JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Waves & Sound

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is

2

times the speed of sound, then the value of

n

will be

Answer: (B)

Solution

The speed of sound is

v_{s} = \frac{γ R T}{M}

and the rms speed is

v_{rms} = \frac{3 R T}{M}

.

Given

v_{rms} = 2 v_{s}

, we have:

\frac{v _{rms}}{v _{s}} = 2 = \frac{3}{γ}

Squaring:

2 = \frac{3}{γ}

, so

γ = \frac{3}{2}

.

For a mixture,

γ = 1 + \frac{2}{f _{mix}}

, where

f_{mix} = \frac{2 \times 3 + n \times 5}{2 + n} = \frac{6 + 5 n}{2 + n}

.

Substituting:

\frac{3}{2} = 1 + \frac{2 ( 2 + n )}{6 + 5 n} = \frac{6 + 5 n + 4 + 2 n}{6 + 5 n} = \frac{10 + 7 n}{6 + 5 n}

Cross-multiplying:

3 (6 + 5 n) = 2 (10 + 7 n)

, giving

18 + 15 n = 20 + 14 n

, so

n = 2

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Single correct option