The root mean square speed of the molecules of a gas is

Options

A. independent of its pressure but directly proportional to its Kelvin temperature
B. directly proportional to two square root of both its pressure and its Kelvin temperature
C. indepdent of its pressure but directly proportional to the square root of its Kelvin temperature.
D. directly proportional to its pressure and its Kelvin temperature.

Answer

To find the root mean square speed (v_rms) of the molecules of a gas, we can use the following formula:

### Step-by-Step Solution:

1. **Understand the Formula**: The root mean square speed (v_rms) is given by the equation:
\[ v_{rms} = \sqrt{\frac{3k_B T}{m}} \]
where:
- \( k_B \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin,
- \( m \) is the mass of a gas molecule.

2. **Identify the Relationships**:
- The v_rms is **directly proportional** to the **square root of the temperature (T)**.
- The v_rms is **inversely proportional** to the **square root of the mass (m)** of the gas molecules.

3. **Consider the Effect of Pressure**:
- The formula does not include pressure (P) directly. Therefore, v_rms is independent of pressure.
- However, it is important to note that in a gas, pressure can be related to temperature and volume through the ideal gas law, but for the purpose of this question, we focus on the direct relationship with temperature.

4. **Conclusion**:
- From the analysis, we conclude that the root mean square speed of the molecules of a gas is independent of pressure and directly proportional to the square root of the temperature.

### Final Answer:
The root mean square speed of the molecules of a gas is independent of pressure but directly proportional to the square root of the temperature.

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