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Q.

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 $\sqrt{2}$ times the speed of sound, then the value of n will be

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a

2

b

1

c

3

d

4

answer is B.

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Detailed Solution

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$\begin{array}{l} v_{s} = \sqrt{\frac{γRT}{M}} \\ v_{rms} = \sqrt{\frac{3 RT}{M}} \\ \frac{v_{s}}{v_{rms}} = \sqrt{\frac{γ}{3}} = \frac{1}{\sqrt{2}} \Rightarrow \frac{γ}{3} = \frac{1}{2} \Rightarrow γ = \frac{3}{2} \\ γ = 1 + \frac{2}{f_{mix.}} \\ f_{mix.} = \frac{2 \times 3 + n \times 5}{n + 2} = \frac{6 + n \times 5}{(n + 2)} \\ γ = 1 + \frac{2 (n + 2)}{6 + n \times 5} = \frac{6 + 5 n + 2 n + 4}{6 + 5 n} \\ γ = \frac{7 n + 10}{6 + 5 n} = \frac{3}{2} \\ 14 n + 20 = 18 + 15 n \\ n = 2 \end{array}$

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