Question:
If c is r.m.s speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases.
Solution:
We know following is the equation for molecules:
$c=\sqrt{\frac{3 P}{\rho}}$
$c=\sqrt{\frac{3 R T}{M}}$
p/ρ = PT/M
Where, M is the molar mass of the gas
$v=\sqrt{\frac{\gamma P}{\rho}}=\sqrt{\frac{\gamma R T}{M}}$
c/v is given as:
$\frac{c}{v}=\frac{\sqrt{\frac{3 R T}{M}}}{\sqrt{\frac{\gamma R T}{M}}}=\sqrt{\frac{3}{\gamma}}$
Therefore,
$\frac{c}{v}=\sqrt{\frac{3}{\frac{7}{5}}}=\sqrt{\frac{15}{7}}=\mathrm{constant}$