During adiabatic process pressure P versus density `roh` equation is

Explanation

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

### Step 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 \( T \) can be described by the equation:
\[ PV^\gamma = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

### Step 2: Relate Volume and Density
Density \( \rho \) is defined as mass \( m \) divided by volume \( V \):
\[ \rho = \frac{m}{V} \]
From this, we can express volume in terms of density:
\[ V = \frac{m}{\rho} \]

### Step 3: Substitute Volume in the Adiabatic Equation
Substituting \( V \) in the adiabatic equation:
\[ P \left(\frac{m}{\rho}\right)^\gamma = \text{constant} \]
This can be rewritten as:
\[ P \cdot \frac{m^\gamma}{\rho^\gamma} = \text{constant} \]

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

### Step 5: Final Formulation
We can express this relationship as:
\[ P \cdot \rho^{-\gamma} = \text{constant} \]
Thus, the final equation relating pressure and density during an adiabatic process is:
\[ P \cdot \rho^{-\gamma} = \text{constant} \]

### Summary
The relationship between pressure \( P \) and density \( \rho \) during an adiabatic process can be summarized as:
\[ P \cdot \rho^{-\gamma} = \text{constant} \]

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