Question

For the reaction $$\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)$$ \(K=1.3 \times 10^{-2}\) at a given temperature. If the system at equilibrium is analyzed and the concentrations of both \(\mathrm{N}_{2}\) and \(\mathrm{H}_{2}\) are found to be \(0.10 \mathrm{M}\), what is the concentration of \(\mathrm{NH}_{3}\) in the system?

Short answer

At equilibrium, the concentration of NH3 in the system is \(3.6 \times 10^{-3}\) M.

Step-by-step solution

Write the equilibrium constant expression for the given reaction.

According to the given reaction: \[ N_{2}(g) + 3H_{2}(g) \rightleftharpoons 2NH_{3}(g) \] The equilibrium constant expression is given by: \[ K = \frac{[NH_{3}]^2}{[N_{2}][H_{2}]^3} \]

Substitute the given values and solve for the concentration of NH3.

We're given that K = 1.3 × 10⁻², [N2] = 0.10M, and [H2] = 0.10M at equilibrium. By substituting these values into the equilibrium constant expression, we get: \[ 1.3 \times 10^{-2} = \frac{[NH_{3}]^2}{(0.10)(0.10)^3} \] Now, let's solve for the concentration of NH3: \[ [NH_{3}]^2 = 1.3 \times 10^{-2} \times (0.10) \times (0.10)^3 \] \[ [NH_{3}]^2 = 1.3 \times 10^{-5} \] To find the concentration of NH3, we need to take the square root of both sides: \[ [NH_{3}] = \sqrt{1.3 \times 10^{-5}} \] \[ [NH_{3}] = 3.6 \times 10^{-3} \mathrm{M} \]

Write the answer.

At equilibrium, the concentration of NH3 in the system is 3.6 × 10⁻³ M.

Key concepts

Equilibrium Constant

In chemical equilibrium, the equilibrium constant, denoted as \(K\), is crucial in indicating the extent of a reaction at equilibrium. This constant is a ratio of the concentrations of products to reactants, each raised to the power of their stoichiometric coefficients. For the reaction \(\mathrm{N}_2(g) + 3\mathrm{H}_2(g) \rightleftharpoons 2\mathrm{NH}_3(g)\), the equilibrium constant expression is:\[ K = \frac{[NH_3]^2}{[N_2][H_2]^3} \]

• The numerator \([NH_3]^2\) represents the concentration of ammonia squared.
• The denominator \([N_2][H_2]^3\) combines the concentration of nitrogen and hydrogen, with hydrogen raised to the third power due to its coefficient in the balanced equation.

The value of \(K\) is determined by temperature and stays constant for a given reaction under the same conditions. Understanding \(K\)'s role helps predict whether reactants or products are favored at equilibrium.

Concentration Calculation

Calculating concentration at equilibrium is an essential skill in chemistry. In this exercise, we need to find the concentration of \(\mathrm{NH}_3\) using the equilibrium constant expression. Given \(K = 1.3 \times 10^{-2}\), \([N_2] = 0.10 \mathrm{M}\), and \([H_2] = 0.10 \mathrm{M}\), we substitute these into the expression:\[ 1.3 \times 10^{-2} = \frac{[NH_3]^2}{(0.10)(0.10)^3} \]Next, solve for \([NH_3]^2\) by multiplying both sides by \((0.10^4)\):\[ [NH_3]^2 = 1.3 \times 10^{-2} \times 10^{-4} = 1.3 \times 10^{-5} \]Taking the square root gives:\[ [NH_3] = \sqrt{1.3 \times 10^{-5}} = 3.6 \times 10^{-3} \mathrm{M} \]This step-by-step calculation demonstrates how to derive the unknown concentration from known equilibrium data.

Reaction Quotient

The reaction quotient, denoted \(Q\), serves as a snapshot of a reaction's status at any point in time, not just at equilibrium. Like \(K\), it is calculated using the concentrations of products and reactants raised to their respective powers. However, \(Q\) allows comparison at conditions different from equilibrium.To determine if a system is at equilibrium:- Calculate \(Q\) using the same formula as \(K\), inserting current concentrations.- If \(Q = K\), the system is at equilibrium.- If \(Q < K\), the reaction will shift towards products to reach equilibrium.- If \(Q > K\), the reaction shifts towards reactants.In our original exercise, this concept wasn't needed directly since equilibrium was already established. However, understanding \(Q\) provides a powerful tool for predicting the direction of reaction shifts when conditions change.