the relationships needed to determine how an ideal gas behaves in a
process involving negligible heat transfer.
The heat q may be kept small during a process by surrounding the system with enough
insulation. Alternatively, the immediate surroundings may be heated or cooled so its tem-
perature follows closely that of the system. Even without such precautions, there would
be negligible heat flow if the process were rapid enough.
Consider a given system subject to an adiabatic change. In each infinitesimal step,
we thus have
dq=O. [4.44]
Let us also consider the process to be reversible with all work done work of compres-
sion. Then
dw=-PdV [4.45]
and
dE=dq+dw=-PdV. [4.46]
Furthermore, consider the system to be an ideal gas. The internal energy then depends
only on the temperature following equation (4.38). Combining this with equation (4.46),

Cv dT=-PdV, [4.47]
and with the ideal gas equation yields

nRT
Cv dT = --- dV. [4.48]
V
Rearrange this equation,