Question

The speed of sound in oxygen ($O_2$) at a certain temperature is 460 $ms^{-1}$. The speed of sound in helium ($He$) at the same temperature will be (assume both gases to be ideal)

[AIEEE 2008]

• A. 500 $ms^{-1}$
• B. 650 $ms^{-1}$
• C. 330 $ms^{-1}$
• D. 1420 $ms^{-1}$

Answer 1

Answer: (D)

1420 $ms^{-1}$

Answer 2

The speed of sound in an ideal gas is given by the formula: $v = \sqrt{\frac{\gamma RT}{M}}$ where $v$ is the speed of sound, $\gamma$ is the adiabatic index ($C_p/C_v$), $R$ is the ideal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass of the gas.

We are given the speed of sound in oxygen ($O_2$) at a certain temperature, $v_{O_2} = 460 \, ms^{-1}$. We need to find the speed of sound in helium ($He$) at the same temperature, $v_{He}$.

Since the temperature ($T$) and the gas constant ($R$) are the same for both gases, the ratio of their speeds of sound is given by: $\frac{v_{He}}{v_{O_2}} = \sqrt{\frac{\gamma_{He}/M_{He}}{\gamma_{O_2}/M_{O_2}}} = \sqrt{\frac{\gamma_{He}}{\gamma_{O_2}} \cdot \frac{M_{O_2}}{M_{He}}}$

For an ideal monatomic gas like Helium ($He$), the adiabatic index $\gamma_{He} = \frac{5}{3}$. The molar mass of Helium is $M_{He} \approx 4 \, g/mol$.

For an ideal diatomic gas like Oxygen ($O_2$), at moderate temperatures (where vibrational modes are not excited), the adiabatic index $\gamma_{O_2} = \frac{7}{5}$. The molar mass of Oxygen ($O_2$) is $M_{O_2} \approx 2 \times 16 = 32 \, g/mol$.

Now, substitute these values into the ratio equation: $\frac{v_{He}}{460} = \sqrt{\frac{5/3}{7/5} \cdot \frac{32}{4}}$ $\frac{v_{He}}{460} = \sqrt{\frac{5}{3} \times \frac{5}{7} \times 8}$ $\frac{v_{He}}{460} = \sqrt{\frac{25}{21} \times 8}$ $\frac{v_{He}}{460} = \sqrt{\frac{200}{21}}$

Now, solve for $v_{He}$: $v_{He} = 460 \sqrt{\frac{200}{21}}$

Calculate the value: $\sqrt{\frac{200}{21}} \approx \sqrt{9.5238}$ $\sqrt{9.5238} \approx 3.08606$

$v_{He} \approx 460 \times 3.08606$ $v_{He} \approx 1419.5876$

Rounding this value to the nearest integer, we get 1420 $ms^{-1}$. Comparing this with the given options, option (d) is the closest value.