A critical reaction in the production of energy to do work or drive chemical reactions in biological systems is the hydrolysis of adenosine triphosphate, ATP, to adenosine diphosphate, ADP, as described by

ATP(aq) +H2O (l) --->ADP(aq) +HPO4^2-

for which ΔG°rxn = –30.5 kJ/mol at 37.0 °C and pH 7.0. Calculate the value of ΔGrxn in a biological cell in which [ATP] = 5.0 mM, [ADP] = 0.90 mM, and [HPO4^2–] = 5.0 mM.

To calculate the value of ΔGrxn in a biological cell, you can use the equation:

ΔGrxn = ΔG°rxn + RT ln(Q)

Where:
- ΔGrxn is the change in Gibbs free energy of the reaction in the biological cell.
- ΔG°rxn is the standard Gibbs free energy change at standard conditions.
- R is the gas constant (8.314 J/(mol⋅K)).
- T is the temperature in Kelvin.
- Q is the reaction quotient, which is the ratio of the concentrations of the products to the reactants.

First, convert the temperature from Celsius to Kelvin:
37.0 °C + 273.15 = 310.15 K

Next, calculate the value of Q using the concentrations provided:
Q = [ADP] * [HPO4^2–] / [ATP]
= (0.90 mM) * (5.0 mM) / (5.0 mM)
= 0.90

Now, substitute the values into the equation:

ΔGrxn = -30.5 kJ/mol + (8.314 J/(mol⋅K)) * (310.15 K) * ln(0.90)

Calculating the natural logarithm of 0.90:

ln(0.90) ≈ -0.105

Substituting this value back into the equation:

ΔGrxn = -30.5 kJ/mol + (8.314 J/(mol⋅K)) * (310.15 K) * (-0.105)

Calculate the term (8.314 J/(mol⋅K)) * (310.15 K) * (-0.105):

(8.314 J/(mol⋅K)) * (310.15 K) * (-0.105) ≈ -261.15 J/mol

Convert J to kJ:

-261.15 J/mol = -0.26115 kJ/mol

Substitute the value into the equation:

ΔGrxn = -30.5 kJ/mol + (-0.26115 kJ/mol)

Finally, calculate the value of ΔGrxn:

ΔGrxn ≈ -30.76 kJ/mol

To calculate the value of ΔGrxn in a biological cell, we can use the equation:

ΔGrxn = ΔG°rxn + RT ln(Q)

Where:
- ΔGrxn is the change in Gibbs free energy under the given conditions
- ΔG°rxn is the standard Gibbs free energy change
- R is the gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
- Q is the reaction quotient, calculated using the concentrations of the reactants and products.

Given:
- ΔG°rxn = -30.5 kJ/mol
- [ATP] = 5.0 mM
- [ADP] = 0.90 mM
- [HPO4^2–] = 5.0 mM

First, we need to convert the concentrations to moles per liter (mol/L):

[ATP] = 5.0 mM = 5.0 × 10^-3 mol/L
[ADP] = 0.90 mM = 0.90 × 10^-3 mol/L
[HPO4^2–] = 5.0 mM = 5.0 × 10^-3 mol/L

Next, we need to calculate the reaction quotient, Q:

Q = [ADP] × [HPO4^2–] / [ATP]

Substituting the given values:
Q = (0.90 × 10^-3 mol/L) × (5.0 × 10^-3 mol/L) / (5.0 × 10^-3 mol/L)
Q = 0.90 × 10^-15 mol^2/L^2

Now, we can calculate the value of ΔGrxn:

ΔGrxn = ΔG°rxn + RT ln(Q)
ΔGrxn = (-30.5 kJ/mol) + (8.314 J/(mol·K) × (37.0 + 273.15) K × ln(0.90 × 10^-15 mol^2/L^2)

Let's calculate ΔGrxn: