The reaction converting glycerol to glycerol-3-phosphate can be represented as: glycerol + hpo42− ⟶ glycerol-3-phosphate + h2o. the gibbs energy for this reaction is δg∘ = 9.2 kj/mol. will this reaction be spontaneous if coupled with the hydrolysis of atp?

Yes, this reaction will be spontaneous.

Explanation:

A reaction is spontaneous if ΔG < 0.

The coupling of glycerol to glycerol-3-phospate with ATP gives:

glycerol + HPO₄²⁻ ⟶ glycerol-3-phosphate + H₂O ΔG = +9,2 kJ/mol

ATP + H₂O ⟶ ADP + HPO₄²⁻ ΔG = -30,5 kJ/mol

glycerol + ATP ⟶ ADP + glycerol-3-phosphate

ΔG = +9,2 kJ/mol - 30,5 kJ/mol = -21,3 kJ/mol

As ΔG<0, this reaction will be spontaneous

I hope it helps!