The conductivity of \(0.001028 \mathrm{M}\) acetic acid is \(4.95 \times 10^{-5} \mathrm{~S} \mathrm{~cm}^{-1}\). Calculate its dissociation constant if \(\Lambda^{0}\) for acetic acid is \(390.5 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\)

Molar conductivity, denoted by \(\Lambda_m\), is a measure of how well an electrolyte conducts electricity when there is one mole of electrolyte dissolved in a specific volume of solution. To compute molar conductivity, we use the formula \(\Lambda_m = \frac{\kappa}{C}\), where \(\kappa\) is conductivity measured in Siemens per centimeter (\(S\cdot cm^{-1}\)) and \(C\) is the concentration of the solution in molarity (M).

For an acetic acid solution with a concentration of \(0.001028\, M\) and a conductivity of \(4.95 \times 10^{-5}\, S\cdot cm^{-1}\), we calculate the molar conductivity to facilitate further dissociation constant calculations. The understanding of molar conductivity is essential as it serves as a bridge to link conductivity measurements with the properties of the solute at a given concentration.

Kohlrausch's Law of Independent Migration of Ions states that the molar conductivity of an electrolyte at infinite dilution, denoted by \(\Lambda^{0}\), is a sum of the contributions from its constituent ions. This law enables us to understand how ions contribute to conductivity when they are sufficiently far apart and there are no inter-ionic interactions.

According to Kohlrausch's Law, as the concentration of an electrolyte decreases (dilutes), the molar conductivity increases and approaches a limiting value, \(\Lambda^{0}\). This concept is pivotal when analyzing weak electrolytes, such as acetic acid, because it helps us calculate the degree of dissociation of the acid by comparing its molar conductivity at a given concentration with \(\Lambda^{0}\).

Ostwald's Dilution Law is applied to weak electrolytes, which only partially dissociate into ions in a solution. The law provides a relationship between the degree of dissociation \(\alpha\), concentration of the solution \(C\), and the dissociation constant \(K_a\). The formula for Ostwald's dilution law is \(K_a = \frac{\alpha^2 \cdot C}{1 - \alpha}\).

This formula is particularly useful because it enables us to determine the dissociation constant for a weak acid or base from measurable quantities like conductivity and concentration. The approximation made in this particular exercise—that the degree of dissociation \(\alpha\) is much less than 1—simplifies the calculation of the dissociation constant, allowing us to ignore the \(\alpha\) in the denominator.

The degree of dissociation \(\alpha\) represents the fraction of the total number of moles of an electrolyte that dissociates into ions in a solution. It is a unitless quantity, providing insight into the extent of dissociation of an electrolyte.

In the context of molar conductivity, we relate \(\alpha\) to \(\Lambda_m\) by the formula \(\Lambda_m = \alpha \cdot \Lambda^{0}\), which indicates that the molar conductivity at a given concentration is the product of the degree of dissociation and the molar conductivity at infinite dilution. By rearranging this relationship, we can calculate \(\alpha\), assisting in the determination of \(K_a\) using Ostwald's dilution law. Understanding the degree of dissociation is fundamental when assessing the behavior of weak acids and bases in solution.