Question

A \(1.0 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) solution is slowly added to \(10.0 \mathrm{~mL}\) of a solution that is \(0.20 \mathrm{M}^{\text {in } \mathrm{Ca}^{++}}\)and \(0.30 \mathrm{M}\) in \(\mathrm{Ag}^{+}\). (a) Which compound will precipitate first: \(\mathrm{CaSO}_{4}\left(K_{\mathrm{sp}}=2.4 \times 10^{-5}\right)\) or \(\mathrm{Ag}_{3} \mathrm{SO}_{4}\left(K_{p}=1.5 \times 10^{-5}\right)\) ? (b) How much \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) solution must be added to initiate the precipitation

Short answer

(a) The compound Ag3SO4 will precipitate first. (b) Approximately \(50.0 \mu L\) of Na2SO4 solution must be added to initiate the precipitation.

Step-by-step solution

Calculate the initial concentrations of the ions

We are given the concentrations of Ca^{2+} and Ag^{+}, 0.20 M, and 0.30 M, respectively. Since there is no sulfate ion present at the beginning, the concentration of SO4^{2-} is 0.

Calculate the Ca^{2+} and Ag^{+} concentrations needed for precipitation

Using the solubility product constant for CaSO4, we can write the equation: \(K_{sp} = [Ca^{2+}][SO_4^{2-}]\) \(2.4 \times 10^{-5} = (0.20)[SO_4^{2-}]\) Solving for SO4^{2-}, we get: \[ [SO_4^{2-}] = \frac{2.4\times10^{-5}}{0.20} = 1.2\times10^{-4} M\] For Ag3SO4, we use the solubility product constant: \(K_{p}= [Ag^{3+}] [SO_4^{2-}]\) Taking the cube root of both sides, we get: \[[SO_4^{2-}] = \frac{1.5\times10^{-5}}{0.30}\] \[ [SO_4^{2-}] = 5.0\times10^{-5} M\] The concentration of sulfate needed for Ag3SO4 precipitation is lower which means it'll precipitate first. (a) The compound Ag3SO4 will precipitate first.

Calculate the amount of Na2SO4 added for initiating precipitation

To calculate the volume of Na2SO4 solution needed to initiate the precipitation of Ag3SO4, we must first find the amount of SO4^{2-} ions needed. \(\frac{5.0\times10^{-5} M SO_4^{2-}}{1.0 M Na_2SO_4} = 5.0\times10^{-5} L\) Initially, there are 0.01 L (10.0 mL) of the solution containing Ca^{2+} and Ag^{+} ions. So, the total volume of the solution after adding Na2SO4 is 0.01 + 5.0x10^{-5} L. Now, we can calculate the new concentration of Ag^{+} ions in the mixed solution: \[ [Ag^{+}]=\frac{0.01L \times 0.30M}{0.01 + 5.0\times10^{-5}L} \] After calculating, we'll find that the new concentration of Ag^{+} ions is still close to 0.30 M. Therefore, we can use this value to calculate the volume of Na2SO4 needed. \([[SO_4^{2-}] = 1.2\times10^{-4} M\) \([[SO_4^{2-}] = \frac{2.4\times10^{-5}}{0.20 M}\) (SO4^{2-} = \frac{V_{Na_2SO_4}}{10\mathrm{mL} + V_{Na_2SO_4}})\] \[1.0 M \times V_{Na_2SO_4} = 5\times10^{-5} M \times (0.01 + V_{Na_2SO_4})\] (b) Solving for \(V_{Na_2SO_4}\), we get approximately \(5.0\times10^{-5}L\) or \(50.0 \mu L\) of Na2SO4 solution must be added to initiate the precipitation.

Key concepts

Ionic Precipitation

Ionic precipitation is a process that occurs when oppositely charged ions in a solution come together to form an insoluble compound, called a precipitate. The likelihood of this occurring depends on the product of the ion concentrations reaching a certain threshold known as the solubility product constant (Ksp). If the ion product of the dissolved substances exceeds Ksp, the solution becomes supersaturated, and excess ions combine to precipitate out.

In the exercise, Ag₃SO₄ precipitates before CaSO₄ because the needed concentration of sulfate ions to reach the point of precipitation is lower for Ag₃SO₄, based on the provided Ksp values. During the mixing with Na₂SO₄, once the concentration of sulfate ions hits that critical value, precipitation occurs, leaving behind a solid silver sulfate while the rest of the ions remain in solution.

Ksp Calculations

The solubility product constant, or Ksp, represents the extent to which a compound can dissolve in water. It's calculated by multiplying the molar concentrations of the ions in a saturated solution, each raised to the power of its coefficient in the balanced equation. The smaller the Ksp value, the less soluble the compound is.

For example, to calculate the Ksp for calcium sulfate (CaSO₄), you would use the formula: \[ K_{sp} = [Ca^{2+}][SO_4^{2-}] \] Unlike most other equilibrium constants, Ksp does not include the concentration of solid precipitates. During the problem-solving process, understanding how to manipulate these values is crucial to predict whether a precipitate will form and determining which compound precipitates first. This is demonstrated in the step-by-step solution where the concentration of the ions was plugged into Ksp expressions to find the saturation point for each ionic compound.

Molar Solubility

Molar solubility refers to the number of moles of a solute that can dissolve in a liter of solution until reaching saturation. It is directly related to the Ksp and is a key concept in determining the amount of substance that can be dissolved before precipitation happens.

To initiate the precipitation in the given exercise, the molar solubility of Ag₃SO₄ in the presence of Ag⁺ and SO₄^2- ions was calculated using Ksp values. This resulted in finding the minimal amount of SO₄^2- needed from the Na₂SO₄ to start precipitation. The calculations revealed the volume of Na₂SO₄ required to reach this molar solubility, helping predict when the precipitate forms. Understanding molar solubility helps in accurately determining such points in diverse chemical applications.