The specific heat of an ideal gas is proportional to

Options

A. `T^(@)`
B. `T^(-1)`
C. `T^(1//2)`
D. `T^(3)`

Answer

To determine the relationship of specific heat with respect to temperature for an ideal gas, we will follow these steps:

### Step-by-Step Solution:

1. **Understand Specific Heat**: Specific heat (C) is defined as the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). For gases, we typically consider two types: specific heat at constant volume (C_v) and specific heat at constant pressure (C_p).

2. **Define C_v**: The specific heat at constant volume (C_v) can be defined mathematically as:
\[ C_v = \left( \frac{dU}{dT} \right)_V \]
where \( U \) is the internal energy and \( T \) is the temperature.

3. **Internal Energy of an Ideal Gas**: For an ideal gas, the internal energy \( U \) is a function of temperature only. Therefore, we can express the change in internal energy as:
\[ dU = \frac{F}{2} R dT \]
where \( F \) is the degrees of freedom of the gas molecules and \( R \) is the universal gas constant.

4. **Substituting into C_v**: Substituting \( dU \) into the equation for \( C_v \):
\[ C_v = \left( \frac{F}{2} R \right) \]
This shows that \( C_v \) is proportional to the degrees of freedom \( F \) of the gas.

5. **Relationship between C_p and C_v**: The relationship between the specific heats at constant pressure and constant volume is given by:
\[ C_p - C_v = R \]
If \( C_v \) is independent of temperature, then \( C_p \) will also be independent of temperature.

6. **Conclusion**: From the above analysis, we conclude that the specific heat of an ideal gas (both \( C_v \) and \( C_p \)) is independent of temperature. Thus, the specific heat of an ideal gas is proportional to the degrees of freedom \( F \) and not to temperature.

### Final Answer:
The specific heat of an ideal gas is independent of temperature.

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