Question:medium

In a mixture of gases, the average number of degrees of freedom per molecule is 6. If the rms speed of the molecule is \(c\), what is the velocity of sound in the gas?

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The rms speed and the velocity of sound are related for ideal gases. Use the degrees of freedom to determine the relationship.
Updated On: Nov 26, 2025
  • \(c/\sqrt{3}\)
  • \(c/\sqrt{2}\)
  • \(2c/3\)
  • \(c\)
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The Correct Option is A

Solution and Explanation

The speed of sound in a gas depends on its temperature and molecular characteristics. The speed of sound (\(v_s\)) is calculated using the formula: \[ v_s = \sqrt{\frac{\gamma R T}{M}} \] In this equation, \(\gamma\) represents the adiabatic index, \(R\) is the universal gas constant, \(T\) denotes the temperature, and \(M\) signifies the molar mass. A monoatomic ideal gas has an average of 3 degrees of freedom (\(f = 3\)). A gas with 6 degrees of freedom behaves as a diatomic gas, with an adiabatic index of \(\gamma = 1.4\). The relationship between the root mean square (rms) speed (\(c\)) and the velocity of sound is: \[ v_s = \frac{c}{\sqrt{3}} \] Therefore, the correct answer is \(c/\sqrt{3}\).
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