Problem 48 - General Chemistry: Principles and Modern Applications

Question

The following reaction is an important reaction in the citric acid cycle: citrate(aq) \(+\mathrm{NAD}_{\mathrm{ox}}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightleftharpoons\) \(\mathrm{CO}_{2}(\mathrm{aq})+\mathrm{NAD}_{\mathrm{red}}+\) oxoglutarate \((\mathrm{aq}) \quad K=0.387\) Write the equilibrium constant expression for the above reaction. Given the following data for this reaction, \([\text { citrate }]=0.00128 \mathrm{M},\left[\mathrm{NAD}_{\mathrm{ox}}\right]=0.00868,\left[\mathrm{H}_{2} \mathrm{O}\right]=\) \(55.5 \mathrm{M},\left[\mathrm{CO}_{2}\right]=0.00868 \mathrm{M},\left[\mathrm{NAD}_{\mathrm{red}}\right]=0.00132 \mathrm{M}\) and [oxoglutarate] \(=0.00868 \mathrm{M},\) 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 is approximately 0.258. Since this is less than the given equilibrium constant Kc of 0.387, the reaction is not at equilibrium and will proceed to the right, in the direction of the products.

Step 1: Write the equilibrium constant (Kc) expression

The equilibrium constant Kc for a general reaction \(aA + bB \rightleftharpoons cC + dD\) is given by \( K_c = \frac{( [C]^c \cdot [D]^d )}{( [A]^a \cdot [B]^b )}\). For the given reaction, the equilibrium constant Kc will be \(K_c = \frac{([CO_2] \cdot [NAD_{red}] \cdot [oxoglutarate])}{([citrate] \cdot [NAD_{ox}] \cdot [H_{2}O])}\)

Step 2: Substitute the given concentrations into the Kc expression

Substitute the given concentrations of the reactants and products into the Kc expression: \(Q = \frac{(0.00868 \cdot 0.00132 \cdot 0.00868)}{(0.00128 \cdot 0.00868 \cdot 55.5)}\)

Step 3: Calculate Q

By performing the above calculation, Q is found to be approximately 0.258.

Step 4: Compare Q with Kc and determine the direction of the reaction

Since \(Q < Kc\), the reaction will proceed in the direction of the products to reach equilibrium, meaning it will move to the right.

Key Concepts

Citric Acid Cycle

The Citric Acid Cycle, also known as the Krebs Cycle or TCA Cycle, is a central metabolic pathway found in the cells of aerobic organisms. This fascinating sequence of chemical reactions takes place in the mitochondria and is crucial for the generation of energy. Through a series of enzyme-catalyzed reactions, the cycle oxidizes acetyl-CoA, derived from carbohydrates, fats, and proteins, to produce carbon dioxide, ATP (a form of energy usable by the cell), NADH, and FADH2.

These latter two molecules carry electrons to the electron transport chain, which is another critical phase in cellular respiration, yielding further ATP via oxidative phosphorylation. Each turn of the cycle not only provides energy currency for the cell but also intermediates that are essential for the biosynthesis of various substances. Understanding each step of this cycle is imperative for students studying biochemistry, as it connects to numerous other metabolic pathways and provides insights into cellular energy production.

Reaction Quotient

The reaction quotient, denoted by Q, plays a pivotal role in determining the direction of a chemical reaction at any point before reaching equilibrium. It is calculated in the same manner as the equilibrium constant, using the formula:
\[Q = \frac{([C]^c \cdot [D]^d )}{([A]^a \cdot [B]^b )}\]
where the concentrations are those at a particular moment in time, not necessarily at equilibrium. It enables chemists to predict which way a reaction will proceed. If the value of Q is less than the equilibrium constant (K), the reaction will move forward, converting reactants into products. Conversely, if Q is greater than K, the reaction tends towards the reactants. When Q equals K, the system is at equilibrium, meaning the rates of the forward and reverse reactions are equal, with no net change in the concentrations of reactants and products.

Calculating the reaction quotient is an essential skill in chemistry as it helps in understanding and predicting the outcome of chemical reactions under varying conditions.

Equilibrium in Chemical Reactions

Equilibrium in chemical reactions refers to the state where the concentrations of reactants and products remain constant over time. It occurs when the forward and reverse reaction rates are equal, leading to no observable changes in the system. The equilibrium constant, K, captures the essence of the reaction's balance, representing the ratio of product to reactant concentrations at equilibrium, each raised to the power of their stoichiometric coefficients.

An important aspect of chemical equilibrium is that it is dynamic; reactants continue to form products and vice versa, but their concentrations do not change. This can be a confusing concept for students because even though the reaction is ongoing, there's no net change observable macroscopically. Nonetheless, understanding equilibrium is vital for several areas of chemistry and related fields, including reaction engineering, environmental science, and pharmacology, as it underpins the behavior of chemically reacting systems under stable conditions.