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

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Explanation

To solve the question regarding the graph between PV and V for an adiabatic process of an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.

2. **Consider the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. For an ideal gas undergoing an adiabatic process, the relationship between temperature \( T \) and volume \( V \) is given by:
\[ TV^{\gamma - 1} = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

3. **Relate \( PV \) and \( T \)**:
Since \( PV = nRT \), we can express \( PV \) in terms of \( T \):
\[ PV = nR \cdot T \]
Thus, the variation of \( PV \) with \( V \) will be the same as the variation of \( T \) with \( V \).

4. **Find the Relationship between \( T \) and \( V \)**:
From the adiabatic condition \( TV^{\gamma - 1} = \text{constant} \), we can express \( T \) as:
\[ T = K V^{1 - \gamma} \]
where \( K \) is a constant.

5. **Analyze the Exponent**:
Since \( \gamma > 1 \) for an ideal gas, \( 1 - \gamma \) will be negative. This means that as \( V \) increases, \( T \) decreases, leading to an inverse relationship.

6. **Graph Characteristics**:
The graph of \( T \) versus \( V \) will be a decreasing curve. Since \( PV \) is proportional to \( T \), the graph of \( PV \) versus \( V \) will also exhibit a similar decreasing trend.

7. **Conclusion**:
Therefore, the graph between \( PV \) and \( V \) for an adiabatic process of an ideal gas will be a curve that decreases as \( V \) increases.