Question

the ratio of the speed of sound in nitrogen gas to that in the helium gas at 300 k is?

Answer 1

speed of sound in air is given by

$v= \sqrt{ \frac{\gamma RT}{M} } \\ \\ \gamma_H_e= \frac{5}{3};\ \gamma_N_2= \frac{7}{5} \\ M_{He} =4;\ M_{N2}=28\\ \\ \frac{v_{N2}}{v_{He}} = \sqrt{ \frac{\frac{\gamma_N_2 RT}{M_N_2} }{ \frac{\gamma_{He} RT}{M_{He}}}} = \sqrt{ \frac{\gamma_N_2}{\gamma_H_e} \times \frac{M_{He}}{M_{N2}} } = \sqrt{ \frac{7 \times 3}{5 \times 5} \times \frac{4}{28} } = \frac{ \sqrt{3} }{5}$

Answer 2

The ratio of the speed of sound in nitrogen gas to that in the helium gas at 300 K is $\frac { \sqrt { 3 } }{ 5 }$

Solution:

Speed of the sound in air is given by the equation,

$v\quad =\quad \sqrt { \frac { \gamma RT }{ M } }$

Where,

$\gamma$ - Adiabatic constant

R - Ideal gas constant

T - Absolute Temperature

M - Molecular mass of gas

Since, Nitrogen gas molecules are diatomic and that of Helium are monoatomic.

So,

$\gamma _{ H_{ e } }\quad =\quad \frac { 5 }{ 3 }$

$\gamma _{ N_{ 2 } }\quad =\quad \frac { 7 }{ 5 }$

$M_{ He }\quad =\quad 4$

$M_{ { N }_{ 2 } }\quad =\quad 28$

$\frac { v_{ { N }_{ 2 } } }{ v_{ He } } \quad =\quad \sqrt { \frac { \frac { \gamma _{ N_{ 2 } }RT }{ M_{ { N }_{ 2 } } }  }{ \frac { \gamma _{ He }RT }{ M_{ He } } } }$

$=\quad \sqrt { \frac { \gamma _{ N_{ 2 } } }{ \gamma _{ He } } \quad \times \quad \frac { M_{ He } }{ M_{ N_{ 2 } } } }$

$=\quad \sqrt { \frac { 7\quad \times \quad 3 }{ 5\quad \times \quad 5 } \quad \times \quad \frac { 4 }{ 28 } }$

$\frac { v_{ { N }_{ 2 } } }{ v_{ He } } \quad =\quad \frac { \sqrt { 3 } }{ 5 }$