Question

Suppose that for the reaction $$\mathrm{CH}_{3} \mathrm{OH}(g) \rightleftharpoons \mathrm{CH}_{2} \mathrm{O}(g)+\mathrm{H}_{2}(g)$$ it is determined that, at a particular temperature, the equilibrium concentrations are \(\left[\mathrm{CH}_{3} \mathrm{OH}(g)\right]=\) \(0.00215 M,\left[\mathrm{CH}_{2} \mathrm{O}(g)\right]=0.441 \mathrm{M},\) and \(\left[\mathrm{H}_{2}(g)\right]=\) 0.0331 M. Calculate the value of \(K\) for the reaction at this temperature.

Short answer

The equilibrium constant (K) for the reaction at this temperature is approximately 6.79.

Step-by-step solution

Write the given information and the balanced equation

The balanced chemical equation for the reaction is: \[ \mathrm{CH}_{3}\mathrm{OH}(g) \rightleftharpoons \mathrm{CH}_{2}\mathrm{O}(g) + \mathrm{H}_{2}(g) \] At equilibrium, the concentrations are: \( \left[\mathrm{CH}_{3}\mathrm{OH}(g)\right] = 0.00215 M \\ \left[\mathrm{CH}_{2}\mathrm{O}(g)\right] = 0.441 M \\ \left[\mathrm{H}_{2}(g)\right] = 0.0331 M \)

Write the expression for the equilibrium constant (K)

The equilibrium constant expression for the given reaction is: \[ K = \frac{\left[\mathrm{CH}_{2}\mathrm{O}(g)\right] \left[\mathrm{H}_{2}(g)\right]}{\left[\mathrm{CH}_{3}\mathrm{OH}(g)\right]} \]

Substitute the known concentrations into the K expression

Insert the equilibrium concentrations into the equilibrium constant expression: \[ K = \frac{(0.441 M)(0.0331 M)}{0.00215 M} \]

Calculate the value of K

Now, we will calculate the value of K using the given concentrations: \[ K = \frac{(0.441)(0.0331)}{0.00215} \approx 6.79 \] The equilibrium constant (K) for the reaction at this temperature is approximately 6.79.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental principle in chemistry that occurs when a chemical reaction and its reverse reaction proceed at the same rate, resulting in a stable ratio of reactants and products. It's like a tug-of-war where both teams are equally strong, and neither side is winning. This state can be reached in both closed systems, where no new reactants are added nor products removed, and involves dynamic processes—not a standstill—where reactants continue to form products and vice versa.

Understanding chemical equilibrium is crucial because it helps us predict how a reaction will behave over time. Factors such as temperature, pressure, and the presence of catalysts can shift the equilibrium position, favoring either the formation of reactants or products.

For students tackling equilibrium problems, visualizing the reaction as a balanced see-saw can be a helpful analogy. This balance is necessary for maintaining life processes, industrial synthesis, and even environmental systems.

Reaction Concentration

In chemical reactions, the concentration of different substances can significantly impact how the reaction proceeds. Reaction concentration specifically refers to the amount of a substance in a given volume of solution, typically expressed in moles per liter (Molarity, M). It's akin to how much sugar you've stirred into your coffee—a higher amount means a sweeter drink.

When dealing with chemical equilibria, the concentrations of reactants and products become central to understanding and calculating the reaction's behavior at equilibrium. For instance, if a problem gives the concentration of each substance at equilibrium, these values can be used to calculate the equilibrium constant, which provides insight into the reaction's extent and direction under specific conditions.

It's essential for students to measure and calculate concentrations accurately, as these values are key ingredients in crafting the final calculation of equilibrium constants. Think of it as following a recipe carefully to ensure your cake rises perfectly.

Equilibrium Constant Expression

The equilibrium constant expression, represented by the letter K, is a formula that quantifies the position of the equilibrium. For a generic reaction where reactants A and B form products C and D, the equilibrium constant expression can be written as:
\[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
Here, the letters a, b, c, and d represent the stoichiometric coefficients of the reactants and products, while the square brackets indicate concentration in molarity.

The value of K itself offers deep insight into the reaction's characteristics at a specific temperature. A large K value suggests products are favored, whereas a small K indicates a preference for reactants. It's crucial for students to understand that K is not just a number—it's a storyteller, revealing the essence of the chemical identity of the reaction under study.

Students should remember that only species in the aqueous or gaseous state are included in the expression, while pure solids and liquids are excluded since their concentrations do not change during a reaction.