Which of the following relations does not give the equation of an adiabatic process, where tems have their usual meaning ?

Options

A. `P^(gamma)T^(1-gamma)` = constant
B. `P^(1-gamma)T^(gamma)` = constant
C. `PV^(gamma)`= constant
D. `TV^(gamma-1)` = constant

Answer

To determine which of the given relations does not represent an adiabatic process, we need to understand the fundamental characteristics of an adiabatic process in thermodynamics.

### Step-by-Step Solution:

1. **Understanding Adiabatic Process**:
- An adiabatic process is one in which there is no heat transfer (Q = 0) between the system and its surroundings.

2. **Key Equations for Adiabatic Processes**:
- The general equation for an adiabatic process for an ideal gas is given by:
\[ PV^\gamma = \text{constant} \]
where \( P \) is pressure, \( V \) is volume, and \( \gamma \) (gamma) is the heat capacity ratio (C_p/C_v).

3. **Deriving Relationships**:
- From the ideal gas law, we know:
\[ PV = nRT \]
- Rearranging gives us:
\[ P = \frac{nRT}{V} \]
- Substituting this expression for \( P \) into the adiabatic equation gives:
\[ \left(\frac{nRT}{V}\right)V^\gamma = \text{constant} \]
- Simplifying this leads to:
\[ T V^{\gamma - 1} = \text{constant} \]
- This shows that the relation \( TV^{\gamma - 1} = \text{constant} \) is valid for an adiabatic process.

4. **Another Relation**:
- We can also derive another relation by keeping pressure and temperature constant:
\[ P V^\gamma = \text{constant} \]
- Rearranging for volume gives:
\[ V = \frac{nRT}{P} \]
- Substituting this into the adiabatic equation leads to:
\[ P^{1 - \gamma} T^\gamma = \text{constant} \]
- This indicates that the relation \( P^{1 - \gamma} T^\gamma = \text{constant} \) is also valid for an adiabatic process.

5. **Identifying the Incorrect Relation**:
- Given the above derivations, we can conclude that the relations \( PV^\gamma = \text{constant} \), \( TV^{\gamma - 1} = \text{constant} \), and \( P^{1 - \gamma} T^\gamma = \text{constant} \) are all valid for adiabatic processes.
- If any relation deviates from these forms, it would not represent an adiabatic process.

### Conclusion:
- The relation that does not represent an adiabatic process is the one that does not conform to the derived equations above.