Question

A reaction mixture that consisted of \(0.20 \mathrm{~mol} \mathrm{} \mathrm{N}_{2}\) and \(0.20 \mathrm{~mol} \mathrm{} \mathrm{H}_{2}\) was introduced into a \(25.0\) - \(\mathrm{L}\) reactor and heated. At equilibrium, \(5.0 \%\) of the nitrogen gas had reacted. What is the value of the equilibrium constant \(K_{c}\) for the reaction \(\mathrm{N}_{2}(\mathrm{~g})+\) \(3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})\) at this temperature?

Short answer

The equilibrium constant, Kc, for the reaction at this temperature is 0.040.

Step-by-step solution

Determine the Moles of N2 that Reacted

To know the amount of nitrogen that has reacted, calculate 5% of the initial moles of N2. This is done by multiplying 0.20 moles by 0.05 (5%).

Calculate the Change in Moles for Each Substance

For every 1 mole of N2 that reacts, 3 moles of H2 react and 2 moles of NH3 are produced. Therefore, if x moles of N2 react, 3x moles of H2 will react and 2x moles of NH3 will be produced.

Set Up the ICE Table

An ICE table will help to visualize the initial concentrations, the change in concentrations, and the equilibrium concentrations of the reactants and products. 'I' stands for initial concentration, 'C' for the change in concentration, and 'E' for the equilibrium concentration.

Calculate the Equilibrium Concentrations

Subtract the change in moles from the initial moles for N2 and H2, and add the change in moles to 0 for NH3 (since NH3 starts at 0 moles) to find the equilibrium concentrations. Then divide these by the volume of the reactor to get molarity.

Write the Equilibrium Constant Expression

The equilibrium constant, Kc, for the reaction is given by the expression: \[K_c = \frac{[\text{NH}_3]^2}{[\text{N}_2][\text{H}_2]^3}\] where the concentration of each gas is raised to the power of its stoichiometric coefficient.

Calculate the Value of Kc

Substitute the equilibrium concentrations into the Kc expression and solve for Kc.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. It's essential to understand that equilibrium does not imply that the reactants and products are present in equal amounts, but rather that their ratios remain constant at equilibrium.

At equilibrium, the system is said to be dynamic because both the forward and reverse reactions are still occurring, but they counterbalance each other. The equilibrium point is dependent on reaction conditions such as temperature and pressure, and it reflects the particular ratio of reactants to products at which the system has stabilized.

ICE Table

An ICE table (Initial, Change, Equilibrium) is a systematic way to calculate the concentrations of reactants and products at chemical equilibrium. The table is set up as follows:

• I (Initial): List the initial concentrations or moles of each reactant and product before the reaction starts.
• C (Change): Indicate the changes in concentrations or moles that occur as the reaction proceeds towards equilibrium. These changes are based on the stoichiometry of the balanced chemical equation.
• E (Equilibrium): List the concentrations or moles of each reactant and product at equilibrium.

The ICE table is a visual aid that can help clarify how reactants are consumed and how products are formed, making the process of solving for equilibrium concentrations more intuitive and organized.

Reaction Quotient

The reaction quotient, Q, is a measure that tells us how far a system is from equilibrium. It uses the same formula as the equilibrium constant, K, but for a system that is not at equilibrium. The formula takes into account the concentrations of the reactants and products at a particular moment in time:
\[Q = \frac{[\text{Products}]}{[\text{Reactants}]}\]
For gases, the concentration is usually expressed in terms of partial pressure. In the case of a reaction at equilibrium, Q equals the equilibrium constant, K. If Q is different from K, it indicates which direction the reaction will shift to reach equilibrium; if Q < K, the forward reaction is favored, and if Q > K, the reverse reaction is favored. Calculating Q is useful when determining the direction of the reaction's shift in response to changes in concentrations.

Equilibrium Constant Expression

The equilibrium constant expression, K, is a mathematical relationship that quantifies the chemical equilibrium state. For the generic reaction aA + bB \rightleftharpoons cC + dD, the equilibrium constant expression is written as:\[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]
where [A], [B], [C], and [D] are the molar concentrations of the reactants and products at equilibrium, and a, b, c, d are their respective stoichiometric coefficients. In our specific exercise of nitrogen reacting with hydrogen to form ammonia:\(N_{2}(g) + 3H_{2}(g) \rightleftharpoons 2NH_{3}(g)\), the equilibrium expression would be:\[K_c = \frac{[\text{NH}_3]^2}{[\text{N}_2][\text{H}_2]^3}\]
This expression allows us to calculate the numerical value of the equilibrium constant, Kc, by substituting the equilibrium concentrations of the reactants and products. The equilibrium constant provides insight into the extent of a reaction at a given temperature and is a crucial value for predicting the behavior of chemical systems.