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 problem regarding the specific heat of an ideal gas that varies with temperature \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Specific Heat**:
The specific heat (\( c \)) of a substance is defined as the amount of heat required to change the temperature of a unit mass of the substance by one degree Celsius (or one Kelvin).

2. **Specific Heat of Ideal Gases**:
For ideal gases, the specific heat is often considered to be constant over a wide range of temperatures. This means that the specific heat does not significantly change with temperature, especially for ideal gases.

3. **Mathematical Representation**:
If we denote the specific heat as \( c(T) \) and assume it is constant, we can express it mathematically as:
\[ c(T) = c_0 \]
where \( c_0 \) is a constant value.

4. **Considering the Variation with Temperature**:
Since the problem states that the specific heat varies with temperature, but for an ideal gas, this variation is negligible, we can represent this as:
\[ c(T) \propto T^0 \]
This implies that the specific heat does not change with temperature, hence the exponent of \( T \) is 0.

5. **Conclusion**:
Therefore, the specific heat of an ideal gas can be expressed as:
\[ c(T) = k \cdot T^0 = k \]
where \( k \) is a constant. Thus, the correct option is that the specific heat is constant, represented mathematically as \( T^0 \).

### Final Answer:
The specific heat of an ideal gas can be considered constant, which can be represented as \( T^0 \).

---