Figure displays the plot of the compression factor `Z` verses `p` for a few gases
Which of the following statements is//are correct for a van-der waals gas:

Options

A. The plot I is applicable provided the vander waals constant a is negligible.
B. The plot II is a applicable provided the vander waals constant `b` is negligible.
C. The plot III is applicable provided the vander waals constants `a` and `b` are negligible.
D. The plot IV is applicable provided the temperature of the gas is much higher than its critical temperature

Explanation

To solve the question regarding the compressibility factor \( Z \) versus pressure \( p \) for van der Waals gases, we need to analyze the provided statements based on the behavior of real gases as described by the van der Waals equation.

### Step-by-Step Solution:

1. **Understanding the Van der Waals Equation**:
The van der Waals equation is given by:
\[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]
where \( a \) and \( b \) are van der Waals constants. The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{nRT} \]

2. **Analyzing Statement A**:
- The first statement claims that the plot is applicable when the van der Waals constant \( A \) is negligible.
- If \( a = 0 \), the equation simplifies to the ideal gas law:
\[ PV = nRT \implies Z = 1 \]
- As pressure increases, \( Z \) increases linearly, confirming that this statement is correct.

3. **Analyzing Statement B**:
- The second statement claims that the plot is applicable when the van der Waals constant \( B \) is negligible.
- If \( b = 0 \), we can analyze the equation:
\[ Z = 1 - \frac{a}{RT} \cdot \frac{n}{V} \]
- As pressure increases (and thus \( V \) decreases), \( Z \) becomes less than 1, confirming that this statement is also correct.

4. **Analyzing Statement C**:
- The third statement claims that the plot is applicable when both van der Waals constants \( A \) and \( B \) are negligible.
- If both \( a \) and \( b \) are zero, we revert to the ideal gas behavior:
\[ Z = 1 \]
- This indicates that this statement is correct as well.

5. **Analyzing Statement D**:
- The fourth statement claims that the plot is applicable when the temperature of the gas is much higher than its critical temperature.
- However, above the critical temperature, all gases behave ideally regardless of their nature (real or ideal). Therefore, this statement is false.

### Conclusion:
Based on the analysis:
- Statements A, B, and C are correct.
- Statement D is incorrect.

Thus, the correct answer is that statements A, B, and C are correct.