Two van der waal's gases have same value of a but different value of b which of the following statement is correct ?

Options

A. The smaller the value of b larger will be compressibility
B. The larger the value of b larger will be compressibility
C. Both have same compressibility
D. All the with smaller value of b will occupy larger volume

Answer

To solve the question regarding two van der Waals gases that have the same value of 'a' but different values of 'b', we will analyze the implications of these parameters in the van der Waals equation and their physical meanings.

### Step-by-Step Solution:

1. **Understand the van der Waals Equation**:
The van der Waals equation is given by:
\[ P + \frac{aN^2}{V^2} (V - Nb) = NRT \]
Here, 'P' is the pressure, 'V' is the volume, 'N' is the number of moles, 'R' is the gas constant, 'T' is the temperature, 'a' is a measure of the attractive forces between molecules, and 'b' is the volume occupied by the gas molecules (excluded volume).

2. **Analyze the given conditions**:
- We have two gases with the same value of 'a'. This indicates that the attractive forces between the molecules of both gases are similar.
- The gases have different values of 'b'. This means that the effective size of the gas molecules is different; a larger 'b' indicates larger molecules occupy more volume, while a smaller 'b' indicates smaller molecules occupy less volume.

3. **Implications of 'b'**:
- A smaller value of 'b' means that the volume occupied by the molecules is less, leading to a larger effective volume (V - Nb) when calculating the real volume available for the gas to occupy.
- Conversely, a larger value of 'b' means that the volume occupied by the molecules is more, leading to a smaller effective volume (V - Nb).

4. **Compressibility Factor (Z)**:
- The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV_{\text{real}}}{RT} \]
- If 'b' is smaller, then \( V - Nb \) is larger, leading to a larger \( V_{\text{real}} \) and hence a larger \( Z \). This indicates that the gas is less compressible.
- If 'b' is larger, then \( V - Nb \) is smaller, leading to a smaller \( V_{\text{real}} \) and hence a smaller \( Z \). This indicates that the gas is more compressible.

5. **Conclusion**:
- Since the question asks which statement is correct, we conclude that a smaller value of 'b' leads to a larger effective volume and thus a greater compressibility factor \( Z \).
- Therefore, the correct statement is: "Smaller value of b results in a larger volume occupied by the gas molecules."

### Final Answer:
The correct statement is: **Smaller value of b results in a larger volume occupied by the gas molecules.**