The specific heat of an ideal gas varies with temperature T as

Options

A. `T^1`
B. `T^2`
C. `T^-2`
D. `T^0`

Answer

To solve the question regarding how the specific heat of an ideal gas varies with temperature \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand Specific Heat**:
Specific heat is defined as the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin).

2. **Behavior of Specific Heat in Ideal Gases**:
For ideal gases, the specific heat does not change significantly over a wide range of temperatures. It is often treated as a constant for practical purposes.

3. **Negligible Dependence on Temperature**:
The specific heat of an ideal gas has a negligible dependence on temperature, meaning that for many calculations, we can assume it remains constant as the temperature changes.

4. **Mathematical Representation**:
If we denote the specific heat as \( C \), we can express its dependence on temperature \( T \) as:
\[ C(T) \approx C_0 \]
where \( C_0 \) is a constant value.

5. **Evaluating the Options**:
- **Option 1**: \( C \propto T \) (Directly proportional to temperature) - Incorrect.
- **Option 2**: \( C \propto T^2 \) - Incorrect.
- **Option 3**: \( C \propto T^{-2} \) - Incorrect.
- **Option 4**: \( C \propto T^0 \) (Constant) - Correct.

6. **Conclusion**:
Since the specific heat of an ideal gas can be treated as a constant over a wide range of temperatures, the correct answer is that the specific heat varies as \( T^0 \), indicating it is independent of temperature.

### Final Answer:
The specific heat of an ideal gas varies with temperature \( T \) as \( C \propto T^0 \) (constant).

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