Consider the van der Waals' constants, a and b, for the following gases:
`{:(gas, Ar, Ne, Kr, Xe),(a//(atm " dm"^6 mol^(-2)),1.3,0.2,5.1,4.1),(b//(10^(-2)dm^3 mol^(-1)),3.2,1.7,1.0,5.0):}`
Which gas is expected to have the highest critical temperature?

Options

A. Xe
B. Ne
C. Kr
D. Ar.

Explanation

To determine which gas has the highest critical temperature among Argon (Ar), Neon (Ne), Krypton (Kr), and Xenon (Xe) using the van der Waals constants \( a \) and \( b \), we can use the formula for critical temperature \( T_c \):

\[ T_c = \frac{8a}{27Rb} \]

Where:
- \( a \) is the van der Waals constant related to the attraction between particles.
- \( b \) is the van der Waals constant related to the volume occupied by the gas particles.
- \( R \) is the universal gas constant (approximately 0.0821 atm·dm³/(mol·K)).

### Step 1: Calculate the ratio \( \frac{a}{b} \) for each gas

1. **Xenon (Xe)**:
- \( a = 4.1 \) atm·dm³/mol²
- \( b = 5.0 \times 10^{-2} \) dm³/mol
- \( \frac{a}{b} = \frac{4.1}{5.0 \times 10^{-2}} = 82 \)

2. **Krypton (Kr)**:
- \( a = 5.1 \) atm·dm³/mol²
- \( b = 1.0 \times 10^{-2} \) dm³/mol
- \( \frac{a}{b} = \frac{5.1}{1.0 \times 10^{-2}} = 510 \)

3. **Argon (Ar)**:
- \( a = 1.3 \) atm·dm³/mol²
- \( b = 3.2 \times 10^{-2} \) dm³/mol
- \( \frac{a}{b} = \frac{1.3}{3.2 \times 10^{-2}} = 40.625 \)

4. **Neon (Ne)**:
- \( a = 0.2 \) atm·dm³/mol²
- \( b = 1.7 \times 10^{-2} \) dm³/mol
- \( \frac{a}{b} = \frac{0.2}{1.7 \times 10^{-2}} = 11.76 \)

### Step 2: Compare the values of \( \frac{a}{b} \)

- Xenon: \( 82 \)
- Krypton: \( 510 \)
- Argon: \( 40.625 \)
- Neon: \( 11.76 \)

### Step 3: Determine which gas has the highest critical temperature

Since the critical temperature \( T_c \) is directly proportional to \( \frac{a}{b} \), the gas with the highest \( \frac{a}{b} \) will have the highest critical temperature.

From our calculations:
- Krypton has the highest \( \frac{a}{b} \) ratio of \( 510 \).

### Conclusion

Thus, the gas expected to have the highest critical temperature is **Krypton (Kr)**.

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