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Following statement was given for boric acid $(\ce{H3BO3}):$

At low concentrations $(\leq\pu{0.02 M})$ essentially $\ce{B(OH)3}$ and $\ce{B(OH)4-}$ are present, but at higher concentration the acidity increases and pH studies are consistent with the formation of polymeric species such as $$\ce{B(OH)3 <=> H+ + [B3O3(OH)4]- + H2O}$$

Please explain the meaning of the italicised statement i.e. pH studies are consistent with the formation of polymeric species?

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    $\begingroup$ Could you please narrow down as to what exactly needs clarification? The following questions could arise: How the pH studies are conducted?; Why polyoxoborates form and what their structure is?; What does Pourbaix diagram say regarding olation/oxolation?; What are corresponding $\mathrm{p}K_\mathrm{a}$ values? It would be nice if you could point out which one interests you in the first place, otherwise the answer can and should be very lengthy. $\endgroup$ – andselisk Dec 23 '19 at 13:09
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    $\begingroup$ @andselisk what is the meaning of the statement pH studies are consistent with the formation of polymeric species? $\endgroup$ – Zenix Dec 23 '19 at 13:21
  • $\begingroup$ I suppose predictions of polyanion hypothesis about H+ concentration agrees with observation. Also note borax existence as the solid evidence of boron polyanions $\endgroup$ – Poutnik Dec 23 '19 at 13:27
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In the equation $\ce{H2O + H3BO3 -> H+ + B(OH)4-}$, one negative charge on the anion is shared among 4 electronegative oxygens. Yes, one positive charge on the proton is also shared with some water molecules, so the orthoborate ion probably shares some negative charge with some waters, but let's keep it simple.

Boric acid is known to complex with sugars to generate higher acidity. If boric acid complexes with itself, as in $$\ce{3 H3BO3 -> H+ + [B3O3(OH)4]- + 2 H2O}$$, the one negative charge on the anion can be considered to be distributed among 7 electronegative oxygens, or maybe still distributed mostly on the OH groups, but since they will be separated more, the stability of the ion will be greater, and the tendency to lose a proton will be greater: greater acidity. The electronegativity of boron is less than that of hydrogen (2.02 vs 2.2, electronegativity, Wikipedia), so when the atoms in the ions are counted, the first ionization equation has one hydrogen and one boron as the metallic atoms getting the positive charge, while the second equation has one hydrogen and 3 borons donating electrons - more sharing of the positive charge here, tho the hydrogen still gets the positive charge.

But this structure may be only a suggestion. Well-known polyborate anions include the triborate(1-), tetraborate(2-) and pentaborate(1-) anions. The condensation reaction for the formation of tetraborate(2-) is as follows Wikipedia

$$\ce{2 B(OH)3 + 2 [B(OH)4]− ⇌ [B4O5(OH)4]2- + 5 H2O}$$

The italicized text mentions an observation of pH that is consistent with the complexation of boric acid with itself. It doesn't prove the existence of the triborate ion only, and perhaps is only a way to keep the text shorter without ignoring complexation completely. Then along comes a student who looks into the sentence and sees that something was omitted or not explained thoroughly. (Keep asking questions!)

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A more precise answer to the polymeric structure, at least at neutral pH, is found in Wikipedia on borate, to quote in part:

"At neutral pH boric acid undergoes condensation reactions to form polymeric oxyanions.... The tetraborate anion (tetramer) includes two tetrahedral and two trigonal boron atoms symmetrically assembled in a fused bicyclic structure. The two tetrahedral boron atoms are linked together by a common oxygen atom and each also bears a negative net charge brought by the supplementary OH− groups laterally attached to them. This intricate molecular anion also exhibits three rings: two fused distorted hexagonal (boroxole) rings and one distorted octagonal ring. Each ring is made of a succession of alternate boron and oxygen atoms. Boroxole rings are a very common structural motif in polyborate ions."

Also:

"Simple bonding theories point to the trigonal planar structure. In terms of valence bond theory the bonds are formed by using sp2 hybrid orbitals on boron. Some compounds termed orthoborates do not necessarily contain the trigonal planar ion, for example gadolinium orthoborate..."

My reading of the above is that it is possible, via a condensation reaction, for Boric acid to create polymeric oxyanions, and in particular Boroxole rings at neutral pH.

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