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I'm wondering why the pH of extreme alkaline solutions, such as the case with $\pu{10 M}$ $\ce{NaOH}$ does not make practical sense and I'm wondering why this is the case? Refer to my calculations below:

$$\ce{NaOH ->[H2O] Na+(aq) + OH-(aq)}$$ $$\mathrm{pH} + \mathrm{pOH} = 14$$ $$\mathrm{pOH} = -\log{\ce{[OH-]}} = -\log(\pu{10M}) = -1$$

Therefore,

$$\mathrm{pH} = 14 - \mathrm{pOH} = 14 - (-1) = 15(?)$$

This doesn’t make sense as I thought the pH scale maxes out at 14? This is even stranger when considering that most commercial NaOH solutions are sold as bulk $\pu{1000 L}$ aqueous $\pu{50 w/w\%}$ NaOH solutions corresponding to approximately $\pu{19 M}$... much higher than $\pu{10 M}$... does the industry not use pH either when it comes to extreme solutions? What am I missing here?

PS Refer below to my calcs which got the $\pu{19 M}$ number in case any of you wanted to check my proof (all symbols used are standard symbols i.e. masses (m), molar masses (M), density ($\rho$), standard gravity (SG) etc.):

Assuming $\pu{50 wt\%}$ caustic (i.e. alkalinity), in 100 g of Solution; $\pu{50 g}$ $\ce{NaOH}$, $\pu{100 g}$ solution.

$\mathrm{SG}_\mathrm{solution} = \pu{1.525 g/mL}$ (from spec at $\pu{20 ^\circ C}$, $M_\ce{NaOH} = \pu{39.997 g/mol}$

\begin{align} \mathrm{M}_\ce{NaOH} &= \frac{n_\mathrm{solute}}{V_\mathrm{solution}} \\ n_\mathrm{solute} &=\frac{m_\mathrm{solute}}{M_\mathrm{solute}} \\ V_\mathrm{solution} &= \frac{m_\mathrm{soln}}{\rho_\mathrm{soln}}\\ \rho_\mathrm{soln} &= \frac{\mathrm{SG_\mathrm{soln}}}{\rho_\ce{H2O}}\\ \end{align}

Solving with said values we get,

$$\mathrm{M}_\ce{NaOH} = \pu{19.12 mol/L} \approx \pu{19 M} $$

For context, I am trying to better understand this behaviour of extreme alkaline systems because I want to keep track of the pH of other solutions I'm making by mixing highly alkaline caustic solutions with sodium silicate systems. Sometimes these systems form precipitates when they've mixed, other times they don't. I thought understanding the pH of everything is a good place to start, hence why I'm here.

Safdar Faisal
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Hendrix13
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    It does make sense, but you will probably not find sensors to measure it, unless incorrectly, with strong trend and short life. Activity coefficients of 10M solutions is a "rocket science". – Poutnik Aug 24 '22 at 04:28
  • I thought the pH scale maxed out at 14 and that anything higher doesn't make sense? Can you elaborate on how the activity coefficients play a role in this example being very complex? – Hendrix13 Aug 24 '22 at 04:32
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    pH scale is not limited to 0-14 range. But value out of this range are not that useful and there is experimental trouble in measuring them. Typically, most of glass oH electrodes do not like much pH > 12, which affects their responses and progressively damages them. Reasonable chemists do not put their pH electrodes to 10 M NaOH. – Poutnik Aug 24 '22 at 04:35
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    https://en.wikipedia.org/wiki/Activity_coefficient // https://en.wikipedia.org/wiki/PH – Poutnik Aug 24 '22 at 04:38
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    Useful links for text and formula formatting: Notation basics / Formatting of math/chem expressions /
    upright vs italic // For more, see Math SE MathJax tutorial. // Keep CH SE Q titles in plain text.     – Poutnik Aug 24 '22 at 04:43
  • @Poutnik, I see, I will look more into activities which seems to be what will make this all practical. Hopefully there'll be more detailed answers posted cheers!. Also, I have used LaTeX here for my equations, aside from the picture i posted because I had done it on word equation a while ago and thought it easier/quicker with the formatting. Regardless, the picture is still clear. What's the issue with formatting which you're pointing out? – Hendrix13 Aug 24 '22 at 04:47
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    Element symbols are typographically never in italic. CH SE uses MathJax mhchem extension by default, involving \ce{} and \pu{} Mathjax syntax. More in links. // Use $\ce{NaOH(aq) <=> Na+(aq) + OH-(aq)}$ to get $\ce{NaOH(aq) <=> Na+(aq) + OH-(aq)}$ // But there is no NaOH(aq), all is dissociated. – Poutnik Aug 24 '22 at 04:50
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    pH and pOH are not in italic either, use \mathrm{} to get them upright. Generally, simplified, only variables are in italic. – Poutnik Aug 24 '22 at 06:21
  • What's a good way to measure pH in these situations then (i.e. highly alkaline caustic conditions)? Either some sort of complex model with activity coefficients or something along the lines of p(NaOH)? – Hendrix13 Sep 13 '22 at 13:49
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    There is no good way and it is not usually used. IF it is used, there is determined pOH, as pH would be highly temperature dependent. For high concentration of strong acids or bases, there is usually used molar or mass concentration, or density. – Poutnik Sep 13 '22 at 18:35

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