Problem 81 - Chemistry The Molecular Science

Question

A natural brine found in Arkansas has a bromide ion concentration of \(5.00 \times 10^{-3} \mathrm{M}\). If \(210 . \mathrm{g} \mathrm{Cl}_{2}\) were added to \(1.00 \times 10^{3} \mathrm{~L}\) of the brine, (a) determine the limiting reactant. (b) calculate the theoretical yield of \(\mathrm{Br}_{2}(d=3.12 \mathrm{~g} / \mathrm{mL})\).

Step-by-Step Solution

Answer

(a) \( \mathrm{Br}^- \) is the limiting reactant. (b) Theoretical yield of \( \mathrm{Br}_2 \) is 399.5 grams.

Step 1: Write the Reaction Equation

The chemical reaction for the displacement of bromine by chlorine is: \[2 \mathrm{Br}^{-} + \mathrm{Cl}_{2} \rightarrow \mathrm{Br}_{2} + 2 \mathrm{Cl}^{-}.\]

Step 2: Calculate Moles of Reactants

First, calculate the moles of \( \mathrm{Br}^- \) ions present. Use the concentration and volume of the brine: \[\text{moles of } \mathrm{Br}^- = 5.00 \times 10^{-3} \mathrm{M} \times 1.00 \times 10^3 \mathrm{~L} = 5.00 \text{ moles}.\] Calculate the moles of \( \mathrm{Cl}_2 \) using its mass and molar mass (70.9 g/mol): \[\text{moles of } \mathrm{Cl}_2 = \frac{210 \text{ g}}{70.9 \text{ g/mol}} = 2.96 \text{ moles}.\]

Step 3: Determine the Limiting Reactant

From the balanced equation, 2 moles of \( \mathrm{Br}^- \) react with 1 mole of \( \mathrm{Cl}_2 \). Determine the number of \( \mathrm{Cl}_2 \) moles needed to react with all the \( \mathrm{Br}^- \) moles: \[\text{required } \mathrm{Cl}_2 = \frac{5.00}{2} = 2.50 \text{ moles}.\] Since we have 2.96 moles of \( \mathrm{Cl}_2 \) available, which is more than 2.50 moles, \( \mathrm{Br}^- \) is the limiting reactant.

Step 4: Calculate Theoretical Yield of Bromine

Based on the limiting reactant \( \mathrm{Br}^- \), use its moles to determine the yield of \( \mathrm{Br}_2 \): \[\text{moles of } \mathrm{Br}_2 = \frac{5.00}{2} = 2.50 \text{ moles}.\] Convert moles of \( \mathrm{Br}_2 \) to grams using its molar mass (159.8 g/mol): \[\text{mass of } \mathrm{Br}_2 = 2.50 \text{ moles} \times 159.8 \text{ g/mol} = 399.5 \text{ g}.\] Thus, the theoretical yield of \( \mathrm{Br}_2 \) is 399.5 grams.

Key Concepts

Bromine Displacement Reaction

In a bromine displacement reaction, chlorine gas (\( \mathrm{Cl}_2 \)) is added to a solution of bromide ions (\( \mathrm{Br}^- \)). The reaction is a fascinating example of redox chemistry where an exchange of atoms occurs. In this specific reaction, chlorine, being more reactive than bromine, displaces bromine from its compound forming bromine gas (\( \mathrm{Br}_2 \)).
This particular reaction can be summarized by the balanced chemical equation:\[\2 \mathrm{Br}^- + \mathrm{Cl}_2 \rightarrow \mathrm{Br}_2 + 2 \mathrm{Cl}^-\\]

• The balanced equation tells us that 2 moles of bromide ions are required to react with 1 mole of chlorine gas.
• Bromine is displaced as molecular Br₂ and chlorine converts into chloride ions.
• This reaction highlights the principle that more reactive halogens can displace less reactive halogens from compounds.

Understanding the nature of such halogen displacement reactions is central to grasping fundamental concepts in chemical reactivity and stoichiometry.

Theoretical Yield Calculation

Theoretical yield calculation helps to predict how much product can be formed from a given amount of reactants in a chemical reaction. It assumes that the reaction goes to completion and there are no losses. In this exercise, it is used to determine how much bromine (\( \mathrm{Br}_2 \)) can potentially be produced.
The first step is to identify the limiting reactant. In this scenario, although there is an excess of chlorine, the available bromide ions limit the reaction completion. Based on our balanced equation, 5 moles of bromide react with 2.5 moles of chlorine to produce 2.5 moles of bromine.
Once the moles of product are known, convert them to grams using the molar mass of bromine gas (159.8~ ext{g/mol}):

• Calculate mass of bromine: \(2.5 \text{ moles} \times 159.8 \text{ g/mol} = 399.5 \text{ g}\).

This theoretical yield represents the maximum amount of bromine that could be expected if the reaction went perfectly without any side reactions or losses.

Chemical Stoichiometry

Chemical stoichiometry allows us to quantify the amounts of substances involved in reactions. It uses the balanced chemical equation to relate proportions of reactants and products. This is very essential for calculating limiting reactants and theoretical yields.
In our bromine displacement example:

• The stoichiometric coefficients in the reaction \(2 \mathrm{Br}^- + \mathrm{Cl}_2 \rightarrow \mathrm{Br}_2 + 2 \mathrm{Cl}^-\) indicates a 2:1 ratio of bromide ions to chlorine gas.
• Moles of substances are calculated using concentration and volume or mass and molar mass for the given reactants.
• The limiting reactant is identified by comparing the mole ratio of substances as indicated by the balanced equation.

By applying chemical stoichiometry concepts, one can accurately predict quantities of materials required for reactions and estimate product formation efficiently. This is crucial in both laboratory and industrial chemical processes.