If c is r.m.s speed of molecules in a gas

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,

$\frac{c}{v}=\sqrt{\frac{3}{\frac{7}{5}}}=\sqrt{\frac{15}{7}}=\mathrm{constant}$

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