Imagine that you have a 0.007 50 M aqueous MgCl2 solu-tion, prepared so that it contains a small amount of radio-active 28Mg. The half-life of 28Mg is 20.91 h, and the initial activity of the MgCl2 solution is 0.112 mCi>mL. Assume that 20.00 mL of this MgCl2 solution is added to 15.00 mL of 0.012 50 M aqueous Na2CO3 solution and that the resultant precipitate is then removed by filtration to yield a clear filtrate. After a long break to go for a run, you find that the activity of the filtrate measured 2.40 h after begin-ning the experiment is 0.029 mCi>mL. What are the molar concentrations of Mg2+ and CO32- in the filtrate, and what is the solubility product constant of MgCO3?

Verified step by step guidance

1. Calculate the initial moles of Mg2+ in the MgCl2 solution by using the concentration and volume of the MgCl2 solution. Use the formula: moles = concentration (M) imes volume (L).
2. Calculate the initial moles of CO32- in the Na2CO3 solution using the same method as in step 1.
3. Determine the limiting reactant by comparing the mole ratio of Mg2+ to CO32-. The reaction between Mg2+ and CO32- forms MgCO3, which precipitates out. The balanced chemical equation is: Mg2+ + CO32- ightarrow MgCO3.
4. Calculate the moles of Mg2+ and CO32- remaining in the solution after the reaction by subtracting the moles of the limiting reactant used from the initial moles of each reactant.
5. Use the final volume of the mixture (sum of the volumes of MgCl2 and Na2CO3 solutions) to find the molar concentrations of Mg2+ and CO32- in the filtrate. Then, use the concentrations to calculate the solubility product constant (Ksp) for MgCO3 using the formula: Ksp = [Mg2+][CO32-].

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. The half-life is a key concept in this process, representing the time required for half of the radioactive atoms in a sample to decay. In this question, the half-life of 28Mg (20.91 hours) is crucial for determining how much of the radioactive isotope remains after a certain period, which affects the activity measured in the filtrate.

Precipitation Reactions

Precipitation reactions occur when two soluble salts react in solution to form an insoluble compound, known as a precipitate. In this scenario, Mg2+ ions from the MgCl2 solution react with CO32- ions from the Na2CO3 solution to form solid MgCO3. Understanding the stoichiometry of this reaction is essential for calculating the concentrations of the ions remaining in the filtrate after the precipitate is removed.

Solubility Product Constant (Ksp)

The solubility product constant (Ksp) is an equilibrium constant that applies to the solubility of sparingly soluble ionic compounds. It is defined as the product of the molar concentrations of the ions in a saturated solution, each raised to the power of their coefficients in the balanced equation. For MgCO3, Ksp can be calculated using the concentrations of Mg2+ and CO32- ions in the filtrate, providing insight into the compound's solubility in water.