Question

Consider the reactions and their respective equilibrium constants. \begin{equation}\mathrm{NO}(g)+\frac{1}{2} \mathrm{Br}_{2}(g) \rightleftharpoons \operatorname{NOBr}(g) \quad K_{p}=5.3\end{equation} \begin{equation} 2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \quad \quad K_{\mathrm{p}}=2.1 \times 10^{30}\end{equation} Use these reactions and their equilibrium constants to predict the equi- librium constant for the following reaction: \begin{equation}\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{NOBr}(g)\end{equation}

Short answer

The equilibrium constant for the reaction \(\mathrm{N}_{2}(g) + \mathrm{O}_{2}(g) + \mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{NOBr}(g)\) is \(K_p = (5.3)^2 = 28.09\).

Step-by-step solution

Identify the Target Reaction

Write down the target reaction for which the equilibrium constant is to be determined. This is the reaction whose equilibrium constant is not provided but can be predicted using the known reactions.

Determine the Correct Combination of Given Reactions

Analyze the provided reactions to determine how they could be combined to yield the target reaction. This might involve reversing and/or multiplying the given reactions to properly align them with the target equation.

Combine the Reactions

Combine the aligned reactions by addition, remembering to do the same with the equilibrium constants (multiplication when reactions are combined, and taking reciprocal when reactions are reversed).

Calculate the Equilibrium Constant for the Target Reaction

Use the relationships between the equilibrium constants for the combined reactions to determine the equilibrium constant for the target reaction. For reactions that are added, multiply their equilibrium constants; for a reaction that is reversed, take the reciprocal of its equilibrium constant; and for reactions that are multiplied by a coefficient, raise the equilibrium constant to the power of that coefficient.

Simplify the Expression

Simplify the expression to find a single value for the equilibrium constant of the target reaction.

Key concepts

Equilibrium Constant

Understanding the equilibrium constant, usually denoted as K or K_eq, is crucial when studying chemical reactions reaching a state of dynamic equilibrium. At equilibrium, the rate of the forward reaction is equal to the rate of the reverse reaction, leading to constant concentrations of reactants and products. The equilibrium constant expression is a ratio that compares the concentration of the products to that of the reactants, each raised to the power of their respective stoichiometric coefficients.

For a generic reaction aA + bB ⇌ cC + dD, the equilibrium expression is written as K = [C]^c [D]^d / [A]^a [B]^b, with [X] representing the molar concentration of species X at equilibrium. If K is much larger than 1, the equilibrium mixture is product-favored; conversely, if K is much less than 1, the mixture is reactant-favored.

For gaseous reactions, partial pressures are used instead, yielding a K_p. It's important to note that equilibrium constants are dimensionless and depend only on the temperature, not on initial concentrations or partial pressures.

Le Chatelier's Principle

Le Chatelier's principle is a qualitative tool to predict the direction in which a system at equilibrium will shift when it experiences a change in concentration, temperature, or pressure. Essentially, the principle states that if an external stress is applied to a system at equilibrium, the system will adjust in a way that counteracts the stress applied.

This principle can be applied in the following ways:

• If the concentration of a reactant or product is changed, the system shifts to oppose that change.
• If the temperature of an exothermic reaction is increased, the system shifts to favor the reactants; for an endothermic reaction, the opposite is true.
• If the pressure of a system is increased by decreasing the volume, the system shifts toward the side with fewer moles of gas.

These shifts are all aimed at re-establishing equilibrium, though the value of the equilibrium constant remains unaffected unless the temperature changes.

Reaction Quotient

The reaction quotient (Q) plays a crucial role in comparing the current state of a reaction to its equilibrium position. It is calculated in the same way as the equilibrium constant but with the concentrations or partial pressures of the reactants and products at any moment in time, not necessarily at equilibrium.

For the general reaction aA + bB ⇌ cC + dD, the reaction quotient is Q = [C]^c [D]^d / [A]^a [B]^b. The comparison of Q to K determines the direction of the shift needed to reach equilibrium. If Q is less than K, the forward reaction is favored, more products will be formed. Conversely, if Q is greater than K, the reverse reaction is favored, and more reactants will be formed. When Q equals K, the system is at equilibrium, with no net change in the concentrations of reactants and products.

Equilibrium Expressions

Equilibrium expressions are algebraic representations of the relationship between the concentrations (or partial pressures) of substances involved in a chemical equilibrium. They are the foundation for calculating the equilibrium constant K. These expressions are specific to each chemical reaction and are derived from the law of mass action.

To construct an equilibrium expression, write down the balanced chemical equation, and use the stoichiometric coefficients as exponents for the concentration or pressure terms of reactants and products. Important points to remember are:

• Pure solids and liquids do not appear in the equilibrium expression because their concentrations are constant and do not affect the equilibrium position.
• The concentrations of the products are placed in the numerator, and the reactants are in the denominator of the quotient.
• Only include species that are gases or are in solution in the expression.

Equilibrium expressions allow for the comparison of different reactant-product systems and provide insights into the extent of the reaction at equilibrium.