Question

Zinc and magnesium metal each react with hydrochloric acid according to the following equations: $$\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ $$\mathrm{Mg}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{MgCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ A 10.00-g mixture of zinc and magnesium is reacted with the stoichiometric amount of hydrochloric acid. The reaction mixture is then reacted with 156 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 mL of HCl was added, what was the concentration of the HCl?

Short answer

a. The mass percentage of magnesium in the original mixture is 20.58%. b. The concentration of the hydrochloric acid added to the mixture is 6.00 M.

Step-by-step solution

Moles of AgNO3

\(moles_{AgNO3} = (concentration_{AgNO3})(volume_{AgNO3})\) To find moles of AgNO3, we need to multiply the given concentration (3.00 M) with the given volume (156 mL). \(moles_{AgNO3} = 3.00\,M \times 0.156\,L = 0.468\,mol\) Since AgNO3 reacts with chloride ions (from ZnCl2 and MgCl2) in a 1:1 ratio, the amount of chloride ions is equal to the amount of AgNO3.

Moles of chloride ions

\(moles_{Cl^-} = moles_{AgNO3} = 0.468\,mol\) **Step 2: Obtain the equations and combine them** Now, let's put the zinc and magnesium reactions equations into the form of moles.

Equations

\(\frac{1}{2} moles_{Cl^-} = moles_{Zn}\) (from Zn reaction) \(\frac{1}{2} moles_{Cl^-} = moles_{Mg}\) (from Mg reaction) Adding both equations:

Combined equation

\(\frac{1}{2} moles_{Cl^-} = moles_{Zn} + moles_{Mg}\) **Step 3: Solve the combined equation** Now, let's solve the combined equation to find moles of Zn and Mg.

Calculate moles of Zn and Mg

\(\frac{1}{2} \times 0.468\, mol = moles_{Zn} + moles_{Mg}\) \(0.234\,mol = moles_{Zn} + moles_{Mg}\) **Step 4: Find the mass of Zn and Mg** Since we found the moles of Zn and Mg combined (0.234 mol), we can calculate their individual masses, which must sum up to the total mass of the mixture (10.00 g). Let mass of Zn = x grams mass of Mg = (10 - x) grams We will now transform the moles equation using the molar mass of Zn (65.38 g/mol) and Mg (24.305 g/mol).

Mass equation

\(\frac{x}{65.38} + \frac{10-x}{24.305} = 0.234\) **Step 5: Solve the mass equation and find mass percentage of Mg** Now, we need to solve the equation for x (= mass of Zn) and find the mass of magnesium.

Solving Equation

Using a calculator, we find that \(x = 7.942\,g\). So, mass of Mg = 10 - 7.942 = 2.058 g Now, let's find the mass percentage of magnesium (Mg) in the mixture:

Mass percentage of Mg

\(MassPercentage_{Mg} = \frac{mass_{Mg}}{(mass_{Mg} + mass_{Zn})} \times 100\%\) \(MassPercentage_{Mg} = \frac{2.058\,g}{(10.00\,g)} \times 100\% = 20.58\%\), a. Thus, the mass percentage of magnesium in the original mixture is 20.58%. **Step 6: Find the concentration of HCl** Finally, we find the concentration of HCl. We know 78.0 mL of HCl was added, and the stoichiometric amount was used. Since both Zn and Mg have reacted with 2 moles of HCl for every mole of the metal, the moles of HCl that reacted with both metals will be twice the total moles of Zn and Mg (0.234 mol).

Moles of HCl

\(moles_{HCl} = 2 \times 0.234\,mol = 0.468\,mol\) Now, we can calculate the concentration of HCl:

HCl Concentration

\(concentration_{HCl} = \frac{moles_{HCl}}{volume_{HCl}}\) \(concentration_{HCl} = \frac{0.468\,mol}{0.078\,L} = 6.00\,M\) b. The concentration of the hydrochloric acid added to the mixture is 6.00 M.

Key concepts

Chemical Reactions

Understanding chemical reactions is crucial for stoichiometry, which involves the quantitative relationships between reactants and products in a chemical transformation. In the provided exercise, zinc (Zn) and magnesium (Mg) react with hydrochloric acid (HCl) to perform distinct reactions. These reactions release chloride ions that subsequently react with silver nitrate (AgNO3) to form silver chloride (AgCl). By balancing the chemical equations, we recognize that each zinc or magnesium atom reacts with two HCl molecules to form respective metal chlorides (ZnCl2 and MgCl2) and hydrogen gas (H2).
This reaction highlights the fundamental concept of stoichiometry: using balanced chemical equations to determine the proportions of reactants and products. The coefficients in the equations allow us to calculate relative amounts of each substance, which we can then convert into measurable quantities like moles.

Mass Percentage

Mass percentage allows us to understand the composition of a mixture by showing how much of a particular substance is present in a given mass of the total mixture. In this exercise, we determine the mass percentage of magnesium in a mixture of zinc and magnesium.
To find this percentage, we calculated the mass of magnesium present in the mixture. With the mixture's total mass known, the formula for mass percentage is applied:

• Mass Percentage = \( \frac{\text{mass of component}}{\text{total mass of mixture}} \times 100\% \)

Applying this formula to magnesium, we determined its contribution to the total mass of the mixture. Understanding mass percentage helps in comparing the proportions of components in a mixture, which can be crucial in various scientific and industrial applications.

Molar Concentration

The concept of molar concentration refers to the number of moles of a solute present in a specific volume of solution. It is expressed as moles per liter (M) and is vital for calculating the amounts of substances in reactions. In the exercise, the concentration of AgNO3 is used to determine how many moles of silver nitrate react.
Molar concentration is also relevant in the second part of the problem, where we determine the concentration of hydrochloric acid (HCl) added to the reaction. The procedure involves calculating the moles of HCl used, which is derived from the stoichiometry of the reaction.
Using the formula for molar concentration:

• Concentration = \( \frac{\text{moles of solute}}{\text{volume of solution in liters}} \)

This allows us to find the molarity of HCl, which aids in understanding how much acid is necessary for complete reaction with metals.

Mole Calculation

Mole calculation is a fundamental practice in chemistry, enabling the transition from the atomic scale to amounts we can measure in the lab. A mole represents a count of Avogadro's number (approximately \(6.022 \times 10^{23} \)) of molecules or atoms. This concept helps in quantifying the substances involved in reactions.
For the exercise, we calculated the total moles of AgNO3 initially and used these to find the chloride ions, since each mole of AgNO3 reacts 1:1 with the chloride ions. Subsequently, we calculated the moles of HCl needed to react with the zinc and magnesium.
The equations used include:

• \(moles = \frac{\text{mass}}{\text{molar mass}} \)
• \(moles = \text{concentration} \times \text{volume} \)

Mastery of mole calculations is vital for translating between mass, volume, and number of particles, facilitating the practical application of chemical knowledge.