In a mixture of gases, the average number of degrees of freedom per molecule is 6. the rms speed of the molecules of the gas is C. the velocity of sound in the gas is

Explanation

To find the velocity of sound in a gas mixture where the average number of degrees of freedom per molecule is 6 and the RMS speed of the molecules is C, we can follow these steps:

### Step 1: Understand the relationship between RMS speed and velocity of sound
The RMS speed (v_rms) of the gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where R is the universal gas constant, T is the absolute temperature, and M is the molar mass of the gas.

The velocity of sound (v_sound) in the gas can be expressed as:
\[ v_{sound} = \sqrt{\frac{\gamma RT}{M}} \]
where \(\gamma\) (gamma) is the ratio of specific heats (Cp/Cv).

### Step 2: Relate the two equations
We can relate the velocity of sound to the RMS speed:
\[ \frac{v_{sound}}{v_{rms}} = \sqrt{\frac{\gamma}{3}} \]

### Step 3: Calculate gamma (γ)
For a gas with an average number of degrees of freedom (n) equal to 6, we can calculate \(\gamma\) using the formula:
\[ \gamma = 1 + \frac{2}{n} \]
Substituting \(n = 6\):
\[ \gamma = 1 + \frac{2}{6} = 1 + \frac{1}{3} = \frac{4}{3} \]

### Step 4: Substitute gamma into the relationship
Now we substitute \(\gamma\) back into the equation relating the velocity of sound and RMS speed:
\[ v_{sound} = v_{rms} \cdot \sqrt{\frac{\gamma}{3}} \]
Substituting \(\gamma = \frac{4}{3}\):
\[ v_{sound} = v_{rms} \cdot \sqrt{\frac{4/3}{3}} = v_{rms} \cdot \sqrt{\frac{4}{9}} = v_{rms} \cdot \frac{2}{3} \]

### Step 5: Substitute the value of RMS speed
Given that \(v_{rms} = C\):
\[ v_{sound} = C \cdot \frac{2}{3} \]

### Final Answer
Thus, the velocity of sound in the gas is:
\[ v_{sound} = \frac{2C}{3} \]

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