Question

The parameter g varies with temperature for molecular gases because of the varying contributions from molecular vibrations. Determine g in the low-temperature and the high- temperature limits for (a) CO2 (g) and (b) H2O (g).

Answer 1

(a) CO2 is a linear molecule
So it behaves like a diatomic molecule and has 2 rotational degree of freedom alongwith
three translational degree of freedom.
So total degree of freedom =3+2 =5 at low temperature
Each degree of freedom has R*T/2 energy associated with it.

At low temperature
So molar heat capacity at constant volume, Cv= 5*R/2
Cp=Cv+R =7*R/2
gamma = Cp/CV=7/5

At high temperature, vibrational degree of freedom exists.
Each vibrational degree of freedom has R*T energy associated with it from both the kinetic and potential part.
vibrational degree of freedom for linear molecule 3*N-5=3*3-5=4
So molar heat capacity at constant volume, Cv= 5*R/2+ 4*R = 13*R/2
Cp=Cv+R =15*R/2
gamma = Cp/CV= 13/15

(a) H2O is a nonlinear molecule
So it has 3 rotational degree of freedom alongwith 3 translational degree of freedom.
So total degree of freedom =3+3 =6 at low temperature
Each degree of freedom has R*T/2 energy associated with it.

At low temperature
So molar heat capacity at constant volume, Cv= 6*R/2=3*R
Cp=Cv+R =4*R
gamma = Cp/CV=4/3

At high temperature, vibrational degree of freedom exists.
Each vibrational degree of freedom has R*T energy associated with it from both the kinetic and potential part.
vibrational degree of freedom for nonlinear molecule 3*N-6=3*3-6=3
So molar heat capacity at constant volume, Cv= 3*R+ 3*R = 6*R
Cp=Cv+R = 7*R
gamma = Cp/CV= 7/6