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 al diatomic gases.

Answer 1

R.m.s speed of molecules of a gas

c = \(\sqrt{\frac{3P}{ρ}}\)

c = \(\sqrt{\frac{3RT}{M}}\) (I) [M = Molar mass]

∵ PV = nRT

n = 1

Or P = \(\frac{RT}{V}\) 

∴ \(\frac{p}{ρ}=\frac{RT}{M}\)  [∴\(\frac{P}{δ}=\frac{\frac{RT}{v}}{\frac{M}{v}}=\frac{RT}{M}\) ] 

Speed of sound wave in gas, v = \(\sqrt{\frac{rP}{ρ}}\) 

v = \(\sqrt{\frac{rRT}{M}}\) (II)

Dividing eqn (II) by eq.n (I),

 \(\frac{c}{v}=\frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{rRT}{M}}}\)

 \(\frac{c}{v}=\sqrt{\frac{3}{r}}\) [r = adiabatic constant for diatomic gas]

r = \(\frac{7}{5}\) 

Thus, \(\frac{c}{v}\) = constant