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. (a) 30K
B. (b) 18K
C. (c) 50K
D. (d) 42K

Explanation

To solve the problem, we will use the principles of thermodynamics, particularly focusing on the heat capacities at constant pressure and constant volume.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Both cylinders contain the same amount of an ideal diatomic gas at an initial temperature of \( T = 300 \, K \).
- The piston in cylinder A is free to move (constant pressure process).
- The piston in cylinder B is fixed (constant volume process).
- The rise in temperature for cylinder A, \( \Delta T_A = 30 \, K \).

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

3. **Equate the Heat Transfer**:
- Since the same amount of heat \( Q \) is added to both cylinders, we can set the 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. **Relate \( C_p \) and \( C_v \)**:
- The relationship between \( C_p \) and \( C_v \) for an ideal gas is given by:
\[ C_p = C_v + R \]
- For a diatomic gas, the ratio \( \frac{C_p}{C_v} \) is denoted as \( \gamma \):
\[ \gamma = \frac{C_p}{C_v} \]
- For diatomic gases, \( \gamma \) is approximately \( 1.4 \).

5. **Substituting the Values**:
- Rearranging the equation from step 3 gives:
\[ \Delta T_B = \frac{C_p}{C_v} \Delta T_A = \gamma \Delta T_A \]
- Substituting the known values:
\[ \Delta T_B = 1.4 \times 30 \, K \]

6. **Calculate \( \Delta T_B \)**:
\[ \Delta T_B = 42 \, K \]

### Conclusion:
The rise in temperature of the gas in cylinder B is \( \Delta T_B = 42 \, K \).

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