Question

For the coupled reactions given below: 
Glucose 6-phosphate + H\(_2\)O → Glucose + Pi (Reaction 1) 
ATP + Glucose → ADP + Glucose 6-phosphate (Reaction 2) The standard free energy change of ATP hydrolysis at 25 °C is __ kJ/mol. 
The equilibrium constants for Reaction 1 and Reaction 2 are 360 and 800, respectively; Gas constant \( R = 8.314 \, {J mol}^{-1} {K}^{-1} \). 
(Round off to two decimal places)

Hint:

The standard free energy change for coupled reactions can be calculated by considering the equilibrium constants and using the relationship \( \Delta G^\circ = -RT \ln K \).

Answer 1

Step 1: Write the relationship for the standard free energy change.
The standard free energy change (\( \Delta G^\circ \)) for a reaction is related to the equilibrium constant (\( K \)) by the equation: \[ \Delta G^\circ = -RT \ln K, \] where:
\( R = 8.314 \, {J mol}^{-1} {K}^{-1} \),
\( T = 298 \, {K} \) (temperature at 25°C),
\( K \) is the equilibrium constant of the reaction.
Step 2: Use the equilibrium constants for the reactions.
We are given the equilibrium constants for Reaction 1 and Reaction 2:
For Reaction 1: \( K_1 = 360 \),
For Reaction 2: \( K_2 = 800 \).
The standard free energy change for each reaction is: \[ \Delta G^\circ_1 = -RT \ln K_1, \quad \Delta G^\circ_2 = -RT \ln K_2. \] Step 3: Calculate the standard free energy change for ATP hydrolysis.
The total standard free energy change for the coupled reactions is the sum of \( \Delta G^\circ_1 \) and \( \Delta G^\circ_2 \): \[ \Delta G^\circ_{{total}} = \Delta G^\circ_2 - \Delta G^\circ_1. \] Substitute the expressions for \( \Delta G^\circ_1 \) and \( \Delta G^\circ_2 \): \[ \Delta G^\circ_{{total}} = -RT \ln K_2 + RT \ln K_1 = -RT \ln \left( \frac{K_2}{K_1} \right). \] Substitute the known values: \[ \Delta G^\circ_{{total}} = -8.314 \times 298 \times \ln \left( \frac{800}{360} \right). \] Calculate the natural logarithm: \[ \ln \left( \frac{800}{360} \right) \approx \ln(2.222) \approx 0.796. \] Now, calculate the free energy change: \[ \Delta G^\circ_{{total}} = -8.314 \times 298 \times 0.796 \approx -1978.25 \, {J/mol} \approx -32.00 \, {kJ/mol}. \] Thus, the standard free energy change of ATP hydrolysis is \( \boxed{-32.00} \, {kJ/mol} \) (rounded to two decimal places).