The ratio of the speed of sound in helium gas | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The speed of sound in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma RT}{M}}$.Since the temperature $T$ is the same for both gases,t

Vedclass · Accepted Answer

$\sqrt{5/3}$

Vedclass · Answer

(A) The speed of sound in an ideal gas is given by the formula $v = \sqrt{\frac{\gamma RT}{M}}$.Since the temperature $T$ is the same for both gases,the ratio of the speed of sound in helium $(v_{He})$ to that in nitrogen $(v_{N_2})$ is given by:$\frac{v_{He}}{v_{N_2}} = \sqrt{\frac{\gamma_{He}}{M_{He}} \cdot \frac{M_{N_2}}{\gamma_{N_2}}}$Substituting the given values $\gamma_{He} = 5/3$,$M_{He} = 4$,$\gamma_{N_2} = 7/5$,and $M_{N_2} = 28$:$\frac{v_{He}}{v_{N_2}} = \sqrt{\left(\frac{5/3}{4}ight) \cdot \left(\frac{28}{7/5}ight)}$$\frac{v_{He}}{v_{N_2}} = \sqrt{\left(\frac{5}{12}ight) \cdot \left(\frac{28 \cdot 5}{7}ight)}$$\frac{v_{He}}{v_{N_2}} = \sqrt{\left(\frac{5}{12}ight) \cdot (4 \cdot 5)} = \sqrt{\frac{5 \cdot 20}{12}} = \sqrt{\frac{100}{12}} = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}}$Wait,re-evaluating the calculation: $\frac{v_{He}}{v_{N_2}} = \sqrt{\frac{5/3}{4} 	imes \frac{28}{7/5}} = \sqrt{\frac{5}{12} 	imes \frac{140}{7}} = \sqrt{\frac{5}{12} 	imes 20} = \sqrt{\frac{100}{12}} = \sqrt{\frac{25}{3}} = \frac{5}{\sqrt{3}}$.Given the options provided in the prompt,there appears to be a discrepancy. Based on the standard calculation,the result is $5/\sqrt{3}$. If we assume the question intended to test the ratio $\sqrt{\frac{\gamma_{He} M_{N_2}}{\gamma_{N_2} M_{He}}}$,the result is $\sqrt{\frac{5/3 	imes 28}{7/5 	imes 4}} = \sqrt{\frac{140/3}{28/5}} = \sqrt{\frac{140}{3} 	imes \frac{5}{28}} = \sqrt{5 	imes \frac{5}{3}} = \sqrt{25/3} = 5/\sqrt{3}$.

The ratio of the speed of sound in helium gas to that in nitrogen gas at the same temperature is $\left(\gamma_{He}=\frac{5}{3}, \gamma_{N_2}=\frac{7}{5}, M_{He}=4, M_{N_2}=28\right)$

Similar Questions

If the pressure, temperature, and density of an ideal gas are denoted by $p, T$ and $\rho$, respectively, the velocity of sound in the gas is

Ultrasonic waves are those waves

The speed of sound in mercury at a certain temperature is $1450 \, m/s$. Given the density of mercury as $13.6 \times 10^3 \, kg/m^3$,the bulk modulus for mercury is:

When an aeroplane attains a speed higher than the velocity of sound in air,a loud bang is heard. This is because

$A$ steel rod $100 \; cm$ long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is given to be $2.53 \; kHz$. What is the speed (in $km/s$) of sound in steel?

Vedclass Test Series

Exam Paper Generator

Online Exam Module