An ideal gas is heated at constant volume until its pressure doubles. Which one of the following statements is correct?

Options

A. The mean speed of the molecule doubles.
B. The number of molecules doubles.
C. The mean square speed of the molecules doubles.
D. The number of molecules per unit volume doubles.

Explanation

To solve the problem, we need to analyze the situation of an ideal gas being heated at constant volume until its pressure doubles. We will use the ideal gas law and the relationships between pressure, volume, and temperature.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

2. **Identify the Given Conditions**:
- The gas is heated at constant volume (\( V \) is constant).
- The pressure doubles (\( P \) changes from \( P \) to \( 2P \)).

3. **Relate Pressure and Temperature**:
Since the volume is constant, we can rearrange the ideal gas law to express the relationship between pressure and temperature:
\[ P \propto T \quad \text{(at constant volume)} \]
Therefore, if the pressure doubles, we can say:
\[ 2P \propto T' \quad \text{(where \( T' \) is the new temperature)} \]
This implies that:
\[ T' = 2T \]
Thus, the temperature also doubles when the pressure doubles.

4. **Mean Speed and Mean Square Speed**:
The mean speed (\( v_{mean} \)) of gas molecules is given by:
\[ v_{mean} = \sqrt{\frac{8RT}{\pi m}} \]
where \( m \) is the mass of a gas molecule. Since \( T \) doubles, the new mean speed becomes:
\[ v_{mean}' = \sqrt{\frac{8R(2T)}{\pi m}} = \sqrt{2} \cdot v_{mean} \]
Therefore, the mean speed increases by a factor of \( \sqrt{2} \).

The mean square speed (\( v_{rms}^2 \)) is given by:
\[ v_{rms}^2 = \frac{3RT}{m} \]
If \( T \) doubles, the new mean square speed becomes:
\[ v_{rms}'^2 = \frac{3R(2T)}{m} = 2 \cdot \frac{3RT}{m} = 2 \cdot v_{rms}^2 \]
Hence, the mean square speed doubles.

5. **Conclusion**:
The correct statement regarding the heating of the ideal gas at constant volume until its pressure doubles is that the mean square speed of the gas molecules doubles.

### Final Answer:
The mean square speed of the gas molecules doubles when the pressure doubles while heating at constant volume.