The adiabatic relation between P and T is

Options

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

Explanation

To find the adiabatic relation between pressure (P) and temperature (T), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- An adiabatic process is one in which no heat is exchanged with the surroundings. This means that the change in internal energy is equal to the work done on or by the system.

2. **Use the Adiabatic Equation**:
- The adiabatic equation is given by:
\[ PV^\gamma = \text{constant} \]
- Here, \( \gamma \) (gamma) is the adiabatic index, defined as:
\[ \gamma = \frac{C_p}{C_v} \]
- Where \( C_p \) is the molar specific heat at constant pressure and \( C_v \) is the molar specific heat at constant volume.

3. **Relate Pressure and Volume**:
- From the ideal gas law, we know:
\[ PV = nRT \]
- We can express volume (V) in terms of pressure (P) and temperature (T):
\[ V = \frac{nRT}{P} \]

4. **Substitute Volume in the Adiabatic Equation**:
- Substitute \( V \) in the adiabatic equation:
\[ P \left(\frac{nRT}{P}\right)^\gamma = \text{constant} \]
- This simplifies to:
\[ P \cdot \frac{(nRT)^\gamma}{P^\gamma} = \text{constant} \]

5. **Rearrange the Equation**:
- Rearranging gives:
\[ P^{1 - \gamma} (nR)^\gamma T^\gamma = \text{constant} \]
- Since \( nR \) is a constant, we can denote it as \( K \):
\[ P^{1 - \gamma} T^\gamma = K \]

6. **Final Form of the Adiabatic Relation**:
- Thus, the adiabatic relation between pressure and temperature can be expressed as:
\[ P^{1 - \gamma} T^\gamma = \text{constant} \]

### Conclusion:
The adiabatic relation between pressure and temperature is given by:
\[ P^{1 - \gamma} T^\gamma = \text{constant} \]