Select the incorrect relation. (Where symboles have their usual meanings):-

Options

A. `C_p = (gammaR)/(gamma-1)`
B. `C_P - C_V = R`
C. `DeltaU = (P_fV_f - P_iV_i)/(1-gamma)`
D. `C_v = R/(gamma-1)`

Answer

To solve the problem of identifying the incorrect relation among the given options, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Symbols**:
- \( C_p \): Molar heat capacity at constant pressure.
- \( C_v \): Molar heat capacity at constant volume.
- \( R \): Universal gas constant.
- \( \gamma \): Ratio of specific heats, defined as \( \gamma = \frac{C_p}{C_v} \).

2. **Option A**:
- The relation states:
\[ C_p = \frac{\gamma R}{\gamma - 1} \]
- We know from thermodynamics that:
\[ C_p - C_v = R \]
- Rearranging gives us:
\[ C_v = C_p - R \]
- Substituting \( C_v \) into the expression for \( \gamma \):
\[ \gamma = \frac{C_p}{C_p - R} \]
- Rearranging this gives:
\[ C_p = \frac{\gamma R}{\gamma - 1} \]
- Therefore, **Option A is correct**.

3. **Option B**:
- The relation states:
\[ C_v = \frac{C_p}{\gamma} \]
- Using the definition of \( \gamma \):
\[ \gamma = \frac{C_p}{C_v} \implies C_v = \frac{C_p}{\gamma} \]
- This is a well-known relation in thermodynamics, thus **Option B is correct**.

4. **Option C**:
- The relation states:
\[ \Delta U = \frac{P_f V_f - P_i V_i}{\gamma - 1} - \gamma \]
- We know that for an ideal gas, the change in internal energy \( \Delta U \) can be expressed as:
\[ \Delta U = C_v \Delta T \]
- From the ideal gas law, we can express \( P_f V_f \) and \( P_i V_i \) in terms of temperature:
\[ P_f V_f = nRT_f \quad \text{and} \quad P_i V_i = nRT_i \]
- For one mole of gas, this gives:
\[ \Delta U = R \Delta T \]
- The correct expression for \( \Delta U \) should not include the term \( -\gamma \) as it does not correspond to the change in internal energy. Thus, **Option C is incorrect**.

5. **Option D**:
- The relation states:
\[ C_v = \frac{R}{\gamma - 1} \]
- Since we have already established that \( C_v = C_p - R \) and using the relation for \( \gamma \):
\[ C_v = \frac{R}{\gamma - 1} \]
- This is also a known relation in thermodynamics, thus **Option D is correct**.

### Conclusion:
The incorrect relation among the options is **Option C**.