Problem 108 - A Complete Resource Book in Chemistry for JEE Main

Question

A \(0.004 \mathrm{M}\) solution of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is isotonic with a \(0.010\) M solution of glucose at same temperature. The apparent degree of dissociation of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is (a) \(25 \%\) (b) \(50 \%\) (c) \(75 \%\) (d) \(85 \%\)

Step-by-Step Solution

Answer

The apparent degree of dissociation of \(\mathrm{Na}_2 \mathrm{SO}_4\) is \(75\%\).

Step 1: Understand Isotonic Solutions

Isotonic solutions have the same osmotic pressure. For ionic solutions, this involves considering the van 't Hoff factor (i), which accounts for the degree of dissociation. For a non-electrolyte like glucose, i = 1.

Step 2: Determine van 't Hoff Factors

For the glucose solution, since glucose doesn't dissociate, the van 't Hoff factor \(i_{\text{glucose}} = 1\). For \(\mathrm{Na}_2\mathrm{SO}_4\), which dissociates into \(2\mathrm{Na}^+\) and \(\mathrm{SO}_4^{2-}\), the complete dissociation would give a factor of 3. Let \(\alpha\) be the degree of dissociation. Thus, the van 't Hoff factor for \(\mathrm{Na}_2\mathrm{SO}_4\) is \(1 + 2\alpha\).

Step 3: Use the Isotonic Condition

The isotonic condition equates the osmotic pressure of the two solutions: \(i_{\text{Na}_2\text{SO}_4} \times 0.004 = i_{\text{glucose}} \times 0.010\). Substituting \(i_{\text{glucose}} = 1\) and \(i_{\text{Na}_2\text{SO}_4} = 1 + 2\alpha\), we have \[(1+2\alpha)\times 0.004 = 0.010.\]

Step 4: Solve for the Degree of Dissociation \(\alpha\)

Simplify and solve the equation for \(\alpha\): \[1 + 2\alpha = \frac{0.010}{0.004} = 2.5.\]Subtract 1 from both sides: \[2\alpha = 1.5.\]Divide by 2: \[\alpha = 0.75.\] Thus, the degree of dissociation is \(75\%\).

Key Concepts

Isotonic Solutions

Isotonic solutions are those that have the same osmotic pressure, meaning they exert equal pressure when separated by a semipermeable membrane. This property is crucial in many biological and chemical processes. When two solutions are isotonic, cells placed in them neither shrink nor swell, keeping cell functions stable. Isotonicity involves comparing the overall solute concentration of the solutions.
For electrolytes, like \(\mathrm{Na}_2\mathrm{SO}_4\), we must consider its dissociation into ions, which increases the number of particles in the solution, thereby affecting osmotic pressure. On the other hand, a non-electrolyte, like glucose, does not dissociate and thus has an isotonic factor of 1.

van 't Hoff Factor

The van 't Hoff factor helps quantify the effect of solute particles on osmotic pressure. It represents how many particles a compound dissociates into in solution, influencing the solution's colligative properties, such as boiling point, freezing point, and osmotic pressure.
For \(\mathrm{Na}_2\mathrm{SO}_4\), complete dissociation results in three particles: two sodium ions (\(2\mathrm{Na}^+\)) and one sulfate ion (\(\mathrm{SO}_4^{2-}\)). Hence, the ideal van 't Hoff factor for complete dissociation is 3. However, not all \(\mathrm{Na}_2\mathrm{SO}_4\) molecules dissociate, so the factor is adjusted to \(1 + 2\alpha\), where \(\alpha\) is the degree of dissociation.
For non-ionic compounds, like glucose, which don't dissociate, the van 't Hoff factor is simply 1.

Osmotic Pressure

Osmotic pressure is the pressure that arises when a solvent passes through a semipermeable membrane to equalize solute concentrations on either side. This pressure is key in maintaining fluid balance in cells and across membranes and is a fundamental concept in chemistry and biology.
The formula for calculating osmotic pressure is \( \, \Pi = iMRT \), where \(\Pi\) is the osmotic pressure, \(i\) is the van 't Hoff factor, \(M\) is the molarity, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

• Osmotic pressure increases with higher solute concentrations and larger van 't Hoff factors due to greater ionic dissociation.
• Isotonic solutions have equal osmotic pressures, which is why their \(iM\) products are equal in calculations.

Ionic Dissociation

Ionic dissociation refers to the process where an ionic compound separates into its individual ions in solution. This process is influenced by factors such as the compound's nature, temperature, and solvent properties.
\(\mathrm{Na}_2\mathrm{SO}_4\), when dissolved in water, dissociates into two sodium ions (\(2\mathrm{Na}^+\)) and one sulfate ion (\(\mathrm{SO}_4^{2-}\)), increasing the number of particles in the solution. The degree of dissociation, \(\alpha\), represents the fraction of the initial compound that undergoes dissociation.
A higher degree of dissociation implies a greater impact on colligative properties, such as osmotic pressure, because more ions in solution enhance these effects.