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) \(85 \%\)

Short answer

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

Step-by-step solution

Understanding Isotonic Solutions

Isotonic solutions have the same osmotic pressure. For solutions, this means they have the same product of van't Hoff factor (i), molarity (M), and the universal gas constant times temperature (R*T), though R*T cancels out if temperature is the same. Thus, we have:\[ i_1 \times M_1 = i_2 \times M_2 \]where 1 is for Na2SO4 and 2 is for glucose.

Van't Hoff Factor for Glucose

Glucose does not dissociate into ions in solution, so its van't Hoff factor is 1. Thus for the glucose solution:\[ i_2 = 1 \]\[ M_2 = 0.010 \]

Setting Up Equation for Na2SO4

Let the apparent degree of dissociation for \(\mathrm{Na}_2\mathrm{SO}_4\) be \(\alpha\). When \(\alpha\) of \(\mathrm{Na}_2\mathrm{SO}_4\) dissociates, it produces 3 ions (2 \(\mathrm{Na}^+\) and 1 \(\mathrm{SO}_4^{2-}\)). Thus, van't Hoff factor \(i\) for \(\mathrm{Na}_2\mathrm{SO}_4\) is:\[ i_1 = 1 + 2\alpha \]

Equating Osmotic Conditions

Using the isotonic condition:\[ i_1 \times 0.004 = 1 \times 0.010 \]Substituting for \(i_1\):\[ (1 + 2\alpha) \times 0.004 = 0.010 \]

Solving for Alpha

Expand and simplify the equation:\[ 0.004 + 0.008\alpha = 0.010 \]Subtract 0.004 from each side:\[ 0.008\alpha = 0.006 \]Divide each side by 0.008:\[ \alpha = \frac{0.006}{0.008} = 0.75 \]

Interpretation of Alpha

Since \(\alpha = 0.75\) indicates that 75% of \(\mathrm{Na}_2\mathrm{SO}_4\) dissociates, the apparent degree of dissociation is 75%. Hence the correct choice is (c) 75%.

Key concepts

Van't Hoff Factor

The van't Hoff factor, denoted by the symbol \(i\), is a crucial concept when studying solutions, particularly when dealing with colligative properties. It represents the number of particles a compound disintegrates into when dissolved. For non-dissociating substances, such as glucose, the van't Hoff factor is \(i = 1\). This means glucose remains as individual glucose molecules when dissolved.

For substances like \( \mathrm{Na}_2 \mathrm{SO}_4\), which dissociate into several ions, the van't Hoff factor changes based on the dissociation extent. If \( \mathrm{Na}_2 \mathrm{SO}_4\) fully dissociates, it breaks into three ions: two \( \mathrm{Na}^+\) and one \( \mathrm{SO}_4^{2-}\). Theoretically, the van't Hoff factor could be 3. However, the actual value depends on the degree of dissociation.

Understanding the van't Hoff factor helps relate the concentration of particles in a solution to its colligative properties, such as osmotic pressure, which is vital for calculations involving isotonic solutions.

Degree of Dissociation

The degree of dissociation, often symbolized by \(\alpha\), measures the extent to which a substance dissociates in solution. Simply put, it indicates the fraction of the original compound that separates into its ionic constituents. For \( \mathrm{Na}_2 \mathrm{SO}_4\), if \(\alpha = 0.75\), it implies that 75% of the original \( \mathrm{Na}_2 \mathrm{SO}_4\) molecules dissociate to form ions.

This percentage is crucial because it directly influences the van't Hoff factor. As more molecules dissociate, the van't Hoff factor increases, capturing the higher number of particles in the solution.

In mathematical terms, the relationship can be expressed as:
\[i = 1 + 2\alpha\]
The degree of dissociation can be determined experimentally or through calculations involving osmotic pressure, making it a significant parameter when assessing the behavior of ionic compounds in a solution.

Osmotic Pressure

Osmotic pressure is a key colligative property that depends on the number of particles in a solution. It describes the pressure needed to stop solvent flow across a semipermeable membrane. For two solutions to be isotonic, they must exhibit the same osmotic pressure. This balance ensures equal solvent movement in and out of both solutions.

In real-world practices, understanding osmotic pressure is essential in contexts like medical treatments with injections, ensuring that intravenous solutions match the body's osmotic pressure.

The equation for osmotic pressure \(\pi\) in a solution is given by:
\[\pi = i \, C \, R \, T\]
where \(i\) is the van't Hoff factor, \(C\) is the molarity, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

In isotonic conditions, this same osmotic pressure principle allows us to equate the van’t Hoff factors and molarities of different solutions, which is precisely how we deduce certain properties like the degree of dissociation.

Molarity

Molarity, symbolized as \(M\), is the concentration measure used to express the amount of solute in a solution. It is defined as the number of moles of solute per liter of solution. For example, when we say a solution is \(0.004 \, \mathrm{M}\), it means there are 0.004 moles of solute per liter of solution.

Molarity is crucial for various calculations in chemistry, including those for osmotic pressure and reaction stoichiometry. It helps determine how concentrated a solution is, which impacts the behavior of the substances involved.

In the context of isotonic solutions, molarity works with the van't Hoff factor to equalize osmotic pressures. As these calculations show in the \(\mathrm{Na}_2\mathrm{SO}_4\) example, the molarity of both \(\mathrm{Na}_2\mathrm{SO}_4\) and glucose solutions, when combined with their respective van't Hoff factors, dictate the osmotic balance.

Dissociation of Na2SO4

The dissociation of \(\mathrm{Na}_2\mathrm{SO}_4\) in water is a classic example of an ionic compound breaking into constituent ions. When \(\mathrm{Na}_2\mathrm{SO}_4\) dissolves, it separates into two sodium ions (\(\mathrm{Na}^+\)) and one sulfate ion (\(\mathrm{SO}_4^{2-}\)). Hence, theoretically, complete dissociation produces three ions.

This dissociation drastically affects colligative properties, making it significant for calculations involving van't Hoff factor and degree of dissociation. The actual dissociation extent, influenced by factors like concentration and temperature, determines how much \(\mathrm{Na}_2\mathrm{SO}_4\) converts into ions.

In calculations, as observed with the isotonic condition, this dissociation provides a foundation for understanding how solutions with differing solutes can exhibit the same osmotic pressure. The relationship between dissociation and osmotic properties illustrates the importance of each concept in determining solution behavior.