Question

The conductivity of a saturated solution of \(\mathrm{AgCl}\) at \(298 \mathrm{~K}\) was found to be \(3.40\) \(\times 10^{-6} \Omega^{-1} \mathrm{~cm}^{-1} ;\) the conductivity of water used to make up the solution was \(1.60\) \(\times 10^{-6} \Omega^{-1} \mathrm{~cm}^{-1} .\) Determine the solubility of \(\mathrm{AgCl}\) in water in mole per litre at \(298 \mathrm{~K}\). The equivalent conductivity of \(\mathrm{AgCl}\) at infinite dilution is \(150.0 \Omega^{-1} \mathrm{~cm}^{-2} \mathrm{eq}^{-1}\). (a) \(1.44 \times 10^{-10}\) (b) \(1.2 \times 10^{-5}\) (c) \(3.33 \times 10^{-5}\) (d) \(1.2 \times 10^{-8}\)

Short answer

The solubility of AgCl in water at 298 K is (b) 1.2 x 10^-5 moles per litre.

Step-by-step solution

Determine the Net Conductivity of the Saturated Solution

To find the solubility of AgCl, first calculate the net conductivity of the saturated AgCl solution by subtracting the conductivity of the water from the total conductivity of the solution. Use the formula: net conductivity = conductivity of solution - conductivity of water.

Calculate the Molar Conductivity of AgCl in the Saturated Solution

Then, determine the molar conductivity of the saturated AgCl solution by using the net conductivity. The molar conductivity is found by dividing the net conductivity of the AgCl solution by the solubility (which is equal to the molar concentration of the AgCl in the saturated solution, since it dissociates completely into Ag+ and Cl- ions).

Find the Solubility of AgCl

The solubility of AgCl can now be calculated by rearranging the modified version of the Kohlrausch's Law: solubility (mol/L) = net conductivity of saturated solution / molar conductivity at infinite dilution.

Match the Calculated Solubility with the Given Options

Finally, solve the equation from Step 3 to find the solubility of AgCl in moles per litre and match it with one of the provided answer options.

Key concepts

Conductivity of Solution

The concept of conductivity of solution is fundamental when exploring the properties of ionic compounds in a solvent. Conductivity refers to a material's ability to conduct an electric current, and in the case of solutions, it's directly related to the presence of ions that can move freely in the solution.

In an educational context, it's important to visualize ions in solution as tiny charged particles that can move under the influence of an electric field. When a voltage is applied across a solution, these ions migrate towards the electrodes, positively charged ions (cations) moving to the cathode, and negatively charged ions (anions) moving to the anode. This movement results in an electric current, which is measured as the solution's conductivity. This concept is pertinent to our exercise involving the conductivity of a saturated solution of AgCl.

Plainly put, when AgCl (silver chloride) dissolves in water, it disassociates into Ag⁺ and Cl^- ions. The conductivity of this solution is partly due to these free-moving ions. To find the solubility of AgCl, understanding that pure water has its own conductivity, which is usually much lower than that of a salt solution, is essential. Subsequently, the net conductivity of the solution—essential for determining solubility—is the difference between the conductivity of the saturated AgCl solution and that of the water used.

Molar Conductivity

Diving into the concept of molar conductivity, it's essentially the conductivity of a solution normalized to the concentration of the electrolyte present. This term helps us understand and compare the ability of different substances to conduct electricity in a solution on a per mole basis.

It is defined as the conductivity of an electrolyte solution divided by the molar concentration of the electrolyte. Mathematically, it looks like this: \( \Lambda_m = \frac{\kappa}{c} \) where \( \Lambda_m \) signifies molar conductivity, \( \kappa \) denotes conductivity of the solution, and \( c \) represents the molar concentration of the electrolyte.

When we speak of AgCl in the given exercise, we're assuming it disassociates completely into ions; the molar conductivity then is a measure of how well these ions carry the electric current. However, in a real scenario, not all compounds fully disassociate, which is why this value is an important descriptor of an electrolyte's behavior in solution. Since the electrolyte's concentration often affects its molar conductivity, scientists frequently refer to the molar conductivity at infinite dilution, where inter-ionic interactions are negligible, for standard comparisons.

Kohlrausch's Law

Kohlrausch's Law of Independent Migration of Ions is a key principle in the study of electrolytic conductivity. It states that at infinite dilution, each ion contributes to the total conductivity of the electrolyte independently of the presence of other ions. This allows for the calculation of the molar conductivity of an electrolyte at infinite dilution by adding the contributions of its individual ions.

Applied to our exercise, Kohlrausch's Law helps in determining the molar conductivity of AgCl at infinite dilution, which is a measure of the highest possible conductivity per mole of electrolyte when interactions between ions are absent. With the molar conductivity at infinite dilution, one can find the solubility of AgCl by rearranging the law as \( \text{solubility} = \frac{\text{net conductivity of saturated solution}}{\text{molar conductivity at infinite dilution}} \) – a process directly applied in the exercise.

Remember, Kohlrausch's Law is pivotal because it allows us to treat each type of ion separately, even in a mixture of different salts. This independence is essential when analyzing complex solutions and makes the study of individual ions' contributions to conductivity straightforward. It's fascinating how such straightforward principles are foundational to understanding complex phenomena in electrochemistry!