For real gases, the relation between P, V and T is given by an van der Waals equation, `(P+(an^(2))/(V^(2)))(V-nb)=nRT`.
For the following gases `CH_(4),CO_(2), O_(2),H_(2)` which gas will have (i) highest value of 'a' and (ii) lowest value of 'b' respectively?

Options

A. `(i) CO_(2), (ii) H_(2)`
B. `(i) CH_(4), (ii) CO_(2)`
C. `(i)H_(2), (ii) CO_(2)`
D. `(i) O_(2), (ii) H_(2)`

Answer

To solve the problem, we need to analyze the van der Waals equation and understand the significance of the constants 'a' and 'b' for the given gases: CH₄ (methane), CO₂ (carbon dioxide), O₂ (oxygen), and H₂ (hydrogen).

### Step-by-Step Solution:

1. **Understanding the van der Waals Equation**:
The van der Waals equation is given by:
\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]
Here, 'a' represents the attraction between gas molecules, and 'b' represents the volume occupied by the gas molecules (excluded volume).

2. **Identifying the Meaning of 'a'**:
The value of 'a' is related to the intermolecular forces. A higher value of 'a' indicates stronger attractive forces between the gas molecules. Generally, larger molecules with more electrons have stronger van der Waals forces.

3. **Comparing the Gases for 'a'**:
- **CH₄ (Methane)**: A small non-polar molecule with weak van der Waals forces.
- **CO₂ (Carbon Dioxide)**: A larger molecule than CH₄, with a higher number of electrons, leading to stronger attractions.
- **O₂ (Oxygen)**: A diatomic molecule, larger than H₂ but smaller than CO₂.
- **H₂ (Hydrogen)**: The smallest molecule among the four, with the weakest attractions.

**Conclusion for 'a'**: CO₂ has the highest value of 'a' due to its larger size and stronger intermolecular forces.

4. **Identifying the Meaning of 'b'**:
The value of 'b' is related to the volume occupied by one mole of gas molecules. A smaller value of 'b' indicates that the gas molecules occupy less volume.

5. **Comparing the Gases for 'b'**:
- The excluded volume is proportional to the size of the gas molecules. Smaller molecules will have a smaller excluded volume.
- **H₂ (Hydrogen)**: The smallest molecule, thus it will have the lowest excluded volume.
- **O₂ (Oxygen)**: Larger than H₂, hence a larger excluded volume.
- **CH₄ (Methane)**: Larger than both H₂ and O₂.
- **CO₂ (Carbon Dioxide)**: The largest among the four gases.

**Conclusion for 'b'**: H₂ has the lowest value of 'b' due to its small size.

### Final Answers:
- (i) The gas with the highest value of 'a': **CO₂**
- (ii) The gas with the lowest value of 'b': **H₂**