It is not necessary to store the solution at 4oC. It is stable at room temperature indefinitely. Crystallization of a1 M solution may occur even at room temperature, but a 0.5 M solution should not have this issue.
Let us refer to the titration curve for H3PO4 with NaOH, presented in some chemistry textbooks. The acid dissociation constants for phosphoric acid correspond to pKa's of 2.15, 7.20, and 12.15. The first equivalent point of phosphoric acid can be detected by about pH = (2.15 + 7.20)/2 = 4.7, depending somewhat on concentration, where 1 mol of H3PO4 is neutralized, to its first equivalent point, by 1 mole of NaOH. The second equivalence point of phosphoric acid occurs by about pH = (7.20 + 12.15)/2 = 9.7. From the first equivalent point of phosphoric acid to the second, just he second proton of H3PO4 is titrated.
About speciation of the phosphate species ― Generically speaking, we expect aq. solutions where both NaH2PO4 and Na2HPO4 are present to have pH between approx. 4.7 and 9.7. The contributions of Na3PO4 and H3PO4 can in principle be neglected within this pH range. Below approx. pH 4.7 the contribution of Na2HPO4 can be neglected, but that of H3PO4 should be considered. Above approx. pH 9.7 the contribution of NaH2PO4 can be neglected, but that of Na3PO4 should be considered.
About pH of a Na2HPO4 aq. sol. ― By approx. pH 9.7 the contributions of H3PO4, NaH2PO4 and Na3PO4 can in principle be neglected, and the solution is essentially of just Na2HPO4, thus moderately basic. The actual pH depends somewhat on concentration.
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A solution of pure sodium phosphate dibasic, Na2HPO4, does not have the pH mentioned in the enunciate (pH 6).
«The pH of disodium hydrogen phosphate water solution is between 8.0 and 11.0, meaning it is moderately basic»: