The mean square speed of the molecules of a gas at absolute temperature T is proportional to

Options

A. (a)(1)/(T)
B. (b)sqrtT
C. (c)T
D. (d)T^(2)

Answer

To solve the question regarding the mean square speed of the molecules of a gas at absolute temperature T, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Mean Square Speed**:
The mean square speed of gas molecules is related to the kinetic energy of the gas. The formula for the root mean square speed (Vrms) of gas molecules is given by:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{m}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( m \) is the molecular mass of the gas.

2. **Identify the Relationship**:
From the formula for Vrms, we can derive the mean square speed (\( V^2 \)):
\[ V^2_{\text{rms}} = \frac{3RT}{m} \]
This indicates that the mean square speed is directly proportional to the absolute temperature \( T \).

3. **Express the Proportionality**:
We can express the relationship as:
\[ V^2_{\text{rms}} \propto T \]
This means that the mean square speed of the gas molecules increases linearly with the increase in temperature.

4. **Choose the Correct Option**:
Given the options:
- \( \frac{1}{T} \)
- \( \sqrt{T} \)
- \( T \)
- \( T^2 \)

Since we established that the mean square speed is proportional to \( T \), the correct answer is:
\[ \text{Option C: } T \]

### Final Answer:
The mean square speed of the molecules of a gas at absolute temperature \( T \) is proportional to \( T \).

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