Question

At \(2000^{\circ} \mathrm{C}\) the equilibrium constant for the reaction $$ 2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) $$ is \(K_{c}=2.4 \times 10^{3}\). If the initial concentration of \(\mathrm{NO}\) is \(0.200 \mathrm{M}\), what are the equilibrium concentrations of NO, \(\mathrm{N}_{2}\), and \(\mathrm{O}_{2} ?\)

Short answer

The equilibrium concentrations are approximately: NO: 0.1102 M N2: 0.0449 M O2: 0.0449 M

Step-by-step solution

Write the balanced chemical equation and the equilibrium constant expression

The balanced chemical equation is already provided in the exercise: \(2 NO(g) \rightleftharpoons N_{2}(g) + O_{2}(g)\) The equilibrium constant expression for this reaction is given by: \(K_c = \frac{[N_{2}][O_{2}]}{[NO]^2}\)

Create an ICE table

An ICE table helps us keep track of the initial concentrations, the change in concentrations, and the final equilibrium concentrations. | | NO | N2 | O2 | |------|-------|------|------| | I | 0.200 | 0 | 0 | | C | -2x | x | x | | E | 0.200-2x | x | x | Here, x represents the change in concentration, with -2x representing the amount of NO reacting and x representing the amounts of N2 and O2 formed at equilibrium.

Write the expression for Kc using the equilibrium concentrations

Using the concentrations from the ICE table and the given value of Kc (2.4 x 10^3), we can write the following expression: \(2.4 \times 10^{3} = \frac{x \cdot x}{(0.200 - 2x)^2}\)

Solve the quadratic equation for x

First simplify the equation: \(2.4 \times 10^{3} = \frac{x^2}{(0.200 - 2x)^2}\) Now multiply both sides by \((0.200 - 2x)^2\): \(x^2 = 2.4 \times 10^{3} (0.200 - 2x)^2\) Solving this quadratic equation, we get: \(x ≈ 0.0449\)

Calculate the equilibrium concentrations

Now, we can calculate the equilibrium concentrations for each species using the values from the ICE table: - [NO] = 0.200 - 2x = 0.200 - 2(0.0449) ≈ 0.1102 M - [N2] = x = 0.0449 M - [O2] = x = 0.0449 M The equilibrium concentrations are approximately: NO: 0.1102 M N2: 0.0449 M O2: 0.0449 M

Key concepts

Understanding Chemical Equilibrium

Chemical equilibrium is a state in the chemical process where the concentrations of reactants and products remain constant over time. It's not that the reactions have stopped; rather, the rate of the forward reaction, where reactants turn into products, is equal to the rate of the reverse reaction, where products revert into reactants. This dynamic balance is fundamental in studying chemical reactions and is indeed a key focus in chemistry education.

When the conditions of the system are such that the concentrations of reactants and products no longer change, we say that the system is at equilibrium. It's important to note that equilibrium does not mean the reactants and products are in equal concentrations, but that their concentrations are stable and do not vary with time.

Equilibrium Constant Expression

The equilibrium constant expression quantifies the balance of a chemical reaction at equilibrium. For a generalized reaction where aA + bB ⇌ cC + dD, the equilibrium constant (K_c) is expressed as:
\[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
In this mathematical expression, the brackets denote the molar concentration of each species at equilibrium, and the letters a, b, c, and d are the stoichiometric coefficients from the balanced chemical equation. The value of K_c helps predict the direction of the reaction and the extent to which reactants are converted into products. A high value of K_c indicates a reaction that favors product formation, while a low K_c favors the reactants.

ICE Table Method

The ICE table method is a systematic tool to calculate equilibrium concentrations. ICE stands for Initial, Change, and Equilibrium – the three stages of a reaction reaching equilibrium. Here's what each stage represents:

• Initial: These are the starting concentrations of reactants and products before the reaction proceeds.
• Change: The amounts by which the concentrations change as the reaction moves towards equilibrium. These changes are often represented by a variable, like 'x', and are based on the stoichiometry of the balanced equation.
• Equilibrium: The final concentrations of reactants and products when the system is at equilibrium.

To use the ICE table, you start by filling in initial concentrations and changes (based on stoichiometry), then solve for 'x'. Once you find 'x', you can calculate equilibrium concentrations. This method provides a clear visualization of how the shift from initial to equilibrium concentrations follows the stoichiometry of the balanced equation, enabling you to solve for unknowns in equilibrium reactions effectively.