Question

Coupling ATP hydrolysis to a thermodynamically unfavorable reaction can shift the equilibrium of a reaction (see Chapter 19). (a) Calculate \(K\) for the hypothetical reaction \(A \longrightarrow B\) when \(\Delta G^{\circ}\) is \(23 \mathrm{kJ} \mathrm{mol}^{-1}\) at \(25^{\circ} \mathrm{C}\) (b) Calculate \(K\) for the same reaction when it is coupled to the hydrolysis of \(\operatorname{ATP}\left(\Delta G^{\circ \prime}=-30 \mathrm{kJ} \mathrm{mol}^{-1}\right)\) Compare with the value obtained in part (a). (c) Many cells maintain [ATP] to [ADP] ratios of 400 or more. Calculate the ratio of \([\mathrm{B}]\) to \([\mathrm{A}]\) when \(\left.[\mathrm{ATP}] /[\mathrm{ADP}]=400 \text { and } \mathrm{P}_{\mathrm{i}}\right]=5 \mathrm{mM} .\) Compare this ratio to that for the uncoupled reaction.

Short answer

The equilibrium constant for the uncoupled reaction is \( K \approx 7.41 \times 10^{-5}\), while for the coupled reaction it is \( K ' \approx 68\). When we consider the ATP/ADP concentration ratio of 400, the [B]/[A] ratio for the coupled reaction significantly increases in comparison to that for the uncoupled reaction. Thus, ATP hydrolysis when coupled to an otherwise unfavorable reaction can shift the reaction equilibrium significantly in the direction of products.

Step-by-step solution

Calculation of equilibrium constant for the uncoupled reaction

For the reaction \(A \longrightarrow B\) with \(\Delta G^\circ = 23 \mathrm{kJ} \mathrm{mol}^{-1}\) at 25°C, we can calculate the equilibrium constant using the formula \(\Delta G^\circ = -RTlnK\). First, we must convert \( \Delta G\) from kJ/mol to J/mol, as well as convert the temperature from Celsius to Kelvin. So, \( \Delta G = 23000 \, J/mol\) and \( T = 273.15 + 25 = 298.15K\). Furthermore, the universal gas constant \( R = 8.314 J/ mol\,K\). Substituting these values, we solve for \( K\).

Calculation of equilibrium constant for the coupled reaction

The reaction \( A \longrightarrow B \) is coupled with ATP hydrolysis, which has a \(\Delta G^\circ ' = -30kJ/mol\). For the coupled reaction, \(\Delta G^\circ_total = \Delta G^\circ + \Delta G^\circ '\). This calculation yields \(\Delta G^\circ_total = -7k J/mol\). We substitute this value into the formula for \(\Delta G^\circ = -RTlnK\) to solve for the new equilibrium constant \(K '\).

Influence of ATP/ADP concentration ratio on equilibrium

The ratio of [B] to [A] is given by \([B]/[A] = K' \times ([ATP]/[ADP][P_{i}])\), where \( K '\) is the equilibrium constant for the coupled reaction, [ATP]/[ADP]is the ratio of ATP to ADP concentration, and [\(P_{i}\)] is the concentration of inorganic phosphate. Given that [ATP]/[ADP] = 400 and [\(P_{i}\)] = 5mM = 0.005M, we substitute these into the formula to calculate the new [B]/[A] ratio. We then compare this ratio to that for the uncoupled reaction, which is simply \( K\).

Key concepts

ATP Hydrolysis

The process of ATP hydrolysis is a fundamental chemical reaction in biology, whereby adenosine triphosphate (ATP) is converted into adenosine diphosphate (ADP) and inorganic phosphate (\( P_i \)). This reaction releases a significant amount of energy, which is utilized in a wide variety of cellular processes, including the coupling to drive thermodynamically unfavorable reactions.

The significance of ATP hydrolysis in thermodynamics lies in its ability to release free energy, a practical measure of energy that can do work at a constant temperature and pressure, which is crucial in biological systems. In quantitative terms, the standard free energy change \( \Delta G^\circ' \) for ATP hydrolysis is approximately -30 kJ/mol, which indicates a spontaneous process under standard conditions.

When ATP is hydrolyzed, it can transfer some of its energy to endergonic reactions, effectively lowering the overall free energy change of the reaction system and making those reactions more likely to occur. This is how cells harness the energy stored in ATP to perform tasks that require energy input, such as building complex molecules or transporting substances across cell membranes.

Thermodynamics in Chemistry

Thermodynamics in chemistry is a branch of science that deals with the study of energy changes, particularly the heat exchange associated with chemical reactions and physical transformations. One of the core principles is the concept of equilibrium, where a chemical reaction reaches a state in which the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products.

The equilibrium constant (\( K \)) is a value that expresses the ratio of the concentrations of products to reactants at equilibrium, with each raised to the power of their stoichiometric coefficients in the balanced equation. The magnitude of the equilibrium constant gives us an indication of the position of equilibrium, whether the reaction favors the formation of products or reactants. Calculating the equilibrium constant involves understanding the changes in Gibbs free energy for the reaction, highlighting the intrinsic link between thermodynamics and the quantitative aspects of chemical equilibrium.

Gibbs Free Energy

Gibbs free energy, denoted as \(G\), is a thermodynamic quantity that represents the maximum reversible work that may be performed by a thermodynamic system at constant temperature and pressure. It combines enthalpy, entropy, and temperature to provide a criterion for spontaneity of processes. In the context of chemical reactions, the change in Gibbs free energy (\( \Delta G \)) helps to predict the direction of the reaction and its equilibrium position.

The relationship between Gibbs free energy and the equilibrium constant is established by the equation \( \Delta G^\circ = -RT\ln K \), where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(K\) is the equilibrium constant. A negative value for \( \Delta G^\circ \) suggests that the reaction is spontaneous in the forward direction under standard conditions, while a positive value implies a non-spontaneous reaction, requiring the input of energy to proceed.

Calculating Equilibrium from Gibbs Free Energy

To calculate the equilibrium constant \(K\) from \( \Delta G^\circ \), first ensure that \( \Delta G^\circ \) is expressed in joules per mole (J/mol) and that temperature is in Kelvin (K). Then rearrange the equation to solve for \(K\): \(K = e^{-\Delta G^\circ / RT}\). When ATP hydrolysis is coupled to a reaction, the total free energy change \( \Delta G^\circ_{total}\) is the sum of the free energy changes for both ATP hydrolysis and the other reaction. This total free energy change can then be used to calculate a new equilibrium constant \(K'\), which will be different from the equilibrium constant for the uncoupled reaction, representing the impact of ATP hydrolysis on shifting the reaction equilibrium.