During adiabatic process pressure (p) versus density `(rho)` equation is

Options

A. `prho^(lambda)="constant"`
B. `prho^(-lambda)="constant"`
C. `p^(lambda)rho^(1+lambda)="constant"`
D. `p^(1//lambda)rho^(lambda)="constant"`

Answer

To solve the question regarding the relationship between pressure (P) and density (ρ) 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) and volume (V) can be expressed as:
\[ P \cdot V^\lambda = \text{constant} \]
where \( \lambda \) is the adiabatic index (ratio of specific heats).

2. **Relate Density to Volume**:
Density (ρ) is defined as mass (m) per unit volume (V):
\[ \rho = \frac{m}{V} \]
This implies that volume is inversely proportional to density:
\[ V = \frac{m}{\rho} \]

3. **Substitute Volume in the Adiabatic Equation**:
Substitute the expression for volume (V) in terms of density (ρ) into the adiabatic equation:
\[ P \cdot \left(\frac{m}{\rho}\right)^\lambda = \text{constant} \]
This simplifies to:
\[ P \cdot \frac{m^\lambda}{\rho^\lambda} = \text{constant} \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ P \cdot \rho^{-\lambda} = \text{constant} \]
This indicates that the product of pressure and density raised to the power of negative lambda is constant.

5. **Conclusion**:
From the derived relationship, we can conclude that during an adiabatic process, the equation relating pressure (P) and density (ρ) can be expressed as:
\[ P \propto \rho^{-\lambda} \]
This means that pressure is inversely proportional to density raised to the power of lambda.

### Final Answer:
The correct relationship for pressure versus density during an adiabatic process is:
\[ P \cdot \rho^{-\lambda} = \text{constant} \]