Question

A \(5.00 \times 10^{2} \mathrm{~mL}\) sample of \(2.00 \mathrm{M} \mathrm{HCl}\) solution is treated with \(4.47 \mathrm{~g}\) of magnesium. Calculate the concentration of the acid solution after all the metal has reacted. Assume that the volume remains unchanged.

Short answer

The concentration of the acid solution is 1.264 M.

Step-by-step solution

Write the Balanced Chemical Equation

When hydrochloric acid (HCl) reacts with magnesium (Mg), they form magnesium chloride (MgCl₂) and hydrogen gas (H₂). The balanced chemical equation is:\[ \text{Mg} + 2 \text{HCl} \rightarrow \text{MgCl}_2 + \text{H}_2 \]

Calculate Moles of Magnesium

We have 4.47 g of magnesium. First, find the molar mass of magnesium, which is 24.31 g/mol. Then, calculate the number of moles:\[ \text{Moles of Mg} = \frac{4.47 \text{ g}}{24.31 \text{ g/mol}} = 0.184 \text{ moles} \]

Determine Moles of HCl Consumed

From the balanced chemical equation, 1 mole of Mg reacts with 2 moles of HCl. Therefore, 0.184 moles of Mg will react with:\[ 0.184 \text{ moles Mg} \times 2 = 0.368 \text{ moles HCl} \]

Calculate Initial Moles of HCl

We have a 500 mL (or 0.500 L) sample of 2.00 M HCl. Calculate the initial moles of HCl:\[ \text{Moles of HCl} = 2.00 \text{ M} \times 0.500 \text{ L} = 1.00 \text{ mole} \]

Calculate Remaining Moles of HCl

Subtract the moles of HCl that reacted from the initial moles to find the moles of HCl remaining:\[ 1.00 \text{ mole HCl} - 0.368 \text{ moles HCl} = 0.632 \text{ moles HCl} \]

Calculate Final Concentration of HCl

The volume of the solution remains unchanged at 0.500 L. Calculate the final concentration of HCl:\[ \text{Concentration} = \frac{0.632 \text{ moles}}{0.500 \text{ L}} = 1.264 \text{ M HCl} \]

Key concepts

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that involves the calculation of reactants and products in chemical reactions. It is based on the principle of the conservation of mass, meaning that matter is neither created nor destroyed in a chemical reaction. Thus, stoichiometry helps us calculate the amount of substances consumed or produced, allowing us to predict the outcomes of reactions.

In the context of the reaction between hydrochloric acid (HCl) and magnesium (Mg), stoichiometry is used to determine how many moles of each reactant are needed and how many moles of product will be formed. By using the balanced chemical equation, we identify the mole ratio between Mg and HCl, which tells us how they combine to form products. For instance, here, 1 mole of Mg reacts with 2 moles of HCl. This mole ratio is crucial for calculating the moles of more specific consumption or production needed in reactions, as demonstrated with calculating the moles from given masses.

Molarity

Molarity is a measure of the concentration of a solution. It is determined by the number of moles of solute dissolved in one liter of solution. Molarity makes it easier to relate volumes and amounts of solute across different solutions and reactions.

For the HCl solution in our exercise, we initially have a 2.00 M solution in 0.500 L. Therefore, we start with 1.00 mole of HCl for the reaction calculation. Knowing the molarity allows us to understand how much of the acid is available to react with magnesium. After the reaction, and calculating how many moles of HCl remain, we can find the new molarity. This involves dividing the moles of HCl remaining by the volume of the solution, which hasn't changed (0.500 L), providing a final molarity of 1.264 M.

Balanced Chemical Equation

A balanced chemical equation is essential for representing a chemical reaction accurately. It ensures that the number of atoms for each element is the same on both the reactants and products sides, adhering to the law of conservation of mass.For the reaction between HCl and Mg, the balanced equation is:\[ \text{Mg} + 2 \text{HCl} \rightarrow \text{MgCl}_2 + \text{H}_2 \]This equation tells us that one atom of magnesium reacts with two molecules of hydrochloric acid to produce one molecule of magnesium chloride and hydrogen gas. The coefficients (numbers before the chemical formulas) are critically important because they provide the exact stoichiometric relationship between reactants and products. Adjustments in balancing are often required to ensure that mass is conserved and the formula reflects reality.

Moles Calculation

Calculating moles is a central aspect of stoichiometry and generally involves converting grams to moles using the substance’s molar mass. In our exercise, to find the moles of magnesium, we used its molar mass: the number of grams per mole. Given 4.47 g of Mg, using its molar mass of 24.31 g/mol, the calculation was:\[ \text{Moles of Mg} = \frac{4.47 \text{ g}}{24.31 \text{ g/mol}} = 0.184 \text{ moles} \]After that, we used the mole ratio from the balanced equation to find how many moles of HCl will react, leading to further calculations of remaining moles and concentration. These calculations are necessary for translating mass into a form that directly relates to chemical quantities in reactions, providing insights into how reactions proceed on a molecular level.