Problem 72 - Chemistry The Central Science

Question

If \(K_{c}=1\) for the equilibrium \(2 \mathrm{A}(g) \rightleftharpoons \mathrm{B}(g),\) what is the relationship between \([\mathrm{A}]\) and \([\mathrm{B}]\) at equilibrium?

Step-by-Step Solution

Answer

The relationship between the equilibrium concentrations of A and B can be expressed as: \([B] = [A]^2\).

Step 1: Write the equilibrium expression

For the given reaction, 2 A(g) ⇌ B(g), the equilibrium constant (Kc) can be expressed in terms of the equilibrium concentrations of the reactants and products. The general form of the expression for Kc is: Kc = \(\frac{[Products]}{[Reactants]}\) For this particular reaction, Kc will be: Kc = \(\frac{[B]}{[A]^2}\) We are given that Kc = 1.

Step 2: Substitute Kc into the equilibrium expression and solve for the relationship

Now we can substitute the given Kc value (1) into the equilibrium expression: 1 = \(\frac{[B]}{[A]^2}\) To find the relationship, rearrange the equation to isolate [B] or [A] on one side. In this case, we can isolate [B]: [B] = [A]^2

Step 3: State the relationship between [A] and [B]

At equilibrium, the concentration of B, [B], is equal to the square of the concentration of A, [A]^2. Therefore, the relationship between the equilibrium concentrations of A and B can be expressed as: [B] = [A]^2

Key Concepts

Chemical Equilibrium

Chemical equilibrium refers to a state in a chemical reaction where the concentrations of reactants and products no longer change over time. This doesn't mean the reaction has stopped, but that the forward and reverse reactions occur at the same rate. As a result, the concentration of each substance remains constant, even though both reactions are still happening.
In simple terms, it's like a busy intersection where cars enter and leave at the same pace, so the number of cars in the intersection remains the same. Achieving equilibrium depends on the reaction conditions, such as temperature and pressure, but not on the initial concentrations of the reactants and products. It's a balance point that every reversible reaction strives to reach under set conditions. Understanding chemical equilibrium helps predict how changes in conditions will affect the concentrations of substances in a chemical reaction.

Concentration Relationship

In equilibrium calculations, understanding the concentration relationship between reactants and products is crucial. For the reaction given in the exercise, 2 A(g) ⇌ B(g), the concentrations of reactants and products have a specific mathematical relationship defined by the equilibrium constant, denoted as \( K_{c} \).
When the given \( K_{c} \) value is 1, it provides insight into the ratio of product concentration to its reactant concentration at equilibrium. In this reaction, it particularly means that at equilibrium, the concentration of \( B \) is equal to the square of the concentration of \( A \), expressed as

• \( [B] = [A]^2 \)

This relationship highlights the importance of stoichiometry, the coefficients in the balanced equation, which show how reactants convert to products. It demonstrates how equilibrium constants help predict the concentrations of substances involved in a reaction at equilibrium.

Equilibrium Expression

An equilibrium expression mathematically represents the relationship between the concentrations of reactants and products for a reversible reaction at equilibrium. For the reaction 2 A(g) ⇌ B(g), the equilibrium expression is written using the equilibrium constant \( K_{c} \), where: \( K_{c} = \frac{[B]}{[A]^2} \).
Here, [B] represents the concentration of product \( B \), and [A] that of reactant \( A \). The coefficients from the balanced chemical equation become exponents in the expression. This expression shows us how to set up ratios that simplify to the value of \( K_{c} \), connecting it directly to the concentration values.

• If \( K_{c} = 1 \), it indicates equal proportions reflected by the concentrations at equilibrium. This specific case demonstrates that the square of \( A's \) concentration must equate that of \( B \), or \([B] = [A]^2\).

Equilibrium expressions thus provide a powerful tool for understanding and predicting how chemical species will present themselves once a reaction reaches equilibrium. It helps chemists and anyone working with chemical reactions to predict and manipulate reaction conditions effectively.