Select the incorrect statement about the specific heat of a gaseous system :

Options

A. Specific heat no exchange condition, `C_A = 0`
B. Specific heat at constant temperature , `C_7 = oo`
C. Specific heat at constant pressure, `C_p = (gammaR)/(gamma-1)`
D. Specific heat at constant volume, `C_v = R/gamma`

Answer

To solve the question regarding the specific heat of a gaseous system and identify the incorrect statement, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Specific Heat**:
- Specific heat (S) is defined as the amount of heat (Q) required to change the temperature of a unit mass (m) of a substance by one degree Celsius (ΔT).
- The formula for specific heat is given by:
\[ S = \frac{Q}{m \Delta T} \]

2. **Analyzing Option A**:
- Option A states: "Specific heat under no exchange condition is that \( C_A = 0 \)".
- Under no heat exchange, \( Q = 0 \). Therefore, if \( Q = 0 \), then:
\[ S = \frac{0}{m \Delta T} = 0 \]
- This statement is correct.

3. **Analyzing Option B**:
- Option B states: "Specific heat at constant temperature \( C_T \) is equal to infinity".
- At constant temperature, the change in temperature \( \Delta T = 0 \). Thus:
\[ S = \frac{Q}{m \Delta T} \rightarrow S = \frac{Q}{m \cdot 0} \]
- Since division by zero is undefined, we can interpret this as \( S \) tending to infinity. This statement is also correct.

4. **Analyzing Option C**:
- Option C states: "Specific heat at constant pressure \( C_p = \frac{\gamma R}{\gamma - 1} \)".
- We know that \( \gamma = \frac{C_p}{C_v} \). Therefore, we can express \( C_p \) in terms of \( C_v \):
\[ C_p = C_v + R \]
- Substituting \( \gamma = \frac{C_p}{C_v} \) into the equation:
\[ C_p = \frac{C_p R}{C_p - R} \]
- After simplifying, we find that this relationship holds true. Thus, this statement is correct.

5. **Analyzing Option D**:
- Option D states: "Specific heat at constant volume \( C_v = \frac{R}{\gamma} \)".
- Using \( \gamma = \frac{C_p}{C_v} \), we can rearrange this to find \( C_v \):
\[ C_v = \frac{R}{\gamma} \rightarrow C_v = \frac{R C_v}{C_p} \]
- This does not hold true as it leads to a contradiction. Thus, this statement is incorrect.

### Conclusion:
The incorrect statement about the specific heat of a gaseous system is **Option D**: \( C_v = \frac{R}{\gamma} \).

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