For real gases, van der Waals' equation is written as
`(P+(an^(2))/(V^(2))) (V-nb)= nRT`
where `a` and `b` are van der Waals' constants.
Two sets of gases are:
`(I) O_(2), CO_(2), H_(2)` and `He(II) CH_(4), O_(2)` and `O_(2) and H_(2)`
The gases given in set `I` in increasing order of `b` and gases given in set `II` in decreasing order of `a` are arranged below. Select the correct order from the following:

Options

A. `(I)O_(2)lt He lt H_(2) lt CO_(2) (II) H_(2)gt O_(2) gt CH_(4)`
B. `(I) H_(2) lt He lt O_(2) lt CO_(2) (II) CH_(4) gt O_(2) gt H_(2)`
C. `(I)H_(2) lt O_(2) lt He lt CO_(2) (II) O_(2) gt CH_(4) gt H_(2)`
D. `(I)He lt H_(2) lt CO_(2) lt O_(2) (II) CH_(4) gt H_(2) gt O_(2)`

Explanation

To solve the problem, we need to analyze the two sets of gases based on the van der Waals constants \( a \) and \( b \).

### Step 1: Understanding the van der Waals Constants
- The constant \( b \) is related to the volume occupied by gas molecules (size of the molecules). A larger \( b \) value indicates larger molecules.
- The constant \( a \) is related to the attractive forces between gas molecules. A larger \( a \) value indicates stronger intermolecular attractions.

### Step 2: Analyzing Set I (Increasing order of \( b \))
The gases in Set I are: \( O_2, CO_2, H_2, \) and \( He \).

1. **Determine the size of the molecules**:
- **Hydrogen (H₂)**: Smallest size, hence smallest \( b \).
- **Helium (He)**: Small size, larger than H₂, hence larger \( b \).
- **Oxygen (O₂)**: Larger than He, hence larger \( b \).
- **Carbon Dioxide (CO₂)**: Largest size, hence largest \( b \).

2. **Order of \( b \)**:
- \( H_2 < He < O_2 < CO_2 \)

### Step 3: Analyzing Set II (Decreasing order of \( a \))
The gases in Set II are: \( CH_4, O_2, H_2 \).

1. **Determine the attractive forces**:
- **Methane (CH₄)**: Strongest intermolecular forces, hence largest \( a \).
- **Oxygen (O₂)**: Moderate intermolecular forces, hence moderate \( a \).
- **Hydrogen (H₂)**: Weakest intermolecular forces, hence smallest \( a \).

2. **Order of \( a \)**:
- \( CH_4 > O_2 > H_2 \)

### Step 4: Final Arrangement
- For Set I (increasing order of \( b \)): \( H_2 < He < O_2 < CO_2 \)
- For Set II (decreasing order of \( a \)): \( CH_4 > O_2 > H_2 \)

### Step 5: Matching with Options
Now, we compare our findings with the provided options:
- **Option A**: Set 1: \( O_2 < He < H_2 < CO_2 \) (Incorrect)
- **Option B**: Set 1: \( H_2 < He < O_2 < CO_2 \) and Set 2: \( CH_4 > O_2 > H_2 \) (Correct)
- **Option C**: Set 1: \( H_2 < O_2 < He < CO_2 \) (Incorrect)
- **Option D**: Set 1: \( He < H_2 < CO_2 < O_2 \) (Incorrect)

### Conclusion
The correct answer is **Option B**.

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