Which of the following statements is // are correct ?

Options

A. `A` gas has two specific heats only
B. A materials will have only one specific heat, if and only if its coefficient of thermal expansion is equal to zero.
C. A gas has infinite number of specific heats.
D. None of these

Answer

To determine which of the statements regarding specific heats and thermal expansion are correct, let’s analyze each statement step by step.

### Step-by-Step Solution:

1. **Statement 1: A gas has two specific heats only.**
- **Analysis**: This statement is true. A gas can have two specific heats:
- \( C_P \) (specific heat at constant pressure)
- \( C_V \) (specific heat at constant volume)
- **Conclusion**: This statement is correct.

2. **Statement 2: A material will have only one specific heat if and only if its coefficient of thermal expansion is equal to zero.**
- **Analysis**: The coefficient of thermal expansion (\( \alpha_T \)) measures how much a material deforms with temperature change. If \( \alpha_T = 0 \), it implies that the material does not expand or contract with temperature changes, which can lead to a scenario where it has a single specific heat value. However, materials can behave differently under various conditions, and this statement is somewhat misleading as it suggests a strict condition that may not universally apply.
- **Conclusion**: This statement is not entirely correct.

3. **Statement 3: A gas has an infinite number of specific heats.**
- **Analysis**: This statement is incorrect. A gas does not have an infinite number of specific heats; it has two specific heats, \( C_P \) and \( C_V \), as mentioned earlier. The specific heat can vary depending on the process (isothermal, adiabatic, etc.), but it does not imply an infinite number.
- **Conclusion**: This statement is incorrect.

4. **Statement 4: A material will have only one specific heat if its coefficient of thermal expansion is not zero.**
- **Analysis**: This statement is misleading. A material can have multiple specific heats depending on the conditions (e.g., phase changes, temperature ranges). The coefficient of thermal expansion being non-zero does not guarantee that there is only one specific heat.
- **Conclusion**: This statement is incorrect.

### Final Conclusion:
- **Correct Statements**:
- Statement 1 is correct.
- Statements 2, 3, and 4 are incorrect.