Question

The specific conductance of \(\mathrm{AgCl}\) solution in water was determined to be \(1.8 \times 10^{-6} \Omega^{-1} \mathrm{~cm}^{-1}\) at \(298 \mathrm{~K}\). The molar conductances at infinite dilution, of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\) are \(67.9\) and \(82.1 \Omega^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\), respectively. What is the solubility of AgCl in water? (a) \(1.2 \times 10^{-8} \mathrm{M}\) (b) \(1.44 \times 10^{-10} \mathrm{M}\) (c) \(1.2 \times 10^{-5} \mathrm{M}\) (d) \(1.44 \times 10^{-16} \mathrm{M}\)

Short answer

The solubility of AgCl in water is \(1.2 \times 10^{-8} \text{M}\) (option (a)).

Step-by-step solution

Calculate the total molar conductance at infinite dilution

Add the molar conductances of Ag+ and Cl- ions at infinite dilution to get the total molar conductance. \[\Lambda_{\infty}(\text{AgCl}) = \Lambda_{\infty}(\text{Ag}^+) + \Lambda_{\infty}(\text{Cl}^-)\]\[\Lambda_{\infty}(\text{AgCl}) = 67.9 + 82.1 \Omega^{-1} \text{cm}^2 \text{mol}^{-1}\]\[\Lambda_{\infty}(\text{AgCl}) = 150 \Omega^{-1} \text{cm}^2 \text{mol}^{-1}\]

Calculate the molar concentration of AgCl

Use the formula for molar conductance to find the molarity of AgCl. The specific conductance (kappa, \(\kappa\)) of the solution relates to the molar conductance (\(\Lambda_m\)) and the concentration (c) as follows:\[\Lambda_m = \frac{\kappa}{c}\]Rearrange to find the concentration (c):\[c = \frac{\kappa}{\Lambda_{\infty}(\text{AgCl})}\]Substitute the given values:\[c = \frac{1.8 \times 10^{-6} \Omega^{-1} \text{cm}^{-1}}{150 \Omega^{-1} \text{cm}^2 \text{mol}^{-1}}\]\[c = 1.2 \times 10^{-8} \text{M}\]

Match the calculated solubility to the given options

Compare the calculated solubility of AgCl with the given answer choices to find the correct answer.\[c = 1.2 \times 10^{-8} \text{M}\]This corresponds to option (a), which states \(1.2 \times 10^{-8} \text{M}\) as the solubility of AgCl in water.

Key concepts

Molar Conductance

Molar conductance is a critical concept in understanding how well an ionic compound conducts electricity when dissolved in a solvent. It is defined as the conductance of a solution containing one mole of electrolyte and is mathematically expressed in units of ohm^{-1} cm^2 mol^{-1}. At infinite dilution, when there are no inter-ionic interactions, this conductance reaches a maximum value denoted as \(\Lambda_{\infty}\). This value is crucial for various calculations, including the determination of solubility as described in the exercise.

In our context, to calculate the total molar conductance of AgCl at infinite dilution, the molar conductances of the individual ions, \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\), are summed. Since individual ions contribute to the overall conductance based on their mobility, the more freely an ion moves through the solution, the greater its impact on molar conductance. Understanding this concept not only helps in calculating solubility but also provides insight into the fundamental properties of ions in a solution.

Electrolytic Conductivity

Electrolytic conductivity, also known as specific conductance, is a measure of a solution’s ability to conduct electric current due to the movement of ions. It is actually what the exercise references when mentioning the 'specific conductance' of AgCl solution. Its unit is \(\Omega^{-1} \mathrm{~cm}^{-1}\) and is symbolized by \(\kappa\). The higher the electrolytic conductivity, the higher the concentration of ions in the solution, and hence, the better the solution conducts electricity.

The relationship between molar conductance \(\Lambda_m\) and specific conductance \(\kappa\) is used in the exercise to uncover the solubility of AgCl. By rearranging the formula, \(c = \frac{\kappa}{\Lambda_{\infty} (\text{AgCl})}\), we can determine the molar concentration of the dissolved substance, which directly correlates with solubility. This application illustrates the practical use of understanding conductivity when dealing with solutions of electrolytes.

Chemical Equilibrium

Chemical equilibrium refers to a state in a chemical reaction where the rates of the forward and reverse reactions are equal, leading to no net change in the concentrations of reactants and products over time. It is an essential concept for comprehensively understanding solubility, as a saturated solution of a substance is an example of a dynamic equilibrium between the dissolved ions and the undissolved solid.

Understanding equilibrium is fundamental when considering solubility product constants, denoted as \(K_{sp}\), which is the product of the molar concentrations of the constituent ions, each raised to the power of its stoichiometric coefficient. This constant aids in calculations of a substance's solubility, as depicted in the exercise where AgCl's solubility is deduced. Recognizing the relation between chemical equilibrium and solubility allows for greater insight into the dissolution processes and the factors that can affect the solubility of a compound.