Question

Equivalent conductivity of \(\mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) is related to molar conductivity by the expression : (a) \(\Lambda_{\mathrm{eq}}=\Lambda_{m}\) (b) \(\Lambda_{\text {eq }}=\Lambda_{m} / 3\) (c) \(\Lambda_{\text {eq }}=3 \Lambda_{m}\) (d) \(\Lambda_{\text {eq }}=\Lambda_{m} / 6\)

Short answer

\(\Lambda_{\text {eq }}=\Lambda_{m} / 6\)

Step-by-step solution

Understand the Concepts of Equivalent and Molar Conductivity

Molar conductivity, denoted as \( \Lambda_m \), is the conductivity of a solution containing one mole of electrolyte dissolved in a given volume of solvent. Equivalent conductivity, denoted as \( \Lambda_{eq} \), on the other hand, is the conductivity of a solution containing one equivalent of the electrolyte dissolved in a given volume of solvent. One equivalent of an electrolyte corresponds to the amount of electrolyte that would dissociate to yield one mole of charge.

Determine the Number of Equivalents for the Given Electrolyte

For the electrolyte \( \mathrm{Fe}_2\left(\mathrm{SO}_4\right)_3 \), the valency factor is the total number of positive charges released by one formula unit upon dissociation. Here, each \( \mathrm{Fe}^{3+} \) ion contributes 3 positive charges, so with two \( \mathrm{Fe}^{3+} \) ions, the valency factor is 6.

Relate Molar Conductivity to Equivalent Conductivity

Equivalent conductivity is obtained by dividing the molar conductivity by the valency factor, which corresponds to the number of equivalents per mole. Thus the relationship is given by \(\Lambda_{\text {eq }}=\frac{\Lambda_m}{n} \), where \(\Lambda_{\text {eq }} \) is the equivalent conductivity and \(\Lambda_m \) is the molar conductivity and n is the valency factor.

Apply the Relationship to the Given Problem

For the given electrolyte \( \mathrm{Fe}_2\left(\mathrm{SO}_4\right)_3 \), the relationship between molar conductivity and equivalent conductivity is \(\Lambda_{\text {eq }} = \frac{\Lambda_m}{6} \) since the valency factor is 6. Therefore, option (d) is the correct answer.

Key concepts

Molar Conductivity

Molar conductivity, represented by the symbol \( \Lambda_m \), is a measure of how well an electrolyte can conduct electricity when there is exactly one mole of it dissolved in a solution. It's important because it helps to understand the efficiency of ions to move through a solvent when they are at a standard concentration.

Molar conductivity increases with dilution, mainly because the ions get more space to move as water molecules keep them further apart. This decrease in ion pairing is critical in helping students comprehend why molar conductivity is not merely about the amount of ions present, but also how freely they can move. For instance, if we take saltwater, the molar conductivity tells us how the flow of current changes when we adjust the amount of salt while keeping the volume constant.

Electrolyte Dissociation

Electrolyte dissociation refers to the process where an electrolyte, which is typically a substance that conducts electricity when dissolved in water, breaks down into its constituent ions. This process is vital for electrical conduction because only the ions are the charge carriers.

When we dissolve salt in water, for instance, the salt crystals break apart into positively charged sodium (Na+) and negatively charged chloride (Cl-) ions. These ions are what allow the solution to conduct electricity. Students must appreciate that not all electrolytes dissociate completely. Strong electrolytes dissociate fully into ions, while weak electrolytes only partially dissociate in solution.

• Strong electrolytes: substances that completely dissociate into ions, such as NaCl and HCl.
• Weak electrolytes: substances that only partially dissociate, such as acetic acid (CH₃COOH).

The extent of dissociation is a factor in the conductivity of a solution and hence affects the molar conductivity as well.

Valency Factor

The valency factor, often denoted by 'n' in equations, corresponds to the charge on the ions that an electrolyte produces upon dissociation. Essentially, it indicates how many charges are carried by one formula unit of the substance.

For instance, in the exercise with \( \mathrm{Fe}_2\left(\mathrm{SO}_4\right)_3 \) the iron ions \( \mathrm{Fe}^{3+} \) have a valency of +3, and since there are two of them, the total valency factor for this compound is 6. This is critical to understand because it influences the equivalent conductivity—how effective an electrolyte is per charge, not just per mole.

• Monovalent ions (such as Na+): valency factor of 1.
• Trivalent ions (such as \( \mathrm{Fe}^{3+} \)): valency factor of 3.

For students, grasping this concept helps with converting molar conductivity to equivalent conductivity and predicting how electrolyte concentration will affect the solution's overall conductivity.