Question

At \(327^{\circ} \mathrm{C}\), the equilibrium concentrations are \(\left[\mathrm{CH}_{3} \mathrm{OH}\right]=0.15 \mathrm{M}\), \([\mathrm{CO}]=0.24 M\), and \(\left[\mathrm{H}_{2}\right]=1.1 M\) for the reaction $$ \mathrm{CH}_{3} \mathrm{OH}(g) \rightleftharpoons \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) $$ Calculate \(K_{\mathrm{p}}\) at this temperature.

Short answer

In this exercise, we have the reaction \(\mathrm{CH}_{3} \mathrm{OH}(g) \rightleftharpoons \mathrm{CO}(g)+2 \mathrm{H}_{2}(g)\) with equilibrium concentrations: \(\left[\mathrm{CH}_{3} \mathrm{OH}\right] = 0.15 M\), \([\mathrm{CO}] = 0.24 M\), and \(\left[\mathrm{H}_{2}\right] = 1.1 M\). The equilibrium constant \(K_p\) can be calculated using the formula \(K_{p} = \frac{[\mathrm{CO}]^{1} [\mathrm{H}_{2}]^{2}}{[\mathrm{CH}_{3} \mathrm{OH}]^{1}}\). Substituting the equilibrium concentrations, we find \(K_p \approx 1.936\) at \(327^\circ \mathrm{C}\).

Step-by-step solution

Identify the stoichiometric coefficients

In the given reaction, the stoichiometric coefficients are: \(\mathrm{CH}_{3}\mathrm{OH}: 1\) \(\mathrm{CO}: 1\) \(\mathrm{H}_{2}: 2\)

Use the equilibrium constant formula

Using the \(K_p\) formula, we have: \[K_{p} = \frac{[\mathrm{CO}]^{1} [\mathrm{H}_{2}]^{2}}{[\mathrm{CH}_{3} \mathrm{OH}]^{1}}\]

Substitute the equilibrium concentrations

Now, substituting the equilibrium concentrations into the formula, we get: \(K_p = \frac{(0.24)^{1} (1.1)^{2}}{(0.15)^{1}}\)

Calculate the equilibrium constant

Perform the calculations to find the equilibrium constant: \(K_p = \frac{(0.24) (1.21)}{(0.15)}\) \(K_p = \frac{0.2904}{0.15}\) \(K_p \approx 1.936\) At \(327^\circ \mathrm{C}\), the equilibrium constant \(K_p\) for this reaction is approximately equal to 1.936.

Key concepts

Equilibrium Concentrations

Equilibrium concentrations refer to the amounts of reactants and products present when a chemical reaction has reached equilibrium. At equilibrium, the concentrations of different species do not change over time, even though they are still actively participating in the reaction. This is because the rate of the forward reaction equals the rate of the reverse reaction.
To calculate equilibrium concentrations, scientific notation is often used and denoted with square brackets, such as \([\text{CH}_3\text{OH}]\), \([\text{CO}]\), and \([\text{H}_2]\). In this exercise, the equilibrium concentrations at 327°C are given, specifically: 0.15 M for \([\text{CH}_3\text{OH}]\), 0.24 M for \([\text{CO}]\), and 1.1 M for \([\text{H}_2]\).

• These values are used to calculate the equilibrium constant, \(K_p\), which provides valuable information on the position of equilibrium.
• Knowing equilibrium concentrations allows chemists to predict how a reaction will respond to changes in concentration, temperature, or pressure based on Le Chatelier's Principle.

Understanding equilibrium concentrations is essential for mastering the concept of chemical equilibrium and predicting the behavior of chemical reactions.

Reaction Stoichiometry

Reaction stoichiometry refers to the quantitative relationships between reactants and products in a chemical reaction. It's essentially the recipe for a chemical reaction, indicating not only what substances are involved but also in what proportions they react and form new substances.
For the reaction \( \text{CH}_3\text{OH}(g) \rightleftharpoons \text{CO}(g) + 2\text{H}_2(g) \), the stoichiometric coefficients are crucial in determining the equilibrium expression. The coefficients for each substance are:

• \(\text{CH}_3\text{OH}: 1\)
• \(\text{CO}: 1\)
• \(\text{H}_2: 2\)

These coefficients indicate that 1 mole of methanol (\(\text{CH}_3\text{OH}\)) decomposes to form 1 mole of carbon monoxide (\(\text{CO}\)) and 2 moles of diatomic hydrogen (\(\text{H}_2\)).
Stoichiometry is used to set up the equilibrium constant expression \(\left( K_p \right)\). In this case, the form is \[ K_{p} = \frac{[\text{CO}]^1 [\text{H}_2]^2}{[\text{CH}_3\text{OH}]^1} \].
Understanding stoichiometry helps in predicting the amounts of reactants needed and products formed, as well as determining the yield of a reaction.

Chemical Equilibrium

Chemical equilibrium is the state of a chemical reaction when the concentrations of reactants and products become constant. It does not mean the reactions have stopped but that their rates in both the forward and reverse directions are equal, creating a dynamic balance.
In the context of the given reaction, \( \text{CH}_3\text{OH}(g) \rightleftharpoons \text{CO}(g) + 2\text{H}_2(g) \), at 327°C, the system has reached equilibrium with the specified concentrations.
One important aspect of chemical equilibrium is the equilibrium constant, \(K\), with \(K_p\) being used when dealing with gaseous reactions.

• \(K_p\) indicates how far the reaction has proceeded towards products at equilibrium under standard conditions.
• A \(K_p\) value greater than 1 suggests that at equilibrium, the products are favored; whereas a value less than 1 indicates the reactants are favored.

The calculated \(K_p\) here is approximately 1.936, showing that products and reactants are present in comparable amounts. Understanding chemical equilibrium is crucial for scientists to manipulate conditions to favor the formation of desired products in reactions.