Page 85 - Modern physical chemistry

74 The First Law for Energy

Here u is the speed of the incoming fluid with respect to the wave. In the laboratory it
would equal the speed of the wave with respect to the fluid at rest. Thus, it can be iden-
tified as the speed of sound in the given fluid.
When the fluid can be approximated as an ideal gas, the adiabatic equation of state
PVY =const [4.63]
applies. Differentiating this leads to the condition

[4.64]

which transforms formula (4.62) to

J = (ypvt = y:
u=V [ y ~ 1I2 2 (J1I2 [4.65]

Here u is the speed of sound, M the molecular weight, T the absolute temperature, R the
gas constant, and r the ratio CJCv.

4.10 Contributions to Energy Capacity Cv

Each degree of freedom of a molecule in a system absorbs energy as the temperature
is raised. Thus each of these contributes to the parameter Cv .
By formula (4.35), the heat capacity at constant volume is given by the correspond-
ing partial derivative of internal energy E:

C =[~~l· [4.66]
v

But the translational energy in an ideal gas is given by formula (3.23). Differentiating
this yields
3
C Vtr =-nR. [4.67]
, 2
Each translational degree of freedom contributes one third of this.
For a diatomic rotator in an ideal gas, we have equation (3.45). Differentiating this yields

CV,rot =nR. [4.68]
For a nonlinear rotator in an ideal gas, equation (3.46) applies. Differentiating it gives us

3
CVrot =-nR. [4.69]
, 2
Each rotational degree of freedom contributes 1/2nR to Cv.
For a classically excited vibrational degree of freedom, formula (3.48) applies. On dif-
ferentiation, we obtain
CV,vib =nR. [4.70]

Some experimentally determined heat capacities appear in table 4.1.