Two cylinders A and B fitted with pistons contain equal amounts of an ideal diatomic gas at 300K. The piston of A is free to move, while that B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in A is 30K, then the rise in temperature of the gas in B is

Options

A. `30K`
B. `18K`
C. `50K`
D. `42K`

Explanation

To solve the problem, we need to analyze the heat transfer in the two cylinders containing an ideal diatomic gas. We will use the concepts of specific heat capacities at constant pressure (Cp) and constant volume (Cv).

### Step-by-Step Solution:

1. **Identify Given Information**:
- Both cylinders A and B contain equal amounts of an ideal diatomic gas.
- Initial temperature (T_initial) = 300 K.
- Rise in temperature of gas in cylinder A (ΔT_A) = 30 K.
- The piston in cylinder A is free to move (constant pressure).
- The piston in cylinder B is fixed (constant volume).

2. **Understand the Heat Transfer**:
- For cylinder A (constant pressure), the heat added (Q_A) can be expressed as:
\[ Q_A = n C_p \Delta T_A \]
- For cylinder B (constant volume), the heat added (Q_B) can be expressed as:
\[ Q_B = n C_v \Delta T_B \]

3. **Equate the Heat Transfer**:
- Since the same amount of heat is given to both gases, we can set the two equations equal to each other:
\[ n C_p \Delta T_A = n C_v \Delta T_B \]
- The number of moles (n) cancels out:
\[ C_p \Delta T_A = C_v \Delta T_B \]

4. **Use the Relation Between Cp and Cv**:
- For a diatomic ideal gas, the ratio of specific heats is given by:
\[ \gamma = \frac{C_p}{C_v} = \frac{7}{5} = 1.4 \]
- Therefore, we can express \(C_p\) in terms of \(C_v\):
\[ C_p = \gamma C_v \]

5. **Substitute Cp in the Equation**:
- Substitute \(C_p\) in the heat transfer equation:
\[ \gamma C_v \Delta T_A = C_v \Delta T_B \]
- Cancel \(C_v\) (assuming it is not zero):
\[ \gamma \Delta T_A = \Delta T_B \]

6. **Calculate the Rise in Temperature for Cylinder B**:
- Substitute the known values:
\[ \Delta T_B = \gamma \Delta T_A = 1.4 \times 30 K \]
- Calculate:
\[ \Delta T_B = 42 K \]

### Final Answer:
The rise in temperature of the gas in cylinder B is **42 K**.