Problem 55 - Chemistry: An Atoms-Focused Approach

Question

If \(K_{c}=22\) for the hypothetical reaction \(\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g)\) at a given temperature, and if \([\mathrm{A}]=0.10 \mathrm{M}\) and \([\mathrm{B}]=2.0 \mathrm{M}\) in a reaction mixture at that temperature, is the reaction at chemical equilibrium? If not, in which direction will the reaction proceed to reach equilibrium?

Step-by-Step Solution

Answer

If not, in which direction will the reaction proceed to reach equilibrium? Answer: The reaction is not at equilibrium. It will proceed in the forward direction (towards the formation of B) to reach equilibrium.

Step 1: Recall the formula for the reaction quotient, \(Q_c\)

For a general reaction \(aA + bB \rightleftharpoons cC + dD\), the reaction quotient, \(Q_c\), is given by: $$Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$$ For our given reaction, A(g) \(\rightleftharpoons\) B(g), the reaction quotient, \(Q_c\), is given by: $$Q_c = \frac{[B]}{[A]}$$

Step 2: Calculate the reaction quotient, \(Q_c\)

We are given the initial concentrations of A and B: \([\mathrm{A}] = 0.10 \, \mathrm{M}\) \([\mathrm{B}] = 2.0 \, \mathrm{M}\) Plug these values into the \(Q_c\) formula: $$Q_c = \frac{[B]}{[A]} = \frac{2.0}{0.10} = 20$$

Step 3: Compare \(Q_c\) to the equilibrium constant, \(K_c\)

Now, compare the calculated reaction quotient (\(Q_c\)) to the given equilibrium constant (\(K_c\)): \(Q_c = 20\) \(K_c = 22\)

Step 4: Determine if the reaction is at equilibrium and the direction it will proceed

If \(Q_c < K_c\), the reaction will proceed in the forward direction (towards the formation of B) to reach equilibrium. If \(Q_c > K_c\), the reaction will proceed in the reverse direction (towards the formation of A) to reach equilibrium. If \(Q_c = K_c\), the reaction is already at equilibrium. In our case, since \(Q_c < K_c\) (20 < 22), the reaction is not at equilibrium and will proceed in the forward direction (towards the formation of B) to reach equilibrium.

Key Concepts

Reaction Quotient

In chemical reactions, understanding the reaction quotient, often denoted as \(Q_c\), helps us determine the current state of a reaction. Essentially, \(Q_c\) provides a snapshot of the reaction's progress based on the concentrations of reactants and products at any given moment. To calculate \(Q_c\), we use the formula:\[Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]This formula is applicable to a general reaction \(aA + bB \rightleftharpoons cC + dD\). Here, the symbols \([A]\), \([B]\), \([C]\), and \([D]\) represent the molar concentrations of the substances involved, while \(a\), \(b\), \(c\), and \(d\) denote their stoichiometric coefficients in the balanced equation. For our specific reaction \(\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g)\), the expression simplifies to:\[Q_c = \frac{[B]}{[A]}\]This means if your reaction's \(Q_c\) is known, it can be compared to the equilibrium constant, \(K_c\), as a way to predict how or if the system will shift to achieve equilibrium.

Equilibrium Constant

The equilibrium constant, \(K_c\), is vital in chemical equilibrium calculations. It indicates the ratio of product concentrations to reactant concentrations at equilibrium, under a given set of conditions, such as temperature. When reactions reach equilibrium, their \(Q_c\) equals \(K_c\), and there is no net change in the concentrations of reactants and products. For the hypothetical reaction \(\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g)\), if \(K_c=22\), this means that at equilibrium, the concentration of \(B\) to \(A\) is 22:1. It captures the intrinsic properties of the reaction system, and will only change with temperature.Understanding \(K_c\) helps in:

• Predicting whether a reaction will favor products or reactants at equilibrium.
• Calculating the concentrations of products or reactants when equilibrium is achieved.

It’s important to note, however, that while \(K_c\) determines the final position of equilibrium, it does not give any details about the rate at which equilibrium will be reached.

Reaction Direction

The concept of reaction direction is crucial in understanding how reactions progress toward equilibrium. By comparing \(Q_c\) with \(K_c\), we can predict which direction the reaction will shift to achieve balance. This is achieved using the following guidelines:- If \(Q_c < K_c\), the reaction proceeds in the forward direction, meaning the formation of products is favored until equilibrium is reached.- If \(Q_c > K_c\), the reaction moves in the reverse direction, indicating that the reaction must produce more reactants to achieve equilibrium.- If \(Q_c = K_c\), the system is already at equilibrium, and no shift will occur.For the given example, where \(Q_c = 20\) and \(K_c = 22\), we have \(Q_c < K_c\). This tells us that the reaction will proceed forward, converting more of \(\mathrm{A}(g)\) into \(\mathrm{B}(g)\), thereby producing more \(B\) until \(Q_c\) equals \(K_c\). Recognizing these trends ensures that we can reliably predict and control the outcomes of chemical processes.