Question

Gaseous ammonia was introduced into a sealed container and heated to a certain temperature: $$ 2 \mathrm{NH}_{3}(g) \rightleftharpoons \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) $$ At equilibrium, \(\left[\mathrm{NH}_{3}\right]=0.0225 M,\left[\mathrm{~N}_{2}\right]=0.114 M,\) and \(\left[\mathrm{H}_{2}\right]=\) \(0.342 M .\) Calculate \(K_{c}\) for the reaction at this temperature.

Short answer

The equilibrium constant \(K_c\) for the reaction at this temperature is 9.00.

Step-by-step solution

Write the equilibrium expression for the reaction

The equilibrium constant in terms of concentration, \(K_c\), for the given reaction is written as follows:\[ K_c = \frac{[{\mathrm{N}_2}] [{\mathrm{H}_2}]^3}{[{\mathrm{NH}_3}]^2} \]

Substitute the equilibrium concentrations into the expression

Now, substitute the given equilibrium concentrations into the \(K_c\) expression:\[[\mathrm{N}_2] = 0.114 \: M\]\[[\mathrm{H}_2] = 0.342 \: M\]\[[\mathrm{NH}_3] = 0.0225 \: M\]Thus, \[K_c = \frac{(0.114) (0.342)^3}{(0.0225)^2}\]

Calculate the equilibrium constant \(K_c\)

First, calculate \( (0.342)^3 \):\[(0.342)^3 = 0.040 \]Next, input the values into the \(K_c\) expression and calculate: \[ K_c = \frac{(0.114) \times (0.040)}{(0.0225)^2} = \frac{0.00456}{0.00050625} \approx 9.00 \] The equilibrium constant \(K_c\) for the reaction at this temperature is 9.00.

Key concepts

Gaseous Equilibrium

Chemical reactions involving gases often reach a state of balance, known as equilibrium. At this point, the rate of the forward reaction (reactants turning into products) equals the rate of the reverse reaction (products turning back into reactants). This balance results in stable concentrations of both reactants and products over time. For the reaction\[2 \text{NH}_3(g) \rightleftharpoons \text{N}_2(g) + 3 \text{H}_2(g)\], equilibrium is established when the intro-gasing levels of NH\textsubscript{3}, N\textsubscript{2}, and H\textsubscript{2} do not change anymore.
Quantifying this state involves an equilibrium constant, termed as \(K_c\), which is calculated using the concentrations of the species involved in the reaction. Understanding gaseous equilibrium is crucial as it helps in predicting how the system responds to changes in conditions like temperature or pressure.

Reaction Rates

In chemical kinetics, reaction rate refers to the speed at which reactants convert into products. Reaction rates are influenced by various factors such as temperature, concentration, and catalysts present in the system. For the ammonia decomposition reaction\[2 \text{NH}_3(g) \rightleftharpoons \text{N}_2(g) + 3 \text{H}_2(g)\], the rate at which NH\textsubscript{3} breaks down equals the rate at which N\textsubscript{2} and H\textsubscript{2} combine back to form the reactants at equilibrium.
It's the balance of these forward and reverse reaction rates that define the equilibrium state of a gaseous system. Reaction rates can be directly observed by measuring how quickly the concentration of one of the reactants or products changes over time.
This understanding aids in controlling industrial processes that rely on specific reaction speeds to optimize product yield.

Concentration in Equilibrium

Equilibrium concentrations refer to the amounts of reactants and products in a reaction mixture once the system has reached equilibrium. For the reaction\[2 \text{NH}_3(g) \rightleftharpoons \text{N}_2(g) + 3 \text{H}_2(g)\], we know that the equilibrium concentrations are given as [NH\textsubscript{3}] = 0.0225 M, [N\textsubscript{2}] = 0.114 M, and [H\textsubscript{2}] = 0.342 M.
These concentrations allow us to compute the equilibrium constant \(K_c\), a measure of the position of equilibrium. By substituting these values into the formula: \(K_c = \frac{[N_2][H_2]^3}{[NH_3]^2}\), you can determine the ratio of product concentrations to reactant concentrations at equilibrium.
This ratio remains constant for a given temperature, providing insight into the system's behavior under different conditions and helping predict shifts in equilibrium according to Le Chatelier's principle.