Question

For the reaction $$2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{CO}_{2}(g)$$ it is found at equilibrium at a certain temperature that the concentrations are \([\mathrm{CO}(g)]=2.7 \times 10^{-4} \mathrm{M}\) \(\left[\mathrm{O}_{2}(g)\right]=1.9 \times 10^{-3} \mathrm{M},\) and \(\left[\mathrm{CO}_{2}(g)\right]=1.1 \times 10^{-1} \mathrm{M}\) Calculate \(K\) for the reaction at this temperature.

Short answer

The equilibrium constant (K) for the reaction at this temperature is approximately 441.

Step-by-step solution

Write the balanced chemical equation.

The balanced chemical equation for the reaction is given: \(2\, CO(g) + O_{2}(g) \rightleftharpoons 2\, CO_{2}(g)\)

Write the equilibrium expression for the reaction.

For the given balanced reaction, the equilibrium expression (K) will be the ratio of the product concentration to the reactant concentration: \(K = \frac{[CO_2]^2}{[CO]^2 \times [O_2]}\)

Substitute the given equilibrium concentrations into the equilibrium expression.

We are given the equilibrium concentrations of CO, O₂, and CO₂: - [CO]: \(2.7 \times 10^{-4}\, M\) - [O₂]: \(1.9 \times 10^{-3}\, M\) - [CO₂]: \(1.1 \times 10^{-1}\, M\) Now, substitute these values into the equilibrium expression: \(K = \frac{(1.1 \times 10^{-1})^2}{(2.7 \times 10^{-4})^2 \times (1.9 \times 10^{-3})}\)

Calculate the equilibrium constant (K).

Now, perform the calculations to get the value of K: \(K = \frac{(1.1 \times 10^{-1})^2}{(2.7 \times 10^{-4})^2 \times (1.9 \times 10^{-3})} = 440.97\)

Round the answer to an appropriate number of significant figures.

In this case, we are given three significant figures for the initial concentrations, so we should round our answer for K to three significant figures as well: \(K = 441\) The equilibrium constant (K) for the reaction at this temperature is approximately 441.

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction proceeds at the same rate as its reverse reaction, resulting in no overall change in the concentrations of the reactants and products. This state can be reached in both closed systems and isolated systems where neither matter nor energy can exit or enter the system.

Imagine a busy intersection where the number of cars entering equals the number of cars leaving. The number of cars within the intersection remains constant over time, similar to how the concentrations of reactants and products in a chemical reaction at equilibrium remain constant. However, 'constant concentration' doesn't mean that the reactants and products are equal in concentration; rather, they have settled into a stable ratio. Understanding this balance is essential in fields like chemistry and engineering, where predicting the outcome of reactions is crucial.

Equilibrium Expression

The equilibrium expression, which represents the ratio of the concentrations of the products and reactants at equilibrium, is unique to each reaction and is dependent on the balanced chemical equation. For a general reaction where aA + bB \( \rightleftharpoons \) cC + dD, the equilibrium expression would look like this:
\[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]

The square brackets denote the molarity, or concentration, of the species, and the exponents correspond to their coefficients in the balanced equation. A key point to remember is that only 'aqui' reactants and products appear in the expression; pure solids and liquids are omitted, while gases and species in solution are included. This expression allows us to calculate the equilibrium constant (K), which provides valuable insights into the extent of the reaction and how shifts in conditions might affect the system.

Equilibrium Concentrations

Equilibrium concentrations are the concentrations of the reactants and products in a reaction mixture when the reaction has reached equilibrium. These values are constant at a given temperature and can be used to calculate the equilibrium constant, which quantitatively describes the composition of the reaction mixture in its equilibrium state.

Returning to the car intersection analogy, equilibrium concentrations are like knowing the number of different types of vehicles in that constant flow—how many are cars, bikes, or trucks. Without the detailed knowledge of these amounts (concentrations), it would be impossible to understand the system's dynamics fully. Similarly, by accurately measuring these values in a chemical system, we can predict how the system will respond to changes like variations in temperature, pressure, or the introduction of additional reactants or products.