Ncert Exercises & Solutions

If c is the RMS 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.

Hint: Use the formula of RMS speed and the speed of the sound wave in a gas.

Step 1: Find the RMS speed.

We know that the RMS speed of molecules of a gas,

c = \sqrt{\frac{3 P}{ρ}} = \sqrt{\frac{3 RT}{M}} . . . (i)

where M= molar mass of the gas.

Step 2: Find the speed of the sound wave in a gas.

Speed of sound wave in gas, v =

\sqrt{\frac{γP}{ρ}} = \sqrt{\frac{γRT}{M}} . . . (i i)

On dividing Eq. (i) by Eq. (ii), we get,

\frac{c}{v} = \sqrt{\frac{3 RT}{M} \times \frac{M}{γRT}} \Rightarrow \frac{c}{v} = \sqrt{\frac{3}{γ}}

where

γ

= adiabatic constant for a diatomic gas

γ = \frac{7}{5} [since γ = \frac{C_{p}}{C_{v}}]

Hence,

\frac{c}{v} = constant

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