Question

The solubility product of \(\mathrm{AgCl}\) is \(10^{-10} \mathrm{M}^{2}\). The minimum volume (in \(\mathrm{m}^{3}\) ) of water required to dissolve \(14.35 \mathrm{mg}\) of \(\mathrm{AgCl}\) is approximately : (a) \(0.01\) (b) \(0.1\) (c) 100 (d) 10

Short answer

Approximately 0.1 m^3 of water is required.

Step-by-step solution

- Convert mass of AgCl to moles

Use the molar mass of AgCl to convert the given mass in milligrams to moles. The molar mass of AgCl (the sum of the atomic masses of silver and chlorine) is approximately 143.5 g/mol. Convert 14.35 mg to grams (by dividing by 1000), and then divide the mass in grams by the molar mass to find the number of moles of AgCl.

- Calculate the molar concentration of AgCl

Use the solubility product constant (Ksp) to calculate the molar concentration of AgCl at equilibrium. Since the solubility product of AgCl is given by \( Ksp = [Ag^+][Cl^-] \), and the stoichiometry of AgCl in water is 1:1, both \( [Ag^+] \) and \( [Cl^-] \) will be equal to the molar solubility (s) of AgCl. Therefore, \( Ksp = s^2 \) and \( s = \sqrt{Ksp} \).

- Calculate the minimum volume of water required

Since the number of moles of AgCl is equal to the product of its molarity and the volume of water in liters (using the molar solubility calculated from the previous step), you can rearrange the formula to solve for the volume: \( V = \frac{n}{s} \). Convert this volume from liters to cubic meters by multiplying by \(10^{-3}\).

- Compare the calculated volume with the options

Compare the calculated minimum volume of water required to dissolve 14.35 mg of AgCl with the given options to find the correct answer.

Key concepts

Understanding AgCl Solubility and Ksp

When we discuss the solubility of AgCl (Silver Chloride), we are looking at how well it can dissolve in water. Its solubility is quantitatively expressed by the solubility product constant (Ksp), which is a special type of equilibrium constant. For AgCl, which dissociates into Ag+ and Cl- ions in water, its Ksp can be represented by the equation \(Ksp = [Ag^+][Cl^-]\). Since these ions are produced in a 1:1 ratio, we can say \(Ksp = s^2\), where \(s\) is the molar solubility, the concentration of each ion in mol/L.

With a Ksp value of 10^-10, we can calculate the molar solubility of AgCl by taking the square root of Ksp. This calculation gives us insight into the amount of AgCl that can dissolve in water to reach equilibrium. It's essential to understand that solubility is an intrinsic property of a substance, meaning that no matter how much you try to dissolve past the equilibrium point, the substance won't further dissolve until some of it is removed from the solution.

Molar Mass Conversion

The molar mass conversion process allows us to go from the mass of a substance to the number of moles it represents. To calculate this, we use the molar mass of the substance, which is the mass of one mole of its particles. For AgCl, the molar mass is approximately 143.5 g/mol. Molar mass acts as a conversion factor between the mass of a sample and the number of moles that sample contains.

For instance, if we have 14.35 mg of AgCl, we first convert this to grams (14.35 mg is 0.01435 g), and then divide by the molar mass of AgCl. This will give us the number of moles of AgCl we're working with. Understanding this conversion is crucial in stoichiometry and plays a significant role in determining the concentration of the substance in a solution.

The Relationship Between Molarity and Volume

Molarity, often denoted as M, is a measure of the concentration of a solute in a solution. It’s defined as the number of moles of solute present in one liter of solution. The relationship between moles, molarity, and volume of a solution is given by the equation \(n = MV\), where \(n\) represents the number of moles of the solute, \(M\) is the molarity, and \(V\) is the volume in liters.

When you solve for volume in this equation, you have \(V = n / M\). This allows us to calculate the volume of solvent needed to achieve a particular concentration given a specific amount of solute. This is especially important when dealing with solutions at equilibrium, such as when a substance like AgCl is dissolving in water up to its solubility limit.

Stoichiometry in Solubility

Stoichiometry is the part of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. In the context of solubility, stoichiometry involves the relationship between the amount of solute and solvent and their concentration in the solution.

For a compound like AgCl in water, we look at how the stoichiometry of its dissociation affects its solubility. As AgCl dissociates into Ag+ and Cl- in a 1:1 molar ratio, the stoichiometry directly informs us about the concentration of ions in the solution at equilibrium. This relationship is vital in calculations involving solubility, like determining how much of the solute can dissolve in a given volume of solvent, or conversely, how much solvent is needed to dissolve a certain amount of solute. Clear understanding of stoichiometry is essential to accurately predict and calculate the outcomes of dissolution processes.