Question

A saturated solution of silver arsenate, \(\mathrm{Ag}_{3} \mathrm{AsO}_{4},\) contains \(8.5 \times 10^{-7} \mathrm{~g} \mathrm{Ag}_{3} \mathrm{AsO}_{4}\) per mL. Calculate the \(K_{\mathrm{sp}}\) of silver arsenate. Assume that there are no other reactions but the \(K_{\mathrm{sp}}\) reaction.

Short answer

The \(K_{sp}\) of silver arsenate is \(4.57 \times 10^{-35}\).

Step-by-step solution

Calculate Molar Mass of Silver Arsenate

First, find the molar mass of silver arsenate (\(\mathrm{Ag}_3\mathrm{AsO}_4\)). Silver's atomic mass is approximately 107.87 g/mol, arsenic is 74.92 g/mol, and oxygen is 16.00 g/mol. Thus, the molar mass is:\[3(107.87) + 74.92 + 4(16.00) = 418.54 \, \text{g/mol}\]

Calculate Molar Concentration

Next, convert the mass of \(\mathrm{Ag}_3\mathrm{AsO}_4\) in one mL to moles to find the molar concentration. Since 1 mL of solution contains \(8.5 \times 10^{-7} \text{ g}\), use the molar mass:\[\frac{8.5 \times 10^{-7} \, \text{g}}{418.54 \, \text{g/mol}} = 2.03 \times 10^{-9} \, \text{mol/L}\]

Write the Dissociation Equation

Silver arsenate dissociates according to the equation:\[\mathrm{Ag}_3\mathrm{AsO}_4 (s) \rightleftharpoons 3\mathrm{Ag}^+ (aq) + \mathrm{AsO}_4^{3-} (aq)\]This means from 1 mole of \(\mathrm{Ag}_3\mathrm{AsO}_4\), we get 3 moles of \(\mathrm{Ag}^+\) and 1 mole of \(\mathrm{AsO}_4^{3-}\).

Calculate Ion Concentrations

Using the stoichiometry from the dissociation equation, calculate the concentrations of the ions at equilibrium:\[[\mathrm{Ag}^+] = 3 \times 2.03 \times 10^{-9} \, \text{mol/L} = 6.09 \times 10^{-9} \, \text{mol/L}\]\[[\mathrm{AsO}_4^{3-}] = 2.03 \times 10^{-9} \, \text{mol/L}\]

Calculate the Solubility Product, \(K_{sp}\)

The solubility product expression for the given reaction is:\[K_{sp} = [\mathrm{Ag}^+]^3 [\mathrm{AsO}_4^{3-}]\]Substitute the equilibrium concentrations:\[K_{sp} = (6.09 \times 10^{-9})^3 (2.03 \times 10^{-9})\]\[K_{sp} = 4.57 \times 10^{-35}\]

Key concepts

Molar Mass Calculation

Calculating the molar mass of a compound is essential for determining how much of the substance is present in a given sample.
Silver arsenate, \(\mathrm{Ag}_3\mathrm{AsO}_4\), is composed of three silver atoms, one arsenic atom, and four oxygen atoms.

• Silver (Ag) has an atomic mass of approximately 107.87 g/mol.
• Arsenic (As) has an atomic mass of about 74.92 g/mol.
• Oxygen (O) has an atomic mass of around 16.00 g/mol.

To find the molar mass of silver arsenate, we combine these atomic masses as follows:\[3 \times 107.87 + 74.92 + 4 \times 16.00 = 418.54 \, \text{g/mol}\] This calculation gives us the molar mass of silver arsenate, ensuring we understand the weight of one mole of the compound.

Saturated Solution

A saturated solution is one where no more solute can dissolve in the solvent at a given temperature and pressure. The concentration of the solute in a saturated solution represents the solubility of the solute.
In this case, the saturated solution of silver arsenate contains \(8.5 \times 10^{-7} \, \text{g} \mathrm{Ag}_3 \mathrm{AsO}_4\) per milliliter.
The term "saturated" indicates that the solution has reached its maximum capacity of dissolved silver arsenate.
Any additional silver arsenate added would remain undissolved and precipitate out of the solution.

Ion Concentrations

When we talk about ion concentrations in a solution, we are referring to the molar amount of ions present per liter of solution. In a saturated solution, the ion concentration is directly influenced by the solubility of the compound.
Once you have the molar concentration of the dissolved compound, \(2.03 \times 10^{-9} \, \text{mol/L}\) for \(\mathrm{Ag}_3\mathrm{AsO}_4\), we calculate the ions formed from its dissociation.

• Each mole of \(\mathrm{Ag}_3\mathrm{AsO}_4\) dissociates into 3 moles of \(\mathrm{Ag}^+\) ions.
• The concentration of \(\mathrm{Ag}^+\) ions is thus \(3 \times 2.03 \times 10^{-9} = 6.09 \times 10^{-9} \, \text{mol/L}\).
• The concentration of \(\mathrm{AsO}_4^{3-}\) ions is \(2.03 \times 10^{-9} \, \text{mol/L}\).

Calculating these concentrations helps us later determine the solubility product constant, \(K_{sp}\), of the compound.

Dissociation Equation

The dissociation equation of a compound shows how it separates into its constituent ions in solution. For silver arsenate, the dissociation process can be represented as:\[\mathrm{Ag}_3\mathrm{AsO}_4 (s) \rightleftharpoons 3\mathrm{Ag}^+ (aq) + \mathrm{AsO}_4^{3-} (aq)\]In this equation, solid silver arsenate dissociates into three silver ions, \(\mathrm{Ag}^+\), and one arsenate ion, \(\mathrm{AsO}_4^{3-}\).
The equation is crucial for understanding how the ions distribute themselves in the solution.
This dissociation effectively defines the stoichiometry of the reaction—meaning the molar ratio in which different ions are produced.
This stoichiometry is used to calculate the equilibrium concentrations of the ions, which feed into the calculation of the solubility product, \(K_{sp}\), altogether providing a full picture of the compound's solubility behavior.