An ideal gas of certain mass is heated in a small vessel and then in a large vessel, such that their volume remains unchanged. The `P - T` curves are :

Options

A. Parabolic with same curvature
B. Parabolic with different curvature
C. Linear with same slope
D. Linear with different slope

Answer

To solve the problem regarding the behavior of an ideal gas in small and large vessels while keeping the volume constant, 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:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles of gas
- \( R \) = Ideal gas constant
- \( T \) = Temperature (in Kelvin)

2. **Constant Volume Condition**: Since the volume of the gas remains unchanged in both the small and large vessels, we can analyze the relationship between pressure and temperature at constant volume.

3. **Relationship Between Pressure and Temperature**: From the ideal gas law, if we rearrange it for constant volume, we get:
\[ P = \frac{nRT}{V} \]
Here, \( \frac{nR}{V} \) is a constant for a given amount of gas in a specific vessel. Therefore, we can express the relationship as:
\[ P \propto T \]
This indicates that pressure is directly proportional to temperature.

4. **Linear Relationship**: The relationship \( P = kT \) (where \( k \) is a constant) shows that the graph of pressure (P) versus temperature (T) will be a straight line. The slope of this line will depend on the specific vessel (small or large), as the constant \( k \) will differ due to the different volumes.

5. **Conclusion**: The \( P-T \) curves for the ideal gas in both vessels will be linear but with different slopes. The curve for the small vessel will have a steeper slope compared to the large vessel due to the smaller volume affecting the constant \( k \).

### Final Answer:
The \( P - T \) curves are linear with different slopes.