Question

Suppose \(3.50 \mathrm{~g}\) of solid \(\mathrm{Mg}(\mathrm{OH})_{2}\) is added to \(30.0 \mathrm{~mL}\) of \(0.500 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{4}\) solution. What will the concentration of \(\mathrm{Mg}^{2+}\) be when all of the acid has been neutralized? How many grams of \(\mathrm{Mg}(\mathrm{OH})_{2}\) will not have dissolved?

Short answer

The concentration of \( \mathrm{Mg}^{2+} \) when all the acid has been neutralized is approximately 0.500 M, and the mass of \( \mathrm{Mg}(\mathrm{OH})_{2} \) that will not have dissolved is approximately 2.62 g.

Step-by-step solution

Write the chemical reaction and balance it

The neutralization reaction between magnesium hydroxide, \( \mathrm{Mg}(\mathrm{OH})_{2} \), and sulfuric acid, \( \mathrm{H}_{2}\mathrm{SO}_{4} \), is \( \mathrm{Mg}(\mathrm{OH})_{2}(s) + \mathrm{H}_{2}\mathrm{SO}_{4}(aq) \rightarrow \mathrm{MgSO}_{4}(aq) + 2 \mathrm{H}_{2}\mathrm{O}(l) \). This reaction is already balanced.

Calculate moles of \( \mathrm{H}_{2}\mathrm{SO}_{4} \)

Use the concentration and volume to calculate the moles of \( \mathrm{H}_{2}\mathrm{SO}_{4} \): moles of \( \mathrm{H}_{2}\mathrm{SO}_{4} \) = 0.500 M \( \times \) 0.030 L = 0.015 moles.

Calculate moles of \( \mathrm{Mg}(\mathrm{OH})_{2} \) initially present

Calculate the initial moles of \( \mathrm{Mg}(\mathrm{OH})_{2} \) using its molar mass: molar mass of \( \mathrm{Mg}(\mathrm{OH})_{2} \approx 58.3 \mathrm{~g/mol} \). Moles of \( \mathrm{Mg}(\mathrm{OH})_{2} \) = \( \frac{3.50 \mathrm{~g}}{58.3 \mathrm{~g/mol}} \approx 0.0600 \) moles.

Determine the limiting reactant

From the balanced equation, 1 mole of \( \mathrm{H}_{2}\mathrm{SO}_{4} \) reacts with 1 mole of \( \mathrm{Mg}(\mathrm{OH})_{2} \). Comparing the moles, \( \mathrm{H}_{2}\mathrm{SO}_{4} \) is the limiting reactant with 0.015 moles. Thus, all of the acid will react.

Calculate moles of \( \mathrm{Mg}^{2+} \) produced

Based on the limiting reactant, 0.015 moles of \( \mathrm{H}_{2}\mathrm{SO}_{4} \) will produce 0.015 moles of \( \mathrm{Mg}^{2+} \) ions.

Calculate the final concentration of \( \mathrm{Mg}^{2+} \)

The final volume is approximately the original volume of sulfuric acid solution since the solid has minimal volume effect, so concentration of \( \mathrm{Mg}^{2+} \) = \( \frac{0.015 \text{ moles}}{0.030 \text{ L}} \approx 0.500 \text{ M} \).

Calculate the mass of \( \mathrm{Mg}(\mathrm{OH})_{2} \) not dissolved

Calculate the unused moles of \( Mg(OH)_2 \): initial moles - used moles = 0.0600 moles - 0.015 moles = 0.0450 moles. Mass of undissolved \( Mg(OH)_2 \) = 0.0450 moles * 58.3 g/mol \approx 2.62 g.

Key concepts

Stoichiometry

Stoichiometry is the section of chemistry that involves using balanced chemical equations to calculate the quantities of reactants and products involved in a chemical reaction. Think of it as the recipe for a chemical reaction, telling you how much of each substance you need and how much you will end up with at the end.

For example, in the neutralization reaction between magnesium hydroxide, \( \text{Mg(OH)_2} \), and sulfuric acid, \( \text{H_2SO_4} \), the balanced chemical equation is: \[ \text{Mg(OH)_2} + \text{H_2SO_4} \rightarrow \text{MgSO_4} + 2\text{H_2O} \]. This tells us that one mole of magnesium hydroxide reacts with one mole of sulfuric acid to produce one mole of magnesium sulfate and two moles of water.

To solve stoichiometry problems, you need to follow these steps:

Limiting Reactant

The concept of the limiting reactant is critical in chemical reactions, as it determines the amount of product that can be formed. The limiting reactant is the substance that is totally consumed in the chemical reaction and thus limits the amount of product formed.

In our exercise, sulfuric acid is the limiting reactant since there are fewer moles of it compared to magnesium hydroxide. Once all of the sulfuric acid is used up, the reaction stops, even if there is some magnesium hydroxide left over. Being able to identify the limiting reactant is essential in predicting the outcome of the reaction and is a staple in stoichiometric calculations.

How to Identify the Limiting Reactant:

Molar Concentration

Molar concentration, often referred to as molarity, is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters. The unit for molarity is moles per liter (M).

Using molarity is helpful when dealing with solutions where the volume is often known, such as in our given problem, where the concentration of sulfuric acid is given as \(0.500\text{ M}\). To find out how the reaction affects the concentration of ions in solution, you simply divide the number of moles of your substance of interest by the total volume of the solution.

Applying Molar Concentration to our Problem:

Once the limiting reactant, sulfuric acid, is fully reacted, we determine the concentration of magnesium ions, \(\text{Mg}^{2+}\), by dividing the moles of \(\text{Mg}^{2+}\) produced by the final volume of the solution.