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  • Two monoatomic ideal gases 1 and 2 of molecular masses,<img alt="\mathrm{M_1 \text { and } M_2}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BM_1%20%5Ctext%20%7B%20and%20%7D%20M

Two monoatomic ideal gases 1 and 2 of molecular masses,\mathrm{M_1 \text { and } M_2} respectively are enclosed in
separate containers kept at the same temperature. The ratio of the speed of sound in gas 1 to that in gas 2
is

Option: 1

\mathrm{\sqrt{\frac{M_1}{M_2}}}


Option: 2

\mathrm{\sqrt{\frac{M_2}{M_1}}}


Option: 3

\mathrm{\frac{M_1}{M_2}}


Option: 4

\mathrm{\frac{M_2}{M_1}}


Answers (1)

best_answer

The speed of sound in gas of bulk modulus B and density ρ is given by

\mathrm{v=\sqrt{\frac{B}{\rho}}}

\mathrm{\text { Bulk modulus } B \text { is given by } B=-\frac{V \Delta P}{\Delta V}}

Now, for a perfect gas, PV = nRT. Differentiating at constant T, we get

\mathrm{\begin{aligned} & P \Delta V+V \Delta P=0 \text { or } \frac{V \Delta P}{\Delta V}=-P \\ & \text { Hence } v=\sqrt{\frac{P}{\rho}} \end{aligned}}

If m is the mass of the gas and M its molecular mass, then

\mathrm{\begin{aligned} & P V=\frac{m}{M} R T \text { or } P M=\frac{m R T}{V}=\rho R T \\ & \text { Or } \frac{P}{\rho}=\frac{R T}{M} \text { or } v^2=\frac{R T}{M} \\ & \text { Or } v=\sqrt{\frac{R T}{M}} \end{aligned}}

\mathrm{\text { Hence } v_1=\sqrt{\frac{R T}{M_1}} \text { and } v_2=\sqrt{\frac{R T}{M_2}} \text { which }}

\mathrm{\text { Give } \frac{v_1}{v_2}=\sqrt{\frac{M_2}{M_1}}}

Posted by

shivangi.bhatnagar

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