For an adiabatic process graph between PV & V for a sample of ideal gas will be

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Explanation

To determine the nature of the graph between \( PV \) and \( V \) for an adiabatic process involving an ideal gas, 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. This means that \( \Delta Q = 0 \).

2. **Use the Ideal Gas Law**:
For an ideal gas, we can use the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

3. **Relate \( PV \) to Temperature**:
Since \( PV \) is proportional to \( T \) (for a given amount of gas at constant \( n \) and \( R \)), we can express this as:
\[ PV \propto T \]

4. **Use the Adiabatic Condition**:
For a reversible adiabatic process, we have:
\[ PV^\gamma = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

5. **Express \( PV \) in terms of Volume**:
Rearranging the equation \( PV^\gamma = \text{constant} \) gives:
\[ P = \frac{\text{constant}}{V^\gamma} \]
This indicates that as \( V \) increases, \( P \) decreases, and vice versa.

6. **Graphing \( PV \) vs. \( V \)**:
Since \( PV \) is constant for a given volume, we can plot \( PV \) on the y-axis and \( V \) on the x-axis. The relationship \( PV \propto \frac{1}{V^\gamma} \) indicates that the graph will be a curve that decreases as \( V \) increases.

7. **Determine the Nature of the Graph**:
The graph between \( PV \) and \( V \) will not be a straight line (which would indicate a linear relationship) but rather a curve that decreases. This is characteristic of an inverse power relationship.

### Conclusion:
The graph between \( PV \) and \( V \) for an adiabatic process of an ideal gas is a decreasing curve, indicating that as the volume increases, the product \( PV \) decreases.