Question

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

Short answer

The relationship between the concentrations of A and B at equilibrium, when \(K_c = 1\) for the given reaction \(2A(g) \rightleftharpoons B(g)\), is \([B] = [A]^2\). This means that the concentration of B at equilibrium is equal to the concentration of A squared.

Step-by-step solution

Write the expression for the equilibrium constant

For a general equilibrium reaction, the equilibrium constant Kc is expressed as follows: Kc = (concentration of products) / (concentration of reactants) For the given equilibrium reaction 2A(g) ⇌ B(g), the products are B, and the reactants are A. Since there are 2 moles of A involved in the reaction, the concentration of A will be squared. Kc = \([B]\) / \([A]^2\) Given Kc = 1, we have: 1 = \([B]\) / \([A]^2\)

Solve for the relationship between [A] and [B]

We want to find a relationship between [A] and [B]. From the expression in Step 1, we can solve for [B] as a function of [A]. \([B]\) = \([A]^2\)

Interpret the result

The relationship \([B]\) = \([A]^2\) indicates that the concentration of B at equilibrium is equal to the concentration of A squared. This means that for any given concentration of A, the concentration of B will always be the square of that value at equilibrium when Kc = 1.

Key concepts

Equilibrium Reactions

In the study of chemistry, equilibrium reactions are processes in which the formation of reactants and products occurs simultaneously and at the same rate. Imagine a busy intersection where the traffic flow into and out of is equal; no side gets congested. This is what happens at the molecular level in an equilibrium reaction.

For an equilibrium reaction represented by the generic equation \( aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) \), the reaction proceeds in both the forward (to produce products C and D) and reverse (to reform reactants A and B) directions. When these two rates are equal, the reaction has reached a state of balance where the concentrations of all reactants and products remain constant over time, though they are not necessarily equal to each other.

It's important to understand that this doesn't mean the reactions have stopped; they are still occurring but are in a dynamic balance. This concept is fundamental in predicting how a system responds to changes in conditions, like changes in concentration, pressure, or temperature, according to Le Chatelier's principle.

Chemical Equilibrium

Chemical equilibrium is a particular type of equilibrium that occurs when the rate of the forward chemical reaction is equal to the rate of the reverse reaction. At this point, the concentrations of reactants and products no longer change with time. Chemical equilibrium does not imply that reactants and products are present in equal concentrations, but rather that their concentrations are stable and proportional based on specific ratios governed by the reaction’s equilibrium constant (\( K_{eq} \)).

The concept of chemical equilibrium is central in chemistry because it allows chemists to understand how chemical reactions occur under different conditions and how to manipulate those conditions to achieve a desired output of products. When you cook or when a plant synthesizes glucose during photosynthesis, these processes involve reactions that strive towards equilibrium.

In the given exercise example, the system described by the reaction \( 2A(g) \rightleftharpoons B(g) \) will reach chemical equilibrium when the rate at which A is converting into B is exactly matched by the rate at which B is converting back into A. The state of equilibrium is reflected in the equilibrium constant, which, in this case, has been mentioned to be 1, indicating a specific ratio of the concentrations of A and B, as detailed in the solution.

Reaction Quotient

The reaction quotient (\( Q_c \)) is a measure that compares the relative amounts of products and reactants present during a reaction at any point in time. It is essentially the equilibrium expression calculated before the system has reached equilibrium. By comparing the reaction quotient to the equilibrium constant (\( K_c \)), we can predict the direction in which the reaction will proceed to reach equilibrium.

If \( Q_c < K_c \) then the forward reaction is favored, and the system will produce more products. If \( Q_c > K_c \) then the reverse reaction is favored, and the system will form more reactants. If \( Q_c = K_c \), the system is already at equilibrium, and no net change will occur.

In our exercise, if we consider different concentrations of A and B at some point during the reaction, we can calculate the reaction quotient \( Q_c = [B] / [A]^2 \) to determine the system's status related to equilibrium. If not at equilibrium, the system will shift to reach the point where \( Q_c \) equals the given equilibrium constant \( K_c = 1 \). Understanding the reaction quotient is crucial for anticipating and controlling the outcome of reactions by adjusting concentrations, pressure, or temperature.