Question

At \(500 .{ }^{\circ} \mathrm{C}, K_{c}=0.061\) for \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})\). If analysis shows that the composition is \(3.00 \mathrm{~mol} \cdot \mathrm{L}^{-1} \mathrm{~N}_{2}\), \(2.00 \mathrm{~mol} \cdot \mathrm{L}^{-1} \mathrm{H}_{2}\), and \(0.500 \mathrm{~mol} \cdot \mathrm{L}^{-1} \mathrm{NH}_{3}\), is the reaction at equilibrium? If not, in which direction does the reaction tend to proceed to reach equilibrium?

Short answer

The reaction is not at equilibrium because Q (0.0208) is less than Kc (0.061), so the reaction will proceed to the right, towards product formation (NH3), to reach equilibrium.

Step-by-step solution

Write the Expression for the Equilibrium Constant (Kc)

Write the expression for the equilibrium constant, Kc, using the balanced chemical equation provided. For the reaction: \(\mathrm{N}_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)\), the expression for Kc is: \(K_c = \frac{[NH_3]^2}{[N_2][H_2]^3}\), where the concentrations of the gases are raised to the power of their stoichiometric coefficients.

Calculate the Reaction Quotient (Q)

Calculate the reaction quotient, Q, using the concentrations of the reactants and products provided. Q is calculated in the same way as Kc, but using the initial concentrations. Using the given concentrations, \(Q = \frac{(0.500)^2}{(3.00)(2.00)^3}\).

Perform the Calculation

Substitute the given concentrations into the Q expression and calculate its value. \(Q = \frac{(0.500)^2}{(3.00)(2.00)^3} = \frac{0.25}{12} = 0.0208\).

Compare Q with Kc

Compare the calculated Q to the given Kc. If Q > Kc, the reaction will proceed to the left (towards reactants). If Q < Kc, the reaction will proceed to the right (towards products). If Q = Kc, the reaction is at equilibrium. Since the given Kc is 0.061 and the calculated Q is 0.0208, Q < Kc.

Determine the Direction of the Reaction

Since Q < Kc, there is too much reactant compared to what is expected at equilibrium. Therefore, the reaction will proceed to the right, towards the formation of more products, to reach equilibrium.

Key concepts

Chemical Reaction

Chemical reactions involve the transformation of substances through the breaking and forming of bonds to generate new products. They can be represented by chemical equations, which show the reactants on the left side and the products on the right. The coefficients in a balanced equation reflect the stoichiometry, indicating the relative amounts of each substance involved.

For the reaction given in our exercise, \( \mathrm{N}_2(g) + 3\mathrm{H}_2(g) \rightleftharpoons 2\mathrm{NH}_3(g) \) , nitrogen and hydrogen gases are reacting to form ammonia. This equation implies that one mole of nitrogen reacts with three moles of hydrogen to produce two moles of ammonia under certain conditions.

Equilibrium Constant (Kc)

The equilibrium constant (Kc) is a numerical value that expresses the ratio of product concentrations to reactant concentrations at equilibrium, with each raised to the power of their coefficients in the balanced equation. It is a crucial concept in understanding chemical equilibrium, as it indicates the position of equilibrium and the extent of the reaction.

The expression for the equilibrium constant for our reaction is derived as follows: \(K_c = \frac{[\mathrm{NH}_3]^2}{[\mathrm{N}_2][\mathrm{H}_2]^3}\). In this equation, the square brackets represent the concentration of the gases in moles per liter. The superscripts are the coefficients from the balanced chemical equation, providing the necessary stoichiometric relationship.

Reaction Quotient (Q)

The reaction quotient (Q) plays a similar role to the equilibrium constant but is calculated using the initial concentrations of reactants and products, rather than the concentrations at equilibrium. It allows us to predict which direction the reaction will need to proceed to reach equilibrium.

In our exercise, we calculated Q using the information provided about the initial composition of the reaction mixture: \(Q = \frac{(0.500)^2}{(3.00)(2.00)^3}\). Comparing the calculated Q to the equilibrium constant, Kc, tells us if the reaction is at equilibrium (Q = Kc), or if it will shift towards products (Q < Kc) or reactants (Q > Kc) to achieve equilibrium.

Le Chatelier's Principle

Le Chatelier's principle helps us understand how a chemical reaction in equilibrium responds to changes in conditions, such as concentration, temperature, and pressure. If an external stress is applied to a system at equilibrium, the system adjusts to partially counteract the change and re-establish equilibrium.

In the context of our exercise, since Q < Kc, this indicates that the reaction is not at equilibrium and will shift to the right, producing more ammonia, to reach the equilibrium state. This is Le Chatelier's principle in action, as the system seeks to reduce the stress of a lower product concentration by forming more products.

Stoichiometry

Stoichiometry is the study of the quantitative relationships between the amounts of reactants and products in a chemical reaction. It's based on the law of conservation of mass, which states that matter is neither created nor destroyed in a reaction. For stoichiometric calculations, it is fundamental to perform mole-to-mole comparisons using the coefficients from the balanced equation.

In our example, stoichiometry dictates that three moles of hydrogen gas reacts with one mole of nitrogen gas to form two moles of ammonia. The step-by-step solution of our problem involves the stoichiometric proportions defined by the balanced equation to correctly determine the reaction quotient and predict the direction the reaction needs to proceed to achieve equilibrium.