Question

Consider the reaction: \(\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)\). a. Write the equilibrium constant expression for this reaction. If some hydrogen is added, before the reaction shifts, b. How will the numerator and denominator of the expression In part a compare to the value at equilibrium? c. Wul Q be larger or smaller than \(K\). Why? d. Will the reaction have to shift forwand or backward to retain equilibrium? Explain. e. Are your answers above consistent with le Chateller's principle? Explain.

Short answer

K is represented by \( \frac{[\mathrm{NH}_{3}]^{2}}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^{3}} \). With added hydrogen, the denominator increases before the reaction shifts, making Q smaller than K. The reaction will shift forward to regain equilibrium, consistent with Le Chatelier's principle.

Step-by-step solution

Write the equilibrium constant expression

Use the balanced chemical equation to write the expression for the equilibrium constant, K. For the reaction \(\mathrm{N}_{2}(g) + 3\mathrm{H}_{2}(g) \rightarrow 2\mathrm{NH}_{3}(g)\), the equilibrium constant expression (K) is: \[ K = \frac{[\mathrm{NH}_{3}]^{2}}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^{3}} \]

Compare the changes in the expression when hydrogen is added

Upon the addition of hydrogen gas, the concentration of \(\mathrm{H}_{2}\) in the denominator of the K expression will increase. Before the reaction shifts to re-establish equilibrium, this will make the denominator larger than it is at equilibrium because the concentrations of reactants and products have not yet adjusted.

Determine the relation of Q to K

The reaction quotient Q has the same expression as K, but for non-equilibrium conditions. Therefore, if the denominator of the K expression increases (due to the addition of \(\mathrm{H}_{2}\)) while the numerator remains the same, Q will be smaller than K because the equilibrium state has been disturbed.

Decide the direction of the shift to regain equilibrium

Since Q is smaller than K after adding more hydrogen, this indicates there are too many reactants and not enough products according to the equilibrium condition. To reach equilibrium, the reaction will shift forward (to the right), producing more ammonia (\(\mathrm{NH}_{3}\)) and reducing the relative amount of hydrogen and nitrogen.

Evaluate consistency with Le Chatelier's principle

Le Chatelier's principle states that if a system at equilibrium is disturbed, the system will adjust itself to diminish the change. Adding hydrogen increases the concentration of a reactant, and the system counteracts this by making more product, which is consistent with our earlier analysis. Thus, the answers are consistent with Le Chatelier's principle.

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression is essential for grasping the delicate balance of a chemical reaction at equilibrium. For a reaction like \(\mathrm{N}_{2}(g) + 3\mathrm{H}_{2}(g) \rightarrow 2\mathrm{NH}_{3}(g)\), the equilibrium constant expression is denoted by \(K\) and calculated using the concentrations of the products and reactants at equilibrium. The expression is derived from the balanced equation and takes the form: \[ K = \frac{[\mathrm{NH}_{3}]^{2}}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^{3}} \].
In this formula, the concentrations of the nitrogen and hydrogen gases are in the denominator, while the concentration of ammonia is in the numerator. Powers correspond to the coefficients of each substance in the balanced equation. Crucially, the expression only holds when the system is at equilibrium, meaning that the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products over time.

Reaction Quotient (Q)

The reaction quotient, denoted by \(Q\), is akin to the equilibrium constant but is evaluated at any point before the reaction reaches equilibrium. It has the same formula as the equilibrium constant expression but uses the current concentrations of the reactants and products. For the given reaction, we have: \[ Q = \frac{[\mathrm{NH}_{3}]_{current}^{2}}{[\mathrm{N}_{2}]_{current}[\mathrm{H}_{2}]_{current}^{3}} \].
When additional \(\mathrm{H}_{2}\) is introduced into the system, \(Q\) changes because the concentration of \(\mathrm{H}_{2}\) in the reaction quotient increases, affecting the ratio of products to reactants. This disturbance leads us to compare \(Q\) to \(K\) to predict which direction the reaction will shift to re-establish equilibrium.

Le Chatelier's Principle

Le Chatelier's principle explains how a chemical system at equilibrium responds to changes in concentration, temperature, or pressure. If the system experiences a disturbance, such as the addition of a reactant or product, it will respond to counteract that disturbance and return to a new equilibrium state. For our nitrogen and hydrogen reaction, adding more \(\mathrm{H}_{2}\) momentarily disrupts the balance. According to Le Chatelier's principle, the system will adjust by shifting the reaction to reduce the excess \(\mathrm{H}_{2}\) and create more \(\mathrm{NH}_{3}\). It's a principle that maintains nature's preference for balance and can predict the behavior of a system undergoing change.
Understanding Le Chatelier's principle helps explain why a chemical reaction may not proceed straightforwardly but instead adapts flexibly to internal and external changes.

Shift in Equilibrium

A shift in equilibrium occurs when the conditions of a chemical reaction change, causing the system to no longer be at equilibrium. Determining the direction of the shift in response to changes is critical for predicting the outcome of a reaction. In the context of our example, after \(\mathrm{H}_{2}\) is added and before the system adjusts, \(Q\) is smaller than \(K\), indicating that the reaction will shift toward the products to restore equilibrium. Thus, the equilibrium will shift forward, producing more ammonia from the added hydrogen. This serves as a practical illustration of how chemical reactions can be coerced into making more of a desired product or how they may adjust naturally to external changes in their environment.