Question

Zinc and magnesium metal each react with hydrochloric acid according to the following equations: $$ \begin{array}{l} \mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g) \\ \mathrm{Mg}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{MgCl}_{2}(a q)+\mathrm{H}_{2}(g) \end{array} $$ A \(10.00-\mathrm{g}\) mixture of zinc and magnesium is reacted with the stoichiometric amount of hydrochloric acid. The reaction mixture is then reacted with \(156 \mathrm{~mL}\) of \(3.00 M\) silver nitrate to produce the maximum possible amount of silver chloride. a. Determine the percent magnesium by mass in the original mixture. b. If \(78.0 \mathrm{~mL}\) of \(\mathrm{HCl}\) was added, what was the concentration of the \(\mathrm{HCl}\) ?

Short answer

a. The percent magnesium by mass in the original mixture is 24.69%. b. The concentration of the hydrochloric acid used was 12.0 M.

Step-by-step solution

Write down the balanced chemical equation for the reactions

First, let's write down the given balanced chemical equations: Zinc reacting with hydrochloric acid: \( \mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g) \) Magnesium reacting with hydrochloric acid: \( \mathrm{Mg}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{MgCl}_{2}(a q)+\mathrm{H}_{2}(g) \) Silver nitrate reacting with zinc chloride and magnesium chloride to produce silver chloride: \( \mathrm{ZnCl}_{2}(a q) + 2 \mathrm{AgNO}_{3}(a q) \longrightarrow 2 \mathrm{AgCl}(s) + \mathrm{Zn(NO}_{3})_{2}(a q) \) \( \mathrm{MgCl}_{2}(a q) + 2 \mathrm{AgNO}_{3}(a q) \longrightarrow 2 \mathrm{AgCl}(s) + \mathrm{Mg(NO}_{3})_{2}(a q) \)

Calculate moles of silver chloride produced

Given that 156 mL of a 3.00 M silver nitrate solution is used, we can first calculate the moles of silver nitrate available to react with the zinc and magnesium chloride solution. Moles of silver nitrate = volume × concentration = 0.156 L × 3.00 mol/L = 0.468 mol Now we know that 1 mol of zinc chloride or magnesium chloride reacts with 2 mol of silver nitrate to produce 2 mol of silver chloride, so we can calculate the total moles of silver chloride produced: Total moles of AgCl = 0.468 mol AgNO₃ × (2 mol AgCl / 2 mol AgNO₃) = 0.468 mol AgCl

Calculate moles of magnesium and zinc in the reaction

Now, we will calculate the moles of magnesium and zinc involved in the reaction. Let x be the moles of magnesium: Moles of zinc = (0.468 - x) mol (since 1 mol of zinc chloride reacts with 1 mol of magnesium chloride to produce 1 mol of silver chloride)

Convert moles to mass and set up an equation

Convert the moles of zinc and magnesium in the reaction to mass: Mass of Mg = x mol × (24.31 g/mol) Mass of Zn = (0.468 - x) mol × (65.39 g/mol) Given that the mass of the mixture of zinc and magnesium is 10.00 grams, we can set up an equation: Mass of Mg + Mass of Zn = 10.00 g 24.31x + 65.39(0.468 - x) = 10.00 Now, solve for x (moles of magnesium):

Solve the equation for x and determine the percentage of magnesium

24.31x + 65.39 × 0.468 - 65.39x = 10.00 x = 0.1016 (mol of Mg) Now, calculate the mass of magnesium in the 10.00-gram mixture: Mass of Mg = 0.1016 mol × 24.31 g/mol = 2.469 g Calculate the percentage of magnesium by mass: Percentage of Mg = (mass of Mg / mass of mixture) × 100% Percentage of Mg = (2.469 g / 10.00 g) × 100% = 24.69% Thus, a. the percent magnesium by mass in the original mixture is 24.69%.

Calculate the concentration of hydrochloric acid

Since magnesium and zinc both react with 2 moles of hydrochloric acid for every mole of metal, we can calculate the total moles of HCl needed for the reaction: Total moles of HCl = 2 × (0.1016 mol Mg + (0.468 - 0.1016) mol Zn) = 2 × 0.468 mol = 0.936 mol HCl Now, use the volume of hydrochloric acid given (78.0 mL) and the total moles of HCl to calculate the concentration of the solution: Concentration of HCl = moles of HCl / volume of HCl = 0.936 mol / 0.078 L = 12.0 M Thus, b. the concentration of the hydrochloric acid used was 12.0 M.

Key concepts

Mole Concept

When studying chemical reactions, it's essential to understand the mole concept. A mole is a unit used in chemistry to express amounts of a chemical substance, defined as the amount of any substance that contains the same number of elemental entities as there are atoms in 12 grams of pure carbon-12. This number is known as Avogadro's number, which is approximately 6.022×10²³ entities per mole.

For example, in a chemical reaction where zinc reacts with hydrochloric acid, the moles of zinc can be calculated by dividing the mass of zinc by its molar mass. Understanding the mole concept allows us to relate masses of substances involved in chemical reactions, which is fundamental in solving stoichiometry problems. It's essentially a bridge between the microscopic world of atoms and the macroscopic world we can measure.

Chemical Equation Balancing

Balancing chemical equations is a critical step in understanding chemical reactions. The law of conservation of mass states that mass is neither created nor destroyed in a chemical reaction. Therefore, to reflect this law, a chemical equation must have the same number of each type of atom on both the reactant and product sides.

In the given equations for zinc and magnesium reacting with hydrochloric acid, the coefficients indicate the stoichiometric ratios necessary to ensure the chemical reaction is balanced. In these cases, each mole of metal reacts with two moles of HCl, a crucial factor when calculating the reactants' amounts or the products formed in a reaction. Balancing equations is the first step in calculating stoichiometric quantities which leads to understanding the entire reaction process.

Percent Composition by Mass

The percent composition by mass is a way to express the relative mass contribution of each element within a compound or a mixture. It is calculated as the mass of a particular component divided by the total mass of the compound or mixture, multiplied by 100%.

In the context of our exercise, which involves a mixture of zinc and magnesium, determining the percent composition by mass of magnesium helps us know the proportion of magnesium in the mixture. This information is vital in various fields, including material science and chemistry, as it influences the properties and the behavior of the mixture. Calculating such percentages aids in adjusting proportions of ingredients in alloys, medicines, and other chemical products.

Molarity

Molarity is a measure of the concentration of a solute in a solution, or in other words, the amount of solute divided by the volume of the solution. It is expressed in moles per liter (M).

In the example we're discussing, the molarity of the hydrochloric acid solution helps us understand how concentrated the acid solution is and therefore influence the reaction with zinc and magnesium. By calculating the molarity, we know the precise amount of substance in a given volume, which is fundamental when mixing solutions for reactions to ensure proper chemical proportions are combined. Molarity is a critical concept not only in chemistry labs but also in industries such as pharmaceuticals and food technology, where precise chemical reactions are key to product quality.