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Q1.41:
Determine the osmotic pressure of a solution prepared by dissolving 25 mg of (K_{2}SO_{4}) in 2 litre of water at (25^{\circ} C), assuming that it is completely dissociated.

Solution :

Step 1: Identifying Given Variables

  • Mass of solute ($K_{2}SO_{4}$), $w = 25 \text{ mg} = 0.025 \text{ g}$
  • Volume of solution, $V = 2 \text{ L}$
  • Temperature, $T = 25^{\circ}C = 25 + 273.15 = 298.15 \text{ K}$
  • Universal Gas Constant, $R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1}$
  • Molar mass of $K_{2}SO_{4}$: $(2 \times 39.1) + 32.06 + (4 \times 16.0) = 174.26 \text{ g mol}^{-1}$

Step 2: Determining the Van't Hoff Factor ($i$)

The salt $K_{2}SO_{4}$ dissociates in aqueous solution as follows: $K_{2}SO_{4}(aq) \rightarrow 2K^{+}(aq) + SO_{4}^{2-}(aq)$

Since the compound is assumed to be completely dissociated, the total number of ions produced per formula unit is $n = 3$ (two $K^{+}$ and one $SO_{4}^{2-}$). Therefore, the Van't Hoff factor $i = 3$. [Per the Van't Hoff theory of electrolytic dissociation]

Step 3: Calculating Molarity ($C$)

Molarity is defined as the number of moles of solute per liter of solution. $n_{solute} = \frac{w}{M} = \frac{0.025 \text{ g}}{174.26 \text{ g mol}^{-1}} \approx 1.4346 \times 10^{-4} \text{ mol}$

$C = \frac{n}{V} = \frac{1.4346 \times 10^{-4} \text{ mol}}{2 \text{ L}} = 7.173 \times 10^{-5} \text{ M}$

Step 4: Calculation of Osmotic Pressure ($\Pi$)

Using the modified Van't Hoff equation for non-ideal solutions: $\Pi = iCRT$ [By the Van't Hoff Law for Osmotic Pressure]

Substituting the known values: $\Pi = 3 \times (7.173 \times 10^{-5} \text{ mol L}^{-1}) \times (0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1}) \times (298.15 \text{ K})$

Osmotic Pressure Calculation Path $\Pi = i \cdot C \cdot R \cdot T$ $\Pi = 3 \cdot 7.173e^{-5} \cdot 0.0821 \cdot 298.15$

$\Pi = 3 \times 7.173 \times 10^{-5} \times 0.0821 \times 298.15$

$\Pi \approx 5.27 \times 10^{-3} \text{ atm}$

Final Solution: The osmotic pressure of the solution is $5.27 \times 10^{-3} \text{ atm}$.


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