Question

Write the equilibrium expression for each of the following reactions. a. \(\operatorname{CO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}(g)\) b. \(2 \mathrm{NO}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) c. \(P_{4}(g)+6 B r_{2}(g) \rightleftharpoons 4 P B r_{3}(g)\)

Short answer

a. Kₐ = \(\frac{[\mathrm{CH}_{3}\mathrm{OH}]}{[\mathrm{CO}][\mathrm{H}_{2}]^2}\) b. K_b = \(\frac{[\mathrm{NO}]^2 [\mathrm{O}_{2}]}{[\mathrm{NO}_{2}]^2}\) c. K_c = \(\frac{[\mathrm{PBr}_{3}]^4}{[\mathrm{P}_{4}][\mathrm{Br}_{2}]^6}\)

Step-by-step solution

Equilibrium Constant Definition

The equilibrium constant (K) for a chemical reaction is the ratio of the concentrations of the products to the concentrations of the reactants, each raised to the power of their stoichiometric coefficients. It can be represented as: K = \(\frac{(C)^c (D)^d}{(A)^a (B)^b}\) where for the reaction aA + bB ⇌ cC + dD, a, b, c, and d are the stoichiometric coefficients of the reactants A and B and the products C and D, respectively. Now, let's write the equilibrium expressions for each of the given reactions:

Reaction a

For the reaction: CO(g) + 2 H₂(g) ⇌ CH₃OH(g) The equilibrium constant expression (Kₐ) will be: Kₐ = \(\frac{[\mathrm{CH}_{3}\mathrm{OH}]}{[\mathrm{CO}][\mathrm{H}_{2}]^2}\)

Reaction b

For the reaction: 2 NO₂(g) ⇌ 2 NO(g) + O₂(g) The equilibrium constant expression (K_b) will be: K_b = \(\frac{[\mathrm{NO}]^2 [\mathrm{O}_{2}]}{[\mathrm{NO}_{2}]^2}\)

Reaction c

For the reaction: P₄(g) + 6 Br₂(g) ⇌ 4 PBr₃(g) The equilibrium constant expression (K_c) will be: K_c = \(\frac{[\mathrm{PBr}_{3}]^4}{[\mathrm{P}_{4}][\mathrm{Br}_{2}]^6}\) In summary, the equilibrium expressions for the given reactions are: a. Kₐ = \(\frac{[\mathrm{CH}_{3}\mathrm{OH}]}{[\mathrm{CO}][\mathrm{H}_{2}]^2}\) b. K_b = \(\frac{[\mathrm{NO}]^2 [\mathrm{O}_{2}]}{[\mathrm{NO}_{2}]^2}\) c. K_c = \(\frac{[\mathrm{PBr}_{3}]^4}{[\mathrm{P}_{4}][\mathrm{Br}_{2}]^6}\)

Key concepts

Equilibrium Expression

An equilibrium expression is a mathematical way to describe a chemical reaction at equilibrium. In any chemical reaction that reaches equilibrium, the rates of the forward and backward reactions become equal. This means that the concentrations of reactants and products remain constant over time.

The equilibrium expression provides a way to quantify this balance using the concentrations of the reactants and products. For a general reaction:

1. aA + bB ⇌ cC + dD

the equilibrium expression is written as:\[K = \frac{[C]^c [D]^d}{[A]^a [B]^b}\]where:

• [A], [B], [C], and [D] are the concentrations of the chemical species.
• K is the equilibrium constant.
• a, b, c, and d are the stoichiometric coefficients from the chemical equation.

Understanding equilibrium expressions helps us grasp how much of each chemical species will be present when a reaction has reached equilibrium.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers placed in front of the compounds in a balanced chemical equation. They represent the ratio of moles of each substance participating in the reaction.

In an equilibrium expression, these coefficients become the powers to which the concentrations are raised. For instance:\[2 NO_2(g) ⇌ 2 NO(g) + O_2(g)\]In this equation:

• The stoichiometric coefficient for NO₂ is 2, which means in the equilibrium expression, the concentration of NO₂ will be squared (i.e., ([NO₂]²)).
• Similarly, for NO, the concentration will be squared, while for O₂, it will be to the first power as its coefficient is 1.

These coefficients ensure the balance of mass and atoms in a reaction, providing a correct ratio for how the reaction occurs at the molecular level.

Equilibrium Constant Expression

The equilibrium constant expression is a crucial part of understanding chemical equilibria. It allows us to predict the direction of a reaction and how much product will be formed at equilibrium. K, or the equilibrium constant, is unique for each reaction at a given temperature.

You calculate K using the concentrations of products and reactants at equilibrium, according to:\[K = \frac{[Products]^n}{[Reactants]^m}\]The numerator of this fraction involves the product concentrations raised to the power of their stoichiometric coefficients. The denominator does the same for the reactants. For example, for the reaction:\[P_4(g) + 6 Br_2(g) ⇌ 4 PBr_3(g)\]The equilibrium constant expression is:\[K = \frac{[PBr_3]^4}{[P_4][Br_2]^6}\]Since these constants differ with temperature, they are only valid at the temperature at which they were determined.

Concentrations of Reactants and Products

In any chemical reaction reaching equilibrium, the concentrations of reactants and products are crucial. These concentrations determine the equilibrium constant's numerical value and describe how far the reaction has proceeded.

At the start, a chemical reaction will have no products, just reactants at their maximum concentration. As the reaction progresses towards equilibrium, reactant concentrations typically decrease, while product concentrations increase. The equilibrium expression utilizes these equilibrium concentrations, as only these values yield the actual equilibrium constant value.

For example, in the reaction:\[CO(g) + 2 H_2(g) ⇌ CH_3OH(g)\]The equilibrium expression is:\[K = \frac{[CH_3OH]}{[CO][H_2]^2}\]Here, [CH₃OH], [CO], and [H₂] represent the concentrations of methanol, carbon monoxide, and hydrogen at equilibrium, respectively. Knowing the equilibrium concentrations allows you to calculate the equilibrium constant and understand the concentration changes that have occurred.