Question

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?

• A. \(c/\sqrt{3}\)
• B. \(c/\sqrt{2}\)
• C. \(2c/3\)
• D. \(c\)

Hint:

The rms speed and the velocity of sound are related for ideal gases. Use the degrees of freedom to determine the relationship.

Answer 1

The Correct Option is A

The velocity of sound in a gas is related to the temperature and the molecular properties of the gas. The speed of sound \(v_s\) is given by: \[ v_s = \sqrt{\frac{\gamma R T}{M}} \] Where \(\gamma\) is the adiabatic index, \(R\) is the universal gas constant, \(T\) is the temperature, and \(M\) is the molar mass. For a monoatomic ideal gas, the average number of degrees of freedom \(f = 3\). If the number of degrees of freedom is 6, it implies that the gas behaves like a diatomic gas, where \(\gamma = 1.4\). The relationship between the root mean square (rms) speed \(c\) and the velocity of sound is: \[ v_s = \frac{c}{\sqrt{3}} \] Thus, the correct answer is \(c/\sqrt{3}\).