Problem 58 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

The relationship between the concentrations of A and B at equilibrium is given by the equation \([B] = [A]^2\). This means that the concentration of B depends on the square of the concentration of A.

Step 1: Write the reaction equation

The given reaction is: \(2A(g) \rightleftharpoons B(g)\)

Step 2: Write the expression for the equilibrium constant

The equilibrium constant (Kc) is given by: \[K_c = \frac{[B]}{[A]^2}\]

Step 3: Substitute the given value of Kc

We are given the value of Kc as 1, so we can substitute this as follows: \[1 = \frac{[B]}{[A]^2}\]

Step 4: Solve for the relationship between [A] and [B]

From the equation in step 3, we have: \[[B] = [A]^2\] This is the relationship between [A] and [B] at equilibrium. When the concentration of A doubles, the concentration of B quadruples, indicating that B depends on the square of the concentration of A.

Key Concepts

Chemical Equilibrium

When a chemical reaction occurs, the reactants transform into products, and these products can sometimes revert back into reactants. This back-and-forth process continues until the system reaches a state called chemical equilibrium. At this point, the rate of the forward reaction equals the rate of the reverse reaction, meaning the concentrations of the reactants and products remain constant over time, not necessarily equal.

It's crucial to recognize that equilibrium describes a dynamic balance. Even though concentrations stay constant, molecules are constantly reacting, maintaining the balance. In our exercise, when 2 moles of gas A are in equilibrium with gas B, we can describe this state mathematically using an equilibrium constant, which in this case is denoted as K_c. It is important to note that changing conditions, such as temperature, can shift the equilibrium, changing the concentrations without changing the value of K_c unless the temperature itself changes.

Reaction Quotient

To predict the direction in which a reaction mixture will proceed to achieve equilibrium, or to check if a system is already at equilibrium, we use the reaction quotient, Q_c. The reaction quotient is calculated using the same expression as the equilibrium constant (K_c) but with the initial concentrations of the reactants and products, not necessarily at equilibrium.

Comparing Q_c and K_c gives us insight into the system's status:

• If Q_c < K_c, the reaction will proceed in the forward direction to reach equilibrium.
• If Q_c > K_c, the reaction will proceed in the reverse direction to reach equilibrium.
• If Q_c = K_c, the system is already at equilibrium.

When we established that K_c = 1 for our reaction, we were implicitly saying that at equilibrium, our reaction quotient would also equal 1. This is a central concept to grasp as it directly relates to the understanding of chemical equilibrium.

Equilibrium Concentration

The equilibrium concentrations of reactants and products are the amounts of substances present in a reaction mixture when the system has reached equilibrium. These concentrations are key to calculating the equilibrium constant, as we've seen in our example. The equilibrium condition for our reaction, 2 A reacts to form B, is expressed by the equation K_c = [B]/[A]².

Because K_c equals 1, we deduced that the equilibrium concentration of B, [B], should equal the square of the equilibrium concentration of A, [A]². This indicates that a small change in [A] would lead to a larger relative change in [B], a good example of how the equilibrium constant provides a specific mathematical relationship between the concentrations of reactants and products in a balanced state.