Question

As shown in Table \(15.2, K_{p}\) for the equilibrium $$\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)$$ is \(4.51 \times 10^{-5}\) at \(450^{\circ} \mathrm{C}\) . For each of the mixtures listed here, indicate whether the mixture is at equilibrium at \(450^{\circ} \mathrm{C} .\) If it is not at equilibrium, indicate the direction (toward product or toward reactants) in which the mixture must shift to achieve equilibrium. (a) 98 atm \(\mathrm{NH}_{3}, 45\) atm \(\mathrm{N}_{2}, 55\) atm \(\mathrm{H}_{2}\) (b) 57 atm \(\mathrm{NH}_{3}, 143\) atm \(\mathrm{N}_{2},\) no \(\mathrm{H}_{2}\) (c) 13 atm \(\mathrm{NH}_{3}, 27 \mathrm{atm} \mathrm{N}_{2}, 82 \mathrm{atm} \mathrm{H}_{2}\)

Short answer

(a) The mixture with 98 atm NH₃, 45 atm N₂, and 55 atm H₂ is not at equilibrium, and the reaction will shift toward product (NH₃) to achieve equilibrium at 450°C. (b) The mixture with 57 atm NH₃, 143 atm N₂, and 0 atm H₂ is not at equilibrium, and the reaction will shift toward reactants (N₂ and H₂) to achieve equilibrium at 450°C. (c) The mixture with 13 atm NH₃, 27 atm N₂, and 82 atm H₂ is not at equilibrium, and the reaction will shift toward product (NH₃) to achieve equilibrium at 450°C.

Step-by-step solution

Write the balanced equation

We have the balanced equation for the given reaction: \[ \mathrm{N}_{2}(g) + 3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) \]

Write the expression for Kₚ and Q

Kₚ and Q are calculated using the partial pressures of the gases involved in the reaction. The expressions for Kₚ and Q are the same: \[ Kₚ = Q = \frac{[\mathrm{NH}_3]^2}{[\mathrm{N}_2] \times [\mathrm{H}_2]^3} \] where [NH₃], [N₂], and [H₂] represent the partial pressures of NH₃, N₂, and H₂ gases, respectively. Given in the exercise, Kₚ = \(4.51 \times 10^{-5}\) at 450°C.

Calculate Q and compare it with Kₚ for each mixture

(a) For the mixture with 98 atm NH₃, 45 atm N₂, and 55 atm H₂: \[ Q=\frac{(98)^2}{(45) \times (55)^3} = 1.73 \times 10^{-5} \] Since Q < Kₚ, the reaction will shift toward product to achieve equilibrium. (b) For the mixture with 57 atm NH₃, 143 atm N₂, and 0 atm H₂: Since there is no H₂ in the mixture, the reaction can't proceed in the forward direction to form NH₃. Hence, the reaction must shift toward reactants to re-establish the equilibrium. (c) For the mixture with 13 atm NH₃, 27 atm N₂, and 82 atm H₂: \[ Q=\frac{(13)^2}{(27) \times (82)^3} = 6.54 \times 10^{-6} \] Since Q < Kₚ, the reaction will shift toward product to achieve equilibrium.

Present the results

(a) The mixture with 98 atm NH₃, 45 atm N₂, and 55 atm H₂ is not at equilibrium, and the reaction will shift toward product (NH₃) to achieve equilibrium at 450°C. (b) The mixture with 57 atm NH₃, 143 atm N₂, and 0 atm H₂ is not at equilibrium, and the reaction will shift toward reactants (N₂ and H₂) to achieve equilibrium at 450°C. (c) The mixture with 13 atm NH₃, 27 atm N₂, and 82 atm H₂ is not at equilibrium, and the reaction will shift toward product (NH₃) to achieve equilibrium at 450°C.

Key concepts

Equilibrium Constant

The equilibrium constant, abbreviated as K, is a numerical value that expresses the ratio of the concentrations of products to reactants at chemical equilibrium for a reversible reaction. In the given exercise, the equilibrium constant, denoted as K_p, is specific to reactions involving gases where the equilibrium concentration is expressed in terms of partial pressure. The value of K_p for the synthesis of ammonia from nitrogen and hydrogen is 4.51 x 10^-5 at 450°C.

For any reaction at equilibrium, this constant value indicates the ratio of the concentrations (or partial pressures) of products to reactants has reached a stable point, where the rate of the forward reaction equals the rate of the reverse reaction. When calculating the equilibrium constant, the concentration or partial pressure of each species is raised to the power of its stoichiometric coefficient from the balanced chemical equation.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept in chemical equilibrium that predicts how a system at equilibrium will respond to external changes. It states that if a dynamic equilibrium is disturbed by changing the conditions, such as concentration, pressure, or temperature, the position of equilibrium will shift to counteract the change and restore a new equilibrium state.

In the context of the provided exercise, if the reaction mixture is not at equilibrium, according to Le Chatelier's Principle, it will adjust itself to re-establish equilibrium by shifting the reaction in the direction that will absorb the added (or remove the decreased) component. For example, if the partial pressure of a reactant is increased, the equilibrium will shift in the direction that consumes more of that reactant, thus producing more products.

Reaction Quotient

The reaction quotient, denoted as Q, represents the ratio of the concentrations (or partial pressures) of products to reactants at any point in time, not necessarily at equilibrium. It is calculated in the same way as the equilibrium constant but can be used to determine the direction in which a reaction will need to shift to reach equilibrium. If Q is less than the equilibrium constant K_p, the reaction will proceed in the forward direction to produce more products. If Q exceeds K_p, the reaction will shift to produce more reactants. In the exercise, we compare Q with K_p to determine which direction the equilibrium will shift.

Partial Pressure

In chemistry, and specifically when dealing with gases, partial pressure refers to the pressure that a single gas in a mixture would exert if it were alone in the entire volume of the mixture. The partial pressure of a gas is directly proportional to its concentration. This concept is critical in chemical equilibrium involving gas-phase reactions because the equilibrium constant K_p is based on the partial pressures of the reactants and products.

When the partial pressures are used in the equilibrium calculation, they must reflect the stoichiometry of the balanced reaction equation. As demonstrated in the exercise, adjusting the partial pressures of reactants or products will alter the reaction quotient Q, and the system will shift according to Le Chatelier's Principle until the reaction quotient equals the equilibrium constant.