Problem 106 - General Chemistry: Principles and Modern Applications

Question

A small piece of zinc is dissolved in \(50.00 \mathrm{mL}\) of \(1.035 \mathrm{M}\) HCl. At the conclusion of the reaction, the concentration of the \(50.00 \mathrm{mL}\) sample is redetermined and found to be \(0.812 \mathrm{M} \mathrm{HCl} .\) What must have been the mass of the piece of zinc that dissolved? $$\mathrm{Zn}(\mathrm{s})+2 \mathrm{HCl}(\mathrm{aq}) \longrightarrow \mathrm{ZnCl}_{2}(\mathrm{aq})+\mathrm{H}_{2}(\mathrm{g})$$

Step-by-Step Solution

Answer

The mass of the piece of zinc that dissolved is approximately \(0.364 \, \text{grams}\).

Step 1: Calculate Initial Moles of Hydrochloric Acid

The initial moles of hydrochloric acid are obtained by multiplying the initial concentration by the volume (Expressed in litres). Which equals to \(1.035 \, \text{M} \times 0.05 \, \text{L} = 0.05175 \, \text{mol}\).

Step 2: Calculate Final Moles of Hydrochloric Acid

Similar to the first step, calculate the final moles of hydrochloric acid. Multiply the final concentration by the volume: \(0.812 \, \text{M} \times 0.05 \, \text{L} = 0.0406 \, \text{mol}\).

Step 3: Calculate Moles of Zinc

The reaction between zinc and hydrochloric acid is a 1:2 reaction, meaning 1 mole of zinc reacts with 2 moles of hydrochloric acid. So, the moles of zinc that reacted would be half of the moles of hydrochloric acid that reacted. Therefore, subtract the final moles of HCl from the initial moles to get the moles of HCl that reacted and divide by 2 to get the moles of zinc that reacted: \((0.05175 - 0.0406) \, \text{mol} / 2 = 0.005575 \, \text{mol}\).

Step 4: Find Mass of Zinc

Finally, convert moles of zinc to grams using the molar mass of zinc (65.38 g/mol): \(0.005575 \, \text{mol} \times 65.38 \, \text{g/mol} = 0.364 \, \text{g}\).

Key Concepts

Understanding Molarity

Molarity is a useful way to describe the concentration of a solution. It measures how many moles of solute are present in one liter of solution. For example, \(1.035 \, \mathrm{M}\) HCl means that there are approximately \(1.035\) moles of hydrochloric acid (HCl) in each liter of the solution. In exercises like these, converting volume from milliliters to liters is essential. This conversion makes it easier to use molarity in calculations:

• Divide milliliters by 1000 to get liters (e.g., \(50.00 \, \mathrm{mL} = 0.05000 \, \mathrm{L}\)
• Use molarity and liters to find moles: \(\text{Molarity} \times \text{Volume in Liters} = \text{Moles of solute}\)

The step-by-step solution uses this principle to calculate the moles of hydrochloric acid at the beginning and end of the reaction.

Diving Into Chemical Reactions

Chemical reactions describe the process where one or more substances (reactants) are transformed into other substances (products). They can be represented by balanced chemical equations. Here, zinc reacts with hydrochloric acid to form zinc chloride and hydrogen gas:

• \(\mathrm{Zn}(\mathrm{s}) + 2 \mathrm{HCl}(\mathrm{aq}) \rightarrow \mathrm{ZnCl}_2(\mathrm{aq}) + \mathrm{H}_2(\mathrm{g})\)

This equation shows that one mole of zinc reacts with two moles of hydrochloric acid. This 1:2 stoichiometric ratio is critical for converting moles of HCl to moles of zinc in the calculations. Always make sure your chemical equations are balanced. Balanced equations ensure that the number of atoms for each element is consistent on both sides, confirming the law of conservation of mass is followed. For this exercise, we see that:

• Each mole of zinc consumes two moles of hydrochloric acid.
• This relationship helps determine how much zinc was consumed in the reaction.

Mastering Mass Calculation

Once the moles of substances involved in a reaction are known, calculating the mass becomes straightforward if you know the substance's molar mass. The molar mass of a substance is the mass of one mole of that substance, measured in grams per mole (g/mol). For zinc, this value is \(65.38 \, \text{g/mol}\). To find the mass of zinc that has reacted, follow these steps:

• Determine the moles of zinc using stoichiometry from the balanced equation.
• Convert these moles to grams with the molar mass formula: \(\text{Moles} \times \text{Molar Mass} = \text{Mass}\).

In this exercise, after finding the moles of zinc, multiply them by zinc's molar mass \(65.38 \, \text{g/mol}\) to obtain the mass that dissolved. This step bridges the stoichiometric relationship and the tangible mass you're tasked to find.