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 the reaction ATP ( aq ) + H 2 O ( l ) ⟶ ADP ( aq ) + HPO 2 − 4 ( aq ) for which Δ G ∘ rxn = − 30.5 kJ/mol at 37.0 °C and pH 7.0. Calculate the value of Δ G rxn in a biological cell in which [ ATP ] = 5.0 mM, [ ADP ] = 0.60 mM, and [ HPO 2 − 4 ] = 5.0 mM.

Reaction quotient (Q) : It is defined as the measurement of the relative amounts of products and reactants present during a reaction at a particular time.

The given balanced chemical reaction is,

[tex]ATP(aq)+H_2O(l)\rightarrow ADP(aq)+HPO_4^{2-}(aq)[/tex]

The expression for reaction quotient will be :

[tex]Q=\frac{[ADP][HPO_4^{2-}]}{[ATP]}[/tex]

In this expression, only gaseous or aqueous states are includes and pure liquid or solid states are omitted.

Given:

[tex][ATP][/tex] = 5.0 mM

[tex][ADP][/tex] = 0.60 mM

[tex][HPO_4^{2-}][/tex] = 5.0 mM

Now put all the given values in this expression, we get

[tex]Q=\frac{(0.60)\times (5.0)}{(5.0)}=0.60mM=0.60\times 10^{-3}M[/tex]

Now we have to calculate the value of [tex]\Delta G_{rxn}[/tex].

The formula used for [tex]\Delta G_{rxn}[/tex] is:

[tex]\Delta G_{rxn}=\Delta G^o+RT\ln Q[/tex] ............(1)

where,

[tex]\Delta G_{rxn}[/tex] = Gibbs free energy for the reaction = ?

[tex]\Delta G_^o[/tex] = standard Gibbs free energy = -30.5 kJ/mol

R = gas constant = [tex]8.314\times 10^{-3}kJ/mole.K[/tex]

T = temperature = [tex]37.0^oC=273+37.0=310K[/tex]

Q = reaction quotient = [tex]0.60\times 10^{-3}[/tex]

Now put all the given values in the above formula 1, we get:

[tex]\Delta G_{rxn}=(-30.5kJ/mol)+[(8.314\times 10^{-3}kJ/mole.K)\times (310K)\times \ln (0.60\times 10^{-3})[/tex]

[tex]\Delta G_{rxn}=-49.6kJ/mol[/tex]

Therefore, the value of [tex]\Delta G_{rxn}[/tex] is -49.6 kJ/mol