Problem 60 - General Chemistry: Principles and Modern Applications

Question

A method of lowering the concentration of \(\mathrm{HCl}(\mathrm{aq})\) is to allow the solution to react with a small quantity of Mg. How many milligrams of Mg must be added to \(250.0 \mathrm{mL}\) of \(1.023 \mathrm{M} \mathrm{HCl}\) to reduce the solution concentration to exactly \(1.000 \mathrm{M} \mathrm{HCl} ?\) $$ \mathrm{Mg}(\mathrm{s})+2 \mathrm{HCl}(\mathrm{aq}) \longrightarrow \mathrm{MgCl}_{2}(\mathrm{aq})+\mathrm{H}_{2}(\mathrm{g}) $$

Step-by-Step Solution

Answer

69.9 mg of Mg must be added to 250.0 mL of 1.023 M HCl to reduce the solution concentration to exactly 1.000 M HCl.

Step 1: Calculate moles of HCl before reaction

First, use the formula for molarity which is \( M = \frac{n}{V} \), where \(M\) is molarity, \(n\) is the number of moles and \(V\) is the volume of the solution in liters. Calculation of moles of 1.023 M HCl in 250.0 ml = 1.023 mole/L \(\times\) 0.250 L = 0.25575 mole. So, there are initially 0.25575 moles of HCl in the solution.

Step 2: Calculate moles of HCl after reaction

Then, calculate the amount of HCl needed to drop the concentration to 1.000 M in the 250 ml solution. Calculation: 1.000 M \(\times\) 0.250 L = 0.250 mole. So, after the reaction, there will be 0.250 moles of HCl left.

Step 3: Calculate moles of HCl reacted

Subtract the moles of HCl after the reaction from the moles of HCl before the reaction to find the moles of HCl reacted. Calculation: 0.25575 mole - 0.250 mole = 0.00575 mole. So, 0.00575 moles of HCl reacted with the magnesium.

Step 4: Calculate moles of Mg needed

Now, use the stoichiometry of the reaction to calculate the moles of Mg needed. From the reaction equation, 1 mole of Mg reacts with 2 mole of HCl. Therefore we get: 0.00575 mole of HCl \(\times\) \(\frac{1\,mole\,{Mg}}{2\,moles\,{HCl}} = 0.002875, which means 0.002875 moles of Mg are needed to lower the concentration of HCl to 1.000 M.

Step 5: Convert moles of Mg to milligrams

Finally, convert the moles of Mg to milligrams using the molar mass of Mg which is 24.3050 g/mol. Calculation: 0.002875 moles of Mg \(\times\) \(\frac{24.3050\,g}{1\,mole}\) \(\times\) \(\frac{1000\,mg}{1\,g} = 69.8775 mg\). Rounding to three significant digits gives us 69.9 mg.

Key Concepts

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, transform into different substances, called products. In the context of this exercise, the reaction between magnesium (Mg) and hydrochloric acid (HCl) is central. The overall balanced equation is:

• \(\text{Mg(s) + 2 HCl(aq) \rightarrow MgCl}_2\text{(aq) + H}_2\text{(g)}\)

This equation shows the stoichiometry: one mole of magnesium reacts with two moles of hydrochloric acid to produce one mole of magnesium chloride and one mole of hydrogen gas. Stoichiometry is vital as it helps us determine the proportions of reactants to use or products to expect from a reaction.
In this particular exercise, the chemical reaction assists in decreasing the concentration of HCl by reacting it with magnesium. Understanding the stoichiometric relationships allows us to calculate exactly how much magnesium is needed to lower the concentration of HCl to a desired level.

Molarity

Molarity is a way to express the concentration of a solution. It is defined as the number of moles of solute per liter of solution. It is expressed in moles per liter (M). In the initial step of our exercise, you calculate the number of moles of HCl in the solution using the molarity formula:

• \(M = \frac{n}{V}\) (where \(M\) is molarity, \(n\) is the number of moles, and \(V\) is the volume in liters).

In this case, for the initial 1.023 M concentration in 250 mL (or 0.250 L) of solution, the number of moles of HCl is calculated as 1.023 mol/L \(\times\) 0.250 L = 0.25575 moles. Molarity helps understand how concentrated a solution is, guiding how much reactant you need or how much product you can expect from a reaction.

Concentration Calculations

Concentration calculations are crucial for mixing solutions and chemical reactions in a precise manner. By performing these calculations, you can determine how much substance is present in a given volume and how these amounts change through reactions. The given exercise demonstrates the importance of this concept in determining the amount of magnesium needed to reach a target concentration of HCl. Initially, the total amount of HCl was 0.25575 moles. To reach the targeted concentration of 1.000 M within the same volume, the new number of moles needs to be 0.250 moles. This requires computing the moles that need to react, which is 0.25575 - 0.250 = 0.00575 moles of HCl. With the stoichiometry of the chemical equation, this converts to half the moles of Mg needed, leading to 0.002875 moles of Mg. Finally, we convert these moles of Mg to a mass in milligrams using its molar mass. Precise concentration calculations are indispensable in chemistry to ensure accurate and predictable reactions.