Which of the following statement is(are) correct?

Options

A. The slope of Z vs P at constant temperature for all real gases, is `(b)/(RT)`
B. The slope of Z vs P at constant temperature for both He and `H_(2)` is `(b)/(RT)`
C. The slope of Z vs P at low pressure for all real gases, at constant temperature is `(b)/(RT)`
D. The slope of Z vs P at high pressure and at constant temperature for real gas is `(-b)/(RT)`

Explanation

To determine which of the statements regarding the compressibility factor (Z) and its relationship with pressure (P) are correct, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding the Compressibility Factor (Z)**:
- The compressibility factor (Z) is defined as \( Z = \frac{PV}{nRT} \), where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = temperature
- Z indicates how much a real gas deviates from ideal gas behavior. For an ideal gas, \( Z = 1 \).

2. **Analyzing the Slope of Z vs P at Constant Temperature**:
- The slope of the graph of Z vs P can be derived from the Van der Waals equation, which accounts for real gas behavior.
- The equation can be rearranged to show that at high pressures, for gases like Helium and Hydrogen (where the attraction forces are negligible, A ≈ 0), the slope is given by \( \frac{B}{RT} \).

3. **Evaluating Each Statement**:
- **Statement 1**: "The slope of Z vs P at constant temperature for all real gases is \( \frac{B}{RT} \)."
- This statement is **incorrect**. The slope \( \frac{B}{RT} \) is specifically applicable to certain gases (like Helium and Hydrogen) at high pressures, not all real gases.

- **Statement 2**: "The slope of Z vs P at constant temperature for both Helium and Hydrogen is \( \frac{B}{RT} \)."
- This statement is **correct**. For Helium and Hydrogen, at high pressures, the slope of Z vs P is indeed \( \frac{B}{RT} \) because A is negligible.

- **Statement 3**: "The slope of Z vs P at low pressure for all real gases at constant temperature is \( \frac{B}{RT} \)."
- This statement is **incorrect**. At low pressures, the behavior of real gases varies, and the slope does not consistently equal \( \frac{B}{RT} \) for all gases.

- **Statement 4**: "The slope of Z vs P at high pressure for all real gases at constant temperature is \( \frac{B}{RT} \)."
- This statement is **incorrect**. While it holds true for certain gases (like Helium and Hydrogen), it does not apply to all real gases, as their behavior can differ significantly.

### Summary of Correct Statements:
- Only **Statement 2** is correct.