During adiabatic process, pressure P and density equation is:

Options

A. `P rho ^(gamma)`= constant
B. `P rho^(- gamma)` = constant
C. `P^(gamma) rho^(gamma+1)` = constant
D. ` P ^(1/gamma) rho^(gamma)` = constant

Explanation

To derive the relationship between pressure \( P \) and density \( \rho \) during an adiabatic process, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure \( P \), volume \( V \), and temperature \( T \) is given by the equation:
\[ PV^\gamma = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

2. **Relate Density to Volume**:
- Density \( \rho \) is defined as mass \( m \) per unit volume \( V \):
\[ \rho = \frac{m}{V} \]
- This implies that volume \( V \) can be expressed in terms of density:
\[ V = \frac{m}{\rho} \]

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

4. **Rearranging the Equation**:
- Rearranging gives:
\[ P = \text{constant} \cdot \rho^\gamma \]
- This shows that pressure \( P \) is proportional to density \( \rho \) raised to the power of \( \gamma \).

5. **Final Relationship**:
- Therefore, the relationship between pressure and density during an adiabatic process can be expressed as:
\[ P \propto \rho^\gamma \]
- This indicates that as the density increases, the pressure also increases, and vice versa, following the power of \( \gamma \).

### Conclusion:
The final equation derived from the adiabatic process shows that pressure \( P \) and density \( \rho \) are related by the equation:
\[ P \rho^{-\gamma} = \text{constant} \]