Question

A mixture of \(1.374 \mathrm{~g}\) of \(\mathrm{H}_{2}\) and \(70.31 \mathrm{~g}\) of \(\mathrm{Br}_{2}\) is heated in a \(2.00\)-L vessel at \(700 \mathrm{~K}\). These substances react according to $$ \mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{HBr}(g) $$ At equilibrium, the vessel is found to contain \(0.566 \mathrm{~g}\) of \(\mathrm{H}_{2}\). (a) Calculate the equilibrium concentrations of \(\mathrm{H}_{2}, \mathrm{Br}_{2}\), and \(\mathrm{HBr}\). (b) Calculate \(K_{\text {. }}\).

Short answer

The equilibrium concentrations of H₂, Br₂, and HBr are 0.142 M, 0.018 M, and 0.404 M, respectively. The equilibrium constant K is approximately 321.

Step-by-step solution

Calculate initial moles of H₂ and Br₂

Use the molar mass of each substance to find the moles. For H₂: molar mass = 2 g/mol Moles of H₂ = mass / molar mass = 1.374 g / 2 g/mol = 0.687 mol For Br₂: molar mass = 160 g/mol Moles of Br₂ = mass / molar mass = 70.31 g / 160 g/mol = 0.439 mol

Calculate change in moles at equilibrium

At equilibrium, 0.566 g of H₂ is remaining. Calculate the moles: Moles of H₂ at equilibrium = 0.566 g / 2 g/mol = 0.283 mol Calculate the change in moles of H₂ (∆n(H₂)) and Br₂ (∆n(Br₂)): ∆n(H₂) = 0.687 mol - 0.283 mol = 0.404 mol Since the reaction has a 1:1 stoichiometry between H₂ and Br₂, the change in moles of Br₂ will be the same as the change in moles of H₂: ∆n(Br₂) = 0.404 mol Now, we find the moles of Br₂ at equilibrium: Moles of Br₂ at equilibrium = 0.439 mol - 0.404 mol = 0.035 mol Since the stoichiometry of HBr is 2 times that of H₂ or Br₂, we calculate the moles of HBr formed at equilibrium: Moles of HBr at equilibrium = 2 * ∆n(H₂) = 2 * 0.404 mol = 0.808 mol

Calculate equilibrium concentrations

Now that we have the moles of each substance at equilibrium, we can calculate their concentrations by dividing by the volume of the vessel (2.00 L): [H₂] = 0.283 mol / 2.00 L = 0.142 M [Br₂] = 0.035 mol / 2.00 L = 0.018 M [HBr] = 0.808 mol / 2.00 L = 0.404 M

Calculate the equilibrium constant K

Use the equilibrium concentrations and the balanced chemical equation to calculate the equilibrium constant K: \( K = \frac{[\mathrm{HBr}]^2}{[\mathrm{H}_{2}][\mathrm{Br}_{2}]} \) Plug in the equilibrium concentrations: \( K = \frac{(0.404 M)^2}{(0.142 M)(0.018 M)} \) K ≈ 321 The equilibrium concentrations of H₂, Br₂, and HBr are 0.142 M, 0.018 M, and 0.404 M, respectively. The equilibrium constant K is approximately 321.

Key concepts

Equilibrium Concentrations

Equilibrium concentrations refer to the amounts of reactants and products in a reaction mixture when the reaction has reached a state of balance. At this point, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of all substances involved remain constant over time. In a chemical equation such as \[ \mathrm{H}_{2}(g) + \mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{HBr}(g) \],the equilibrium concentrations are denoted using square brackets, such as [HBr], to indicate the molarity (M), which is the number of moles of a substance per liter of solution. Calculating equilibrium concentrations involves understanding the stoichiometry of the reaction, the initial amounts of reactants, and the changes that occur as the reaction reaches equilibrium. By identifying the initial moles, calculating the change in moles at equilibrium, and considering the reaction's volume, students can determine the equilibrium concentrations of all species involved.

Equilibrium Constant

The equilibrium constant, typically represented as K, is a numerical value that expresses the ratio of the concentration of products to the concentration of reactants for a reversible chemical reaction at equilibrium, each raised to the power of their respective coefficients in the balanced equation. For the given reaction\[ \mathrm{H}_{2} + \mathrm{Br}_{2} \rightleftharpoons 2 \mathrm{HBr} \],the equilibrium constant is defined as \[ K = \frac{[\mathrm{HBr}]^2}{[\mathrm{H}_{2}][\mathrm{Br}_{2}]} \].The calculated value of K provides insight into the reaction's behavior; a large K suggests that the reaction tends to favor the formation of products, while a small K suggests a tendency to favor the reactants. It does not change with the amounts of reactants or products and only depends on the temperature of the system. Understanding the equilibrium constant is crucial as it helps predict the direction and extent of a chemical reaction under different conditions.

Stoichiometry

Stoichiometry is the aspect of chemistry that concerns the quantitative relationship between reactants and products in a chemical reaction. It involves using balanced chemical equations to determine the proportions of substances consumed and produced. For example, in the reaction\[ \mathrm{H}_{2} + \mathrm{Br}_{2} \rightleftharpoons 2 \mathrm{HBr} \],the ratio of hydrogen to bromine is 1:1, and the ratio of hydrogen to hydrogen bromide is 1:2, reflecting the stoichiometry of the reaction. When calculating equilibrium concentrations, stoichiometry allows us to determine changes in the moles of reactants and products when the reaction reaches equilibrium. By using the initial amount of substances and the stoichiometric coefficients, one can precisely calculate the number of moles at equilibrium, which is fundamental for determining both equilibrium concentrations and the equilibrium constant.

Reaction Quotient

The reaction quotient, Q, is a measure that indicates the direction in which a reaction will shift to reach equilibrium. It uses the same formula as the equilibrium constant (K), but with the initial concentrations rather than the equilibrium concentrations. For the reaction \[ \mathrm{H}_{2} + \mathrm{Br}_{2} \rightleftharpoons 2 \mathrm{HBr} \],the reaction quotient is given by\[ Q = \frac{[\mathrm{HBr}]^2}{[\mathrm{H}_{2}][\mathrm{Br}_{2}]} \].Comparing the values of Q and K helps predict whether the reaction will proceed in the forward direction (if Q < K), remain at equilibrium (if Q = K), or shift backward (if Q > K). It's an essential concept for understanding chemical equilibrium as it helps determine the conditions needed for a reaction to reach equilibrium from any starting point. Estimating the reaction quotient can provide valuable insights when first approaching an equilibrium problem.