Question

For the reaction: $$3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g)$$ \(K=1.8 \times 10^{-7}\) at a certain temperature. If at equilibrium \(\left[\mathrm{O}_{2}\right]=0.062 M,\) calculate the equilibrium \(\mathrm{O}_{3}\) concentration.

Short answer

The equilibrium concentration of O₃ is \(2.06 \times 10^{-6}\,M\).

Step-by-step solution

Write down the equilibrium constant expression

Based on the chemical reaction, we can write the equilibrium constant as: \[K = \frac{[\mathrm{O}_3]^2}{[\mathrm{O}_2]^3}\] where \([\mathrm{O}_3]\) and \([\mathrm{O}_2]\) are the equilibrium concentrations of O₃ and O₂, respectively.

Plug in the given values

We are given that at equilibrium, \([\mathrm{O}_2] = 0.062\, M\) and the equilibrium constant \(K = 1.8 \times 10^{-7}\). Substitute these values into the equilibrium constant expression: \(1.8 \times 10^{-7} = \frac{[\mathrm{O}_3]^2}{(0.062)^3}\)

Calculate the equilibrium concentration of O₃

Solve for \([\mathrm{O}_3]\): \[\begin{aligned} [\mathrm{O}_3]^2 &= 1.8 \times 10^{-7} \times (0.062)^3 \\ [\mathrm{O}_3]^2 &= 4.23 \times 10^{-11} \\ [\mathrm{O}_3] &= \sqrt{4.23 \times 10^{-11}} \end{aligned}\] Calculate the square root: \[[\mathrm{O}_3] = 2.06 \times 10^{-6}\,M\]

State the equilibrium concentration of O₃

The equilibrium concentration of O₃ is \(2.06 \times 10^{-6}\,M\).

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fascinating concept in chemistry. It refers to a state in a chemical reaction where the rates of the forward and reverse reactions are equal. As a result, the concentrations of the reactants and products remain constant over time. This doesn't mean that the reactions stop; rather, they continue to occur, but at the same pace, maintaining a balance.
This state of balance is dynamic, not static, and can be represented by a double arrow in the chemical equation, like in the reaction: \(3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g)\).
Once a reaction reaches equilibrium, the system's macroscopic properties remain unchanged. The conditions required to achieve equilibrium can vary and often depend on factors such as concentration, temperature, and pressure. Understanding chemical equilibrium is crucial for calculating concentrations of reactants or products in a mixture.

Reaction Quotient

The reaction quotient \(Q\) is a vital concept connected to chemical equilibrium. It provides a snapshot of a reaction's state by comparing the concentration of products and reactants at any point in time to the equilibrium constant \(K\).
The reaction quotient is calculated using the same formula as the equilibrium constant but with the current concentrations. If \(Q = K\), the system is at equilibrium. If \(Q < K\), the system will shift towards products to reach equilibrium. Conversely, if \(Q > K\), it will shift towards reactants.

• \(Q = \frac{\text{[products]}}{\text{[reactants]}}\)

This concept helps predict the direction in which a reaction mixture needs to change to reach equilibrium. It provides insights into how far a system is from equilibrium, allowing chemists to anticipate and control reactions more effectively.

Chemical Reaction

A chemical reaction involves the transformation of reactants into products through a process that rearranges chemical bonds. This process is at the heart of chemistry and can be represented symbolically by a balanced chemical equation. For example, in the reaction \(3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g)\), oxygen molecules are converted into ozone.
Chemical reactions follow the law of conservation of mass, which states that mass cannot be created or destroyed in a closed system. Therefore, the total mass of reactants equals the total mass of products.

• Reactions can be categorized into several types, including synthesis, decomposition, single replacement, and double replacement.
• Each reaction type has distinct characteristics and occurs under specific conditions.

Understanding chemical reactions allows us to predict the behavior of substances under different conditions and develop methods to harness them for practical applications.

Equilibrium Concentrations

Equilibrium concentrations are the stable concentrations of reactants and products in a chemical reaction at equilibrium. When a reaction reaches this state, the amounts of each substance no longer change.
To calculate equilibrium concentrations, we use the equilibrium constant expression. For instance, in the reaction \(3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g)\), the equilibrium constant \(K\) is expressed as:
\[K = \frac{[\mathrm{O}_3]^2}{[\mathrm{O}_2]^3}\]
Given the equilibrium constant, you can determine the unknown concentrations if you know the concentrations of other species involved. By substituting values into the equation, you can solve for the unknowns.

• If \([\mathrm{O}_2]\) is known, \([\mathrm{O}_3]\) can be calculated, and vice versa.
• This calculation is crucial in predicting the composition of reaction systems under given conditions.

Mastering the determination of equilibrium concentrations is essential for understanding and controlling chemical processes.