Problem 47 - General Chemistry: Principles and Modern Applications

Question

One important reaction in the citric acid cycle is citrate(aq) \(\rightleftharpoons\) aconitate \((\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \quad K=0.031\) Write the equilibrium constant expression for the above reaction. Given that the concentrations of \([\text { citrate }(\mathrm{aq})]=0.00128 \mathrm{M},[\text { aconitate }(\mathrm{aq})]=4.0 \times\) \(10^{-5} \mathrm{M},\) and \(\left[\mathrm{H}_{2} \mathrm{O}\right]=55.5 \mathrm{M},\) calculate the reaction quotient. Is this reaction at equilibrium? If not, in which direction will it proceed?

Step-by-Step Solution

Answer

The reaction quotient (\(Q\)) for the given reaction is calculated to be 1.738. As this is greater than the equilibrium constant (\(K = 0.031\)), the reaction is not at equilibrium and will proceed in the reverse direction.

Step 1: Write the equilibrium constant expression

The equilibrium constant expression for a chemical reaction is defined as the product of the concentrations of the products raised to their stoichiometric coefficients, divided by the product of the concentrations of the reactants raised to their stoichiometric coefficients. In the reaction given, citrate is the reactant and aconitate and water are the products. Therefore, the equilibrium constant expression \(K\) for the reaction is \(K = \frac{[aconitate][H_{2}O]}{[citrate]}\).

Step 2: Calculate the reaction quotient

The reaction quotient \(Q\) is calculated the same way as the equilibrium constant: it is the ratio of the products' concentrations to the reactants' concentrations. Given the concentrations of citrate, aconitate, and water, we have \(Q = \frac{[aconitate][H_{2}O]}{[citrate]} = \frac{(4.0 \times 10^{-5})(55.5)}{0.00128}\). Solving this gives \(Q = 1.738\).

Step 3: Determine if the reaction is at equilibrium

To determine if a reaction is at equilibrium, we compare the reaction quotient \(Q\) to the equilibrium constant \(K\). If \(Q = K\), the reaction is at equilibrium. If \(Q > K\), the reaction will proceed in the reverse direction to attain equilibrium. If \(Q < K\), the reaction will proceed in the forward direction to reach equilibrium. In this case, \(Q = 1.738\) and \(K = 0.031\). Since \(Q > K\), the reaction will proceed in the reverse direction to reach equilibrium.

Key Concepts

Reaction Quotient

The reaction quotient, denoted as \(Q\), is a measure used to determine the direction in which a chemical reaction will proceed to reach equilibrium. It's calculated using the concentrations of reactants and products at any point in the reaction, not just at equilibrium.
For the given reaction in the citric acid cycle, the equation can be written as:

• \(Q = \frac{[\text{aconitate}][\text{H}_2\text{O}]}{[\text{citrate}]}\)

Similar to how we calculate the equilibrium constant \(K\), \(Q\) compares the products and reactants' concentrations. In our example, by substituting the given concentrations:

• \([\text{aconitate}] = 4.0 \times 10^{-5} \text{ M}\)
• \([\text{H}_2\text{O}] = 55.5 \text{ M}\)
• \([\text{citrate}] = 0.00128 \text{ M}\)

We find \(Q\) to be 1.738. After calculating \(Q\), it's compared with \(K\) to understand if the reaction is currently balanced or needs to shift to attain equilibrium.

Equilibrium State

The equilibrium state of a reaction occurs when the rate of the forward reaction equals the rate of the reverse reaction, resulting in constant concentrations of reactants and products.
In the context of our reaction, we use the equilibrium constant \(K\) to assess this state. Here, \(K = 0.031\). By comparing \(Q\) and \(K\):

• If \(Q = K\), the system is at equilibrium.
• If \(Q > K\), the reaction will move in the reverse direction to attain equilibrium.
• If \(Q < K\), the forward reaction will dominate to achieve equilibrium.

Since the calculated \(Q = 1.738\) is greater than \(K = 0.031\), the reaction is not at equilibrium currently and will shift in the reverse direction to reach that state.

Citric Acid Cycle

The citric acid cycle, also known as the Krebs cycle, is a crucial part of cellular respiration. It's a series of enzyme-catalyzed chemical reactions that takes place in the mitochondria of cells. This cycle is essential for the production of energy, providing electrons to the electron transport chain where most ATP is generated.
Key points about the citric acid cycle include:

• The cycle begins with a molecule of acetyl-CoA combining with oxaloacetate to form citrate.
• Our example reaction is part of this cycle, where citrate is transformed into aconitate and water.
• This cycle not only helps in energy production but also supplies important intermediates for biosynthetic pathways.

Again, understanding these reactions within this cycle is vital for grasping how energy flows through biological systems and how balance (or equilibrium) in these reactions can affect cellular functions.