Problem 98 - Chemistry: An Atoms-Focused Approach

Question

At \(2000^{\circ} \mathrm{C}, K_{\epsilon}=1.0\) for the following reaction: $$ 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{CO}_{2}(g) $$ What is the ratio of \([\mathrm{CO}]\) to \(\left[\mathrm{CO}_{2}\right]\) in an atmosphere in which \(\left[\mathrm{O}_{2}\right]=0.0045 M ?\)

Step-by-Step Solution

Answer

The ratio of the concentration of CO to CO₂ in the atmosphere is approximately 14.9.

Step 1: Write the equilibrium expression

Considering the balanced chemical equation given, the equilibrium constant expression can be written as: $$ K_\epsilon = \frac{[\mathrm{CO}_2]^2}{[\mathrm{CO}]^2 \cdot [\mathrm{O}_{2}]} $$ Now we are given the values of \(K_\epsilon = 1.0\) and \([\mathrm{O}_2]=0.0045\,\mathrm{M}\).

Step 2: Introduce variables for unknown concentrations

Let \(x\) be the concentration of CO and \(y\) be the concentration of CO₂ at equilibrium. We need to find the ratio \(\frac{x}{y}\). Substitute the unknown concentrations and the given values in the equilibrium constant expression: $$ 1.0 = \frac{y^2}{x^2 \cdot 0.0045} $$

Step 3: Rearrange the equation to find the ratio

Now we want to isolate \(\frac{x}{y}\) (the desired ratio) from the above equation. First, isolate \(\frac{y^2}{x^2}\): $$ \frac{y^2}{x^2} = 0.0045 $$ Now take the square root of both sides: $$ \frac{y}{x} = \sqrt{0.0045} $$ Then, to find the ratio between CO and CO₂ concentrations, we must reciprocate the equation: $$ \frac{x}{y} = \frac{1}{\sqrt{0.0045}} $$

Step 4: Calculate the ratio

Now we can calculate the ratio by solving the last equation: $$ \frac{x}{y} = \frac{1}{\sqrt{0.0045}} \approx 14.9 $$ Therefore, the ratio of the concentration of CO to the concentration of CO₂ in the given atmosphere is approximately 14.9.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a fascinating concept encountered in reactions where the forward and backward processes occur at the same rate. In this state, the concentrations of reactants and products remain constant over time. For a reaction, reaching equilibrium doesn't mean the reactants and products are present in equal amounts; instead, it refers to a balance between the two opposing reactions.
In the reaction given in the exercise, the equilibrium state means that the production of carbon dioxide from carbon monoxide and oxygen occurs at the same rate as the reverse reaction. By studying the equilibrium constant, we can understand the ratio between the concentrations of these substances when they have reached equilibrium, without having to measure each concentration individually.

Reaction Quotient

The reaction quotient, denoted as \(Q\), is a calculation used to determine the direction a reaction will shift to reach equilibrium. While similar to the equilibrium constant \(K\), \(Q\) is used when a system is not at equilibrium. By comparing \(Q\) and \(K\), we can predict whether the reaction will proceed forward or backward.
The reaction quotient is calculated using the same expression as the equilibrium constant, substituting the current concentrations into the formula. If the value of \(Q\) matches \(K\), the system is at equilibrium. If \(Q\) is less than \(K\), the reaction will proceed forward to form more products. Conversely, if \(Q\) is greater than \(K\), the reaction will shift backward to form more reactants.

Concentration

In chemical reactions, concentration refers to the amount of a substance in a certain volume. It is a critical factor in determining how reactions occur and how they reach equilibrium. Concentration affects the rate of reaction; higher concentrations generally lead to faster reactions due to more frequent collisions between reacting particles.
In the provided exercise, the concentrations of carbon monoxide, carbon dioxide, and oxygen are crucial for calculating the equilibrium constant. The exercise gives us the concentration of oxygen, allowing us to determine the ratio between carbon monoxide and carbon dioxide using the equilibrium constant. Understanding concentrations helps us predict how changes will affect the balance of a chemical reaction.

Stoichiometry

Stoichiometry is the study of the quantitative relationships between reactants and products in a chemical reaction. It involves using balanced chemical equations to determine the proportions of substances required for reactions to occur.
In the reaction \(2 \text{CO}(g) + \text{O}_2(g) \rightleftharpoons 2 \text{CO}_2(g)\), stoichiometry tells us that two molecules of carbon monoxide react with one molecule of oxygen to produce two molecules of carbon dioxide. This balanced equation is essential for setting up the equilibrium expression and also helps calculate the unknown concentrations, as shown in the step-by-step solution. Mastering stoichiometry is fundamental to solving problems involving chemical reactions and equilibrium.