Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is at `100^(@)C`. While the other one is at `0^(@)C`. If the two bodies are brought into contact, then assuming no heat loss, the final common temperature is

Options

A. `50^(@) C`
B. more than `50^(@) C`
C. less than `50^(@) C` but greater than `0^(@) C`
D. `0^(@)C`

Answer

To solve the problem, we need to find the final common temperature when two identical bodies at different temperatures come into contact, assuming no heat loss.

### Step-by-step Solution:

1. **Identify the Initial Conditions**:
- Body A (hot body) is at \( T_A = 100^\circ C \).
- Body B (cold body) is at \( T_B = 0^\circ C \).

2. **Define the Final Temperature**:
- Let the final common temperature be \( \theta \).

3. **Understand Heat Transfer**:
- According to the principle of calorimetry, the heat lost by the hot body will equal the heat gained by the cold body:
\[ Q_{\text{lost}} = Q_{\text{gained}} \]

4. **Express Heat Transfer in Terms of Heat Capacity**:
- The heat lost by the hot body can be expressed as:
\[ Q_{\text{lost}} = C_h (T_A - \theta) \]
- The heat gained by the cold body can be expressed as:
\[ Q_{\text{gained}} = C_c (\theta - T_B) \]
- Where \( C_h \) and \( C_c \) are the heat capacities of the hot and cold bodies, respectively.

5. **Recognize the Relationship of Heat Capacities**:
- Since the heat capacity increases with temperature, we have \( C_h > C_c \) because \( T_A > T_B \).

6. **Set Up the Equation**:
- From the principle of calorimetry, we have:
\[ C_h (100 - \theta) = C_c (\theta - 0) \]

7. **Rearranging the Equation**:
- Rearranging gives:
\[ C_h (100 - \theta) = C_c \theta \]
- Expanding this gives:
\[ C_h \cdot 100 - C_h \theta = C_c \theta \]
- Combining terms yields:
\[ C_h \cdot 100 = (C_h + C_c) \theta \]

8. **Solve for Final Temperature \( \theta \)**:
- Thus, we can express \( \theta \) as:
\[ \theta = \frac{C_h \cdot 100}{C_h + C_c} \]

9. **Analyze the Result**:
- Since \( C_h > C_c \), it follows that \( \theta > 50^\circ C \). Therefore, the final temperature will be greater than \( 50^\circ C \).

### Conclusion:
The final common temperature \( \theta \) will be greater than \( 50^\circ C \).