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 `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) He lt H_2 lt CO_2 lt O_2 " "
(II) CH_(4) gt H_2 gt O_2`
B. `(I) O_(2) lt He lt H_2 lt CO_(2) " "
(II) H_(2) gt O_2 gt CH_4`
C. `(I) He lt H_(2) lt O_(2) lt CO_(2) " " (II) CH_(4) gt O_(2) gt H_2`
D. `(I) H_2 lt O_2 lt He lt CO_2
" " (II) O_2 gt CH_4 gt H_2`

Answer

To solve the problem, we need to analyze the van der Waals constants \( a \) and \( b \) for the gases in the two sets provided and arrange them accordingly.

### Step-by-Step Solution:

1. **Understanding the van der Waals Constants**:
- The van der Waals constant \( b \) represents the volume occupied by one mole of gas particles, which is related to the size of the molecules. A larger \( b \) value indicates larger molecules.
- The van der Waals constant \( a \) accounts for the attractive forces between gas molecules. A larger \( a \) value indicates stronger intermolecular forces.

2. **Identifying the Gases and Their Constants**:
- For Set I: \( O_2, CO_2, H_2, He \)
- For Set II: \( CH_4, O_2, H_2 \)

We will look up the values of \( a \) and \( b \) for these gases:
- **Set I**:
- \( He: b = 0.0237 \)
- \( H_2: b = 0.0266 \)
- \( O_2: b = 0.318 \)
- \( CO_2: b = 0.427 \)

- **Set II**:
- \( CH_4: a = 2.25 \)
- \( O_2: a = 1.36 \)
- \( H_2: a = 0.244 \)

3. **Arranging Set I in Increasing Order of \( b \)**:
- From the values:
- \( He < H_2 < O_2 < CO_2 \)
- Therefore, the increasing order of \( b \) for Set I is:
- \( He < H_2 < O_2 < CO_2 \)

4. **Arranging Set II in Decreasing Order of \( a \)**:
- From the values:
- \( CH_4 > O_2 > H_2 \)
- Therefore, the decreasing order of \( a \) for Set II is:
- \( CH_4 > O_2 > H_2 \)

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

6. **Selecting the Correct Option**:
- Based on the arrangements derived, we can compare with the provided options to find the correct order.