Problem 3 - Biochemistry

Question

The standard-state free energy of hydrolysis for acetyl phosphate is \\[ \begin{aligned} \Delta G^{\circ}=-42.3 \mathrm{kJ} / \mathrm{mol} \\ \text { Actyl-P }+\mathrm{H}_{2} \mathrm{O} \longrightarrow \text { acctate }+\mathrm{P}_{\mathrm{i}} \end{aligned} \\] Calculate the free energy change for acetyl phosphate hydrolysis in a solution of \(2 \mathrm{m} M\) acctate, \(2 \mathrm{m} M\) phosphate, and \(3 \mathrm{n} M\) acetyl phosphate.

Step-by-Step Solution

Answer

The free energy change for acetyl phosphate hydrolysis in the given solution is approximately -32.4 kJ/mol.

Step 1: Understanding the Problem and the Gibbs-Helmholtz Equation

The problem asks for the free energy change for the hydrolysis of acetyl phosphate under non-standard conditions. The free-energy change under non-standard conditions,\(\Delta G\), can be calculated from the standard free-energy change, \(\Delta G^{\circ}\), using the Gibbs-Helmholtz equation: \(\Delta G = \Delta G^{\circ} + RT\ln Q_c\), where \(R\) is the universal gas constant, \(T\) is the absolute temperature in Kelvin and \(Q_c\) is the reaction quotient, which is the ratio of the concentrations of the products to the reactants at any point in time.

Step 2: Identify Given Variables and the Reaction Quotient

In this exercise, the standard-state free energy change is given as -42.3 kJ/mol. The absolute temperature is not provided, but we will use the value at room temperature of 298K. For this reaction, the reaction quotient \(Q_c\) is given by: \([Acetate][Pi] / [Acetyl-P]\). The given concentrations in the problem are: [Acetate] = 2 mM, [Pi] = 2 mM and [Acetyl-P] = 3 nM. We can use these values to calculate the reaction quotient \(Q_c\).

Step 3: Calculate the Reaction Quotient

Simply substitute the concentration of reactants and products to calculate \(Q_c\). So, \(Q_c = (2mmol/L * 2mmol/L) / (3nmol/L) = 1.33 x 10^6\).

Step 4: Calculate Delta G

Substitute the values of \(Q_c\), \(\Delta G^{\circ}\), \(R\) (8.3145 J/K.mol) and \(T\) (298 K) into the Gibbs-Helmholtz equation to calculate the free energy change. So \(\Delta G = -42.3 kJ/mol + (8.3145 J/K.mol * 298 K) ln (1.33 x 10^6) = -32.4 kJ/mol\).

Key Concepts

Gibbs Free Energy

Gibbs free energy, typically denoted by \(\Delta G\), is a thermodynamic quantity that indicates the amount of reversible work obtainable from a chemical reaction at constant temperature and pressure. It's a central concept in chemical thermodynamics and provides critical insights into whether a process will occur spontaneously. The sign of \(\Delta G\) tells us about the spontaneity of the reaction: if it’s negative, the reaction proceeds spontaneously, positive values indicate a non-spontaneous reaction, and a value of zero signifies a system in equilibrium.

When we talk about hydrolysis reactions like that of acetyl phosphate, understanding Gibbs free energy helps predict the direction of the reaction, and aids the decision-making process in biological and industrial processes. For example, many of the energy-transferring reactions in metabolism involve changes in Gibbs free energy. Understanding \(\Delta G\) is vital for students to grasp the energetics of reactions and predict the behavior of biochemical systems.

Reaction Quotient (Qc)

The reaction quotient, \(Q_c\), represents the ratio of product concentrations to reactant concentrations at any point during a reaction, each raised to the power of their stoichiometric coefficients in the balanced chemical equation. For a generic reaction \( aA + bB \rightleftharpoons cC + dD \), the reaction quotient is defined as \( Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \).

In the context of acetyl phosphate hydrolysis, \(Q_c\) is instrumental in determining how far the reaction has proceeded from equilibrium under non-standard conditions. By comparing \(Q_c\) to the equilibrium constant \(K\), students can discern whether the reaction will proceed forward to create more products, remain at equilibrium, or move in the reverse direction to create more reactants. Calculating \(Q_c\) and understanding its significance in the reaction context are crucial learning objectives.

Gibbs-Helmholtz Equation

The Gibbs-Helmholtz equation is a foundational equation in thermodynamics that relates Gibbs free energy change \(\Delta G\) for a process to its standard state free energy change \(\Delta G^\circ\), the temperature \(T\), and the reaction quotient \(Q_c\). Expressed mathematically as \(\Delta G = \Delta G^\circ + RT\ln Q_c\), this equation allows us to calculate the free energy change under non-standard conditions.

The equation incorporates the universal gas constant \(R\) and \(T\), the absolute temperature, essentially adjusting for variations due to different reaction environments. When students learn how to utilize this equation, they gain the power to predict reaction spontaneity at varying concentrations and temperatures, aiding in a deeper understanding of chemical processes beyond the idealized standard conditions.

Standard-State Free Energy

Standard-state free energy change \(\Delta G^\circ\) is the change in Gibbs free energy when a reaction occurs under standard conditions, meaning each reactant and product is at a concentration of 1 molar, the pressure is 1 atm, and the reaction is carried out at a specified temperature, usually 25°C or 298K.

Referencing the exercise with acetyl phosphate, \(\Delta G^\circ\) provides a baseline for the energy change. It is a fixed value that scientists have determined through experimental measurements under these strict conditions. For students tackling thermodynamics problems, recognizing that \(\Delta G^\circ\) is distinct from \(\Delta G\), which varies with actual reaction conditions, is paramount. Understanding this concept assists students in connecting the theoretical aspects of chemical thermodynamics with real-world applications, where conditions are often non-standard.