1. Moles of $SO_2$ passed

Given:

• Volume of $SO_2 = 224\text{ mL} = 0.224\text{ L}$
• $T = 298\text{ K}$, $P = 1\text{ atm}$

Using ideal gas equation:

$n = \frac{PV}{RT} = \frac{1\times 0.224}{0.0821\times 298}$

$n \approx \frac{0.224}{24.46} \approx 9.16\times 10^{-3}\text{ mol}$

So,

$n(SO_2) \approx 0.00916\text{ mol}$

2. Moles of $NaOH$ present

Given:

• Volume of solution $= 100\text{ mL} = 0.1\text{ L}$
• Molarity $= 0.1\text{ M}$

$n(NaOH) = M\times V = 0.1\times 0.1 = 0.01\text{ mol}$

3. Reaction between $SO_2$ and $NaOH$

Possible reactions are:

$SO_2 + NaOH \rightarrow NaHSO_3$

and with excess base,

$SO_2 + 2NaOH \rightarrow Na_2SO_3 + H_2O$

Now compare mole ratio:

$\frac{n(NaOH)}{n(SO_2)} = \frac{0.01}{0.00916} \approx 1.09$

Since this ratio is greater than 1 but less than 2, the products will be a mixture of $NaHSO_3$ and $Na_2SO_3$.

Let moles of $Na_2SO_3 = a$ and moles of $NaHSO_3 = b$.

Then:

• Sulfur balance: $a+b = 0.00916$

• Sodium hydroxide balance: $2a+b = 0.01$

Subtracting,

$a = 0.01-0.00916 = 0.00084$

Then,

$b = 0.00916-0.00084 = 0.00832$

So non-volatile solutes formed are:

• $0.00084$ mol $Na_2SO_3$
• $0.00832$ mol $NaHSO_3$

4. Total moles of solute particles

Since solution is dilute, use van't Hoff factor by complete dissociation:

$Na_2SO_3 \rightarrow 2Na^+ + SO_3^{2-} \quad (3\text{ particles})$ $NaHSO_3 \rightarrow Na^+ + HSO_3^- \quad (2\text{ particles})$

Hence total moles of solute particles:

$n_{solute} = 3(0.00084) + 2(0.00832)$

$n_{solute} = 0.00252 + 0.01664 = 0.01916\text{ mol}$

5. Moles of solvent (water)

Water given $=36\text{ g}$

$n_{water} = \frac{36}{18} = 2\text{ mol}$

6. Lowering of vapour pressure

For dilute solution:

$\Delta P = P^0 X_{solute}$

Since dilute,

$X_{solute} \approx \frac{n_{solute}}{n_{solvent}} = \frac{0.01916}{2} = 0.00958$

Thus,

$\Delta P = 24\times 0.00958 \approx 0.2299\text{ mm Hg}$

$\Delta P \approx 0.23\text{ mm Hg}$

Now,

$\Delta P = x\times 10^{-2}\text{ mm Hg}$

So,

$0.23 = x\times 10^{-2}$

$x \approx 23$

7. Final answer

$\boxed{23}$

This lies within the stored range $18$ to $24$.