What is the molar specific heat capacity of a gas undergoing an adiabatic process ?

Options

A. Zero
B. 1
C. `oo`
D. None of these

Answer

To find the molar specific heat capacity of a gas undergoing an adiabatic process, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Adiabatic Process**:
- In an adiabatic process, there is no heat exchange with the surroundings. This means that the heat added to the system (dq) is zero.
- Mathematically, we express this as:
\[ dq = 0 \]

2. **Relating dq to Molar Specific Heat Capacity**:
- The change in heat (dq) can also be expressed in terms of the number of moles (n), the molar specific heat capacity (C), and the change in temperature (dt):
\[ dq = nC dt \]

3. **Setting the Equation**:
- Since we know that dq = 0 for an adiabatic process, we can set up the equation:
\[ nC dt = 0 \]

4. **Analyzing the Equation**:
- In the equation \(nC dt = 0\), we have three factors:
- \(n\) (number of moles) cannot be zero (as we are dealing with a gas).
- \(dt\) (change in temperature) cannot be zero if there is a change in the state of the gas.
- Therefore, the only possibility left is that the molar specific heat capacity \(C\) must be zero:
\[ C = 0 \]

5. **Conclusion**:
- Thus, the molar specific heat capacity of a gas undergoing an adiabatic process is:
\[ C = 0 \]

### Final Answer:
The molar specific heat capacity of a gas undergoing an adiabatic process is **0**.

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