Today I came across a "thought experiment" regarding $\mathrm{pH}$ of a diluted acid solution. This idea looks somewhat logical but obviously contradicts the reality. Can you show me where is the error?

Let's suppose we have $\pu{100 g}$ of concentrated 95% (mass) sulfuric acid. That means we have a mixture of $\pu{95 g}$ of H2SO4 and $\pu{5 g}$ of $\ce{H2O}.$ Let's calculate the amount of both compounds.

$$n = \frac{m}{M}$$

$$n(\ce{H2SO4}) = \frac{\pu{95 g}}{\pu{98.08 g mol-1}} = \pu{0.9686 mol}$$

$$n(\ce{H2O}) = \frac{\pu{5 g}}{\pu{18.02 g mol-1}} = \pu{0.2778 mol}$$

Since the amount of the solvent molecules is much much lower than the amount of the $\ce{H2SO4}$ molecules, it is not possible for the acid to be completely dissolved, which leads to the relatively small amount of $\ce{H3O+}$ cations, and $\mathrm{pH}$ of the solution is relatively high $(\mathrm{pH} = -\log [\ce{H3O+}]).$

$$\ce{H2SO4 + H2O -> HSO4^- + H3O+}$$

If we increase the amount of the $\ce{H2O}$ molecules (dilute solution), more $\ce{H3O+}$ cations can be formed, and the $\mathrm{pH}$ will decrease.

Conclusion: dilution of acid-water solutions decreases $\mathrm{pH}.$

In reality it is completely opposite. So what is the catch?

  • 2
    $\begingroup$ 95% sulfuric acid isn't an "aqueous" solution. // At higher concentrations, certainly anything beyond 1 molar, one needs to use activities of the ions not their concentrations. $\endgroup$ – MaxW Jul 9 '19 at 0:10
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    $\begingroup$ Also note, that activity coefficients will be much higher than 1, and that sulphuric acid will dissociate just to hydrogensulphate. $$\ce{H2SO4 + H2O -> HSO4- + H3O+}$$ $\endgroup$ – Poutnik Jul 9 '19 at 4:11
  • $\begingroup$ Related: chemistry.stackexchange.com/questions/108842/… $\endgroup$ – Karsten Theis Jul 10 '19 at 18:08

You run into all sorts of problems when the pH is outside of the range of 1 to 13. For example, in Environ. Sci. Technol. (2000) 342, p. 254-258, they say:

[...] the former National Bureau of Standards (NBS) established a set of conventions that limits measurements to 1 < pH < 13 and to ionic strength, I < 0.1. The main limitations are the activity coefficient expression, the range of defined standard pH buffers, and interferences with the reversible response of the glass H+-sensitive membrane electrode.

They go on to use standard of concentrated sulfuric acid to define the pH, using a Pitzer model to predict activity coefficients (figure from researchgate) :

enter image description here

In reality it is completely opposite. So what is the catch?

If you do the math, the extreme pH of -5 shown in the figure corresponds to a hydrogen ion activity of hundred-thousand. The concentration of water for pure water is "only" ~55 mol/L, so this must reflect an activity coefficient far from one.

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