In a closed rigid container an ideal gas is filled. If the gas is heated, the graph of pressure (P) v/s root mean square speed (rms) will be :

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the relationship between pressure (P) and root mean square speed (C_rms) of an ideal gas in a closed rigid container when heated, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the Conditions**:
In this problem, we have a closed rigid container. This means that the volume \( V \) is constant.

3. **Relate Pressure to Temperature**:
Since the volume is constant, we can rearrange the ideal gas law to express pressure in terms of temperature:
\[ P = \frac{nRT}{V} \]
This shows that pressure \( P \) is directly proportional to temperature \( T \):
\[ P \propto T \]

4. **Relate Root Mean Square Speed to Temperature**:
The root mean square speed \( C_{rms} \) of an ideal gas is given by the equation:
\[ C_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( M \) is the molar mass of the gas. From this equation, we can see that \( C_{rms} \) is also directly proportional to the square root of temperature:
\[ C_{rms} \propto \sqrt{T} \]

5. **Square the Relationship of RMS Speed**:
To relate pressure and \( C_{rms} \), we can square the relationship for \( C_{rms} \):
\[ C_{rms}^2 \propto T \]
Thus, we can express temperature in terms of \( C_{rms} \):
\[ T \propto C_{rms}^2 \]

6. **Combine the Relationships**:
Since we have established that \( P \propto T \) and \( T \propto C_{rms}^2 \), we can combine these relationships:
\[ P \propto C_{rms}^2 \]

7. **Conclusion on the Graph**:
The relationship \( P \propto C_{rms}^2 \) indicates that pressure is proportional to the square of the root mean square speed. This means that if we plot pressure \( P \) against \( C_{rms} \), the graph will be a parabola opening upwards, not a straight line.

### Final Answer:
The graph of pressure (P) versus root mean square speed (C_rms) will be a curve that shows that pressure increases with the square of the root mean square speed.