Problem 70 - 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

At equilibrium, the relationship between the concentrations of A and B in the given reaction is \([B] = [A]^2\).

Step 1: Write down the balanced chemical equation

For the given reaction, we have: \(2A(g) \rightleftharpoons B(g)\)

Step 2: Write the expression for the equilibrium constant (K)

For a reversible reaction with the form \(aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)\), the equilibrium constant expression is: \[K = \frac{[C]^c [D]^d}{[A]^a [B]^b}\] In our case, the reaction is \(2A(g) \rightleftharpoons B(g)\), so the equilibrium constant expression becomes: \[K_c = \frac{[B]}{[A]^2}\]

Step 3: Substitute the given value of the equilibrium constant

It is given that \(K_c = 1\), so we can substitute this value in our equilibrium constant expression: \[1 = \frac{[B]}{[A]^2}\]

Step 4: Rearrange the equation to find the relationship between [A] and [B]

To find the relationship, we need to rearrange the equation to solve for one variable in terms of the other. Here, we will leave [B] on the left side and solve for [A] on the right side: \[[B] = [A]^2\]

Step 5: Conclusion

At equilibrium, the relationship between the concentrations of A and B in the given reaction is \([B] = [A]^2\).

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K_c \), is a crucial concept in understanding chemical reactions that have reached equilibrium. It provides a quantitative measure of the ratio of concentrations of products to reactants for a given reaction at a set temperature. For a general reversible reaction of the form:\[ aA + bB \rightleftharpoons cC + dD \]The equilibrium constant expression is represented mathematically as:\[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]Knowing the value of \( K_c \) helps chemists predict how the concentration of different species in a reaction will relate to each other when the system reaches equilibrium.

A \( K_c \) value of 1, as seen in the example exercise, implies a balanced relationship where neither reactants nor products are favored at equilibrium. This means that the "forward" reaction from reactants to products, and the "reverse" reaction from products back to reactants, happen at comparable rates.

Concentration Relationship

Understanding the relationship between concentrations at equilibrium is key to mastering chemical equilibria. When given an equilibrium constant expression, such as:\[ K_c = \frac{[B]}{[A]^2} \]You can determine how the concentrations of the reactants and products interrelate. Substituting a known value for \( K_c \), such as 1, allows you to establish explicit concentration relationships. In this case:\[ 1 = \frac{[B]}{[A]^2} \]From rearranging this equation, it becomes apparent that:\[ [B] = [A]^2 \]This equation tells us that, at equilibrium, the concentration of \( B \) is the square of the concentration of \( A \).

In practical terms, for every unit increase in \([A]\), \([B]\) increases quadratically, emphasizing the power of balancing chemical equations and recognizing patterns.

Reversible Reactions

Reversible reactions are a central theme in chemical equilibrium, where both the forward and reverse processes occur. This duality leads to the establishment of a dynamic equilibrium where reaction rates, rather than concentrations, equalize. For example, in a reversible equation like\[ 2A \rightleftharpoons B \],both the formation of \( B \) from \( A \) and the reformation of \( A \) from \( B \) proceed simultaneously.

Characteristics of reversible reactions include:

• Reaching a state where the concentrations of reactants and products remain constant over time.
• Dependence on temperature, pressure, and concentration, which can shift the equilibrium position.
• Reduction of observable changes, despite ongoing molecular interaction.

Understanding these processes is essential for predicting how changes to system conditions (such as pressure or concentration) might affect the position of equilibrium in a reversible reaction.