Since you're dealing with a buffer, you can use the Henderson-Hasselbalch equation to calculate the new pH of the solution when the concentrations of dihydrogen phosphate, H_2PO_4^(-), and hydrogen phosphate, HPO_4^(2-), are equal.
The balanced chemical equations for this buffer are
H_2PO_4^(-) + H_2O rightleftharpoons HPO_4^(2-) + H_3^(+)O, pKa_2 = 7.21
HPO_4^(2-) + H_2O rightleftharpoons PO_4^(3-) + H_3^(+)O, pKa_3 = 12.7
Equal concentrations of H_2PO_4^(-) and HPO_4^(2-) will establish the first equilibrium, which implies that the pH of the solution will now be
pH_("solution") = pKa_2 + log (([HPO_4^(2-)])/([H_2PO_4^(-)]))
pH_("solution") = 7.21 + log ("5.25 mmol/L"/"5.25 mmol/L") = 7.21 + log(1) = 7.21
You can determine the GIbbs free energy by using the reaction's pKa_2 by using
DeltaG = - RT ln(K_(a2)), where
K_(a2) is the acid dissociation constant for the established equilibrium reaction.
You can use the mathematical identity ln(x) = 2.303 * log(x) to rewrite the above equation as
DeltaG = -RT * 2.303 log(K_(a2))
If you plug pK_(a2) = - log(K_(a2)) into the equation, you'll get
DeltaG = 2.303 RT * pK_(a2)
Therefore,
DeltaG = 2.303 * 8.3145"J"/("mol" * "K") * (273.15 + 25)"K" * 7.21
DeltaG = "41,162.3 J/mol" = "+41.2 kJ/mol" -> rounded to three sig figs.
SIDE NOTE. DeltaG will be positive because the acid dissociation constant for the established equilibrium is smaller than 1.
