Question

Consider the reaction for the formation of ammonia and its equilibrium constant at \(400 \mathrm{~K}\) : $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \Longrightarrow 2 \mathrm{NH}_{3}(g) \quad K_{\mathrm{eq}}=224 $$ If \(0.050 \mathrm{~mol}\) of each reactant and product is mixed into a \(1.0\)-L container, will the reaction proceed in the forward or reverse direction, or is it already at equilibrium?

Short answer

The reaction will proceed in the forward direction.

Step-by-step solution

Write Reaction Quotient Expression

The reaction quotient (Q) for the reaction is given by expression: \[Q = \frac{[NH_3]^2}{[N_2][H_2]^3}\] where the square brackets denote the molar concentrations of the gases at any given moment.

Substitute the Concentrations

The given initial concentrations of \(N_2\), \(H_2\), and \(NH_3\) are all \(0.050M\). Substitute these into the Q expression. That gives: \[Q = \frac{(0.050)^2}{(0.050)(0.050)^3} = 4\]

Compare Q and K

Compare the calculated Q value with the provided equilibrium constant (K). If \(Q < K\), the reaction will proceed in the forward direction to reach equilibrium. If \(Q > K\), the reaction will go in the reverse direction. If \(Q = K\), then the reaction is already at equilibrium. In this problem, \(K = 224\) and \(Q = 4\). Since \(Q < K\), the reaction will proceed in the forward direction to reach equilibrium.

Key concepts

Reaction Quotient

The reaction quotient, commonly represented as \(Q\), helps us understand the current state of a reaction mixture relative to its equilibrium state. It has a similar form to the equilibrium constant (\(K\)), but while \(K\) is determined under equilibrium conditions, \(Q\) can be calculated at any point during the reaction.

For the reaction \[\text{N}_2(g) + 3\text{H}_2(g) \Longrightarrow 2\text{NH}_3(g)\]the expression for \(Q\) is:\[Q = \frac{[NH_3]^2}{[N_2][H_2]^3}\]

By using the concentrations of reactants and products at any given moment, \(Q\) allows us to judge the direction in which a reaction needs to proceed to achieve equilibrium. If \(Q < K\), the reaction will move forward (producing more products) to reach equilibrium. If \(Q > K\), the reaction will shift backward (producing more reactants). This makes \(Q\) a valuable tool in predicting and understanding chemical behavior in dynamic systems.

Equilibrium Constant

The equilibrium constant, denoted as \(K_{eq}\), is a fundamental aspect of chemical reactions at equilibrium. It provides a relationship between concentrations of products and reactants at equilibrium, specific to a given temperature.

For the synthesis of ammonia, the equilibrium constant at \(400 \text{ K}\) is given as \(K_{eq} = 224\).

Its expression for the reaction is:\[K_{eq} = \frac{[NH_3]^2}{[N_2][H_2]^3}\]

\(K_{eq}\) values can vary greatly, with large values indicating a reaction strongly favoring products, while small values suggest the formation of reactants is favorable. By comparing \(Q\) to \(K\), one can quickly ascertain whether a reaction is at equilibrium or predict its shift either forward or reverse to attain balance.

Understanding \(K\) helps chemists and students evaluate how changes in conditions impact the reaction dynamics and provides essential insights into reaction efficiency and yield.

Ammonia Formation

The formation of ammonia (\[\text{N}_2(g) + 3\text{H}_2(g) \Longrightarrow 2\text{NH}_3(g)\]) is a major chemical process, particularly important in industries for fertilizers and other nitrogen-containing products. This reaction showcases the application of the equilibrium concepts in real-world chemical engineering.

The reaction is exothermic and involves the reaction between nitrogen and hydrogen gas to produce ammonia. The equilibrium constant and reaction quotient play key roles in maximizing the yield of ammonia, as achieving a high concentration of products is desirable for industrial applications.

Adjustments in temperature and pressure, along with catalysts, are often used to manipulate the reaction towards an optimal production of ammonia. This process, exemplified by the Haber-Bosch method, highlights the practical significance of equilibrium knowledge in optimizing and controlling chemical reactions for beneficial outcomes.