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We know that a reaction could have different rates at equilibrium. For example, lets take the reaction:

$$\ce{A + B -> C }$$

At equilibrium, the rates of formation and dissociation will be equal:

$$k_\mathrm{r}\mathrm{[A][B]} = k_\mathrm{c}\mathrm{[C]}$$

But that means at various concentrations of $\mathrm{[A],[B]}$ and $\mathrm{[C]}$, the actual velocity (rate) of the reaction at equilibrium will be different.

In other words, for the above example: If I were to put additional [A] into the reaction after its reached equilibrium once, the new equilibrium it would reach would have a faster rate. Also, this means that a fold change increase in $\mathrm{[A]}$ doesn't require an equivalent fold change decrease in $\mathrm{[B]}$ to reach the new equilibrium.

Now, the autoionization of water is unique because it is independent of the concentration of water: $$K_\mathrm{w} = \ce{[H+][OH-]}$$

This means that a fold change increase in $\ce{[H+]}$ requires an equivalent fold change decrease in $\ce{[OH-]}$

Thus, I am guessing the rate would be equivalent to some constant: $$k_\mathrm{r}\ce{[H+][OH-]} = constant$$

Does the rate of auto-ionization of water change when you add a strong acid or base? An explanation at the level of molecules would be appreciated.

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    $\begingroup$ The autoionization of water is written as $\ce{H2O + H2O -> H3O+ + OH-}$. As written, the forward direction does not depend on the acidity or basicity of the solution. $\endgroup$
    – Zhe
    Jul 25 '20 at 20:16
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    $\begingroup$ @ Zhe. Yes, it does. If the water contains some acid or some base in solution, the autoionization decreases. The autoioniozation of water is very low in pure water. It is even worse in presence or an acid or a base in solution. Don't forget that the product of the concentration of the ions H+ and OH- must be constant, whatever their origin. If one mole of acid or base is dissolved in 1 liter water, the auto ionization produces only $\ce{10^{-14}}$ moles H+ and OH- . $\endgroup$
    – Maurice
    Jul 25 '20 at 20:37
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    $\begingroup$ @Maurice I don't understand how that relates to the kinetics of autoionization. I suppose I am assuming (possibly incorrectly) that the equilibrium I wrote is the elementary reaction step. But if that's correct, the kinetics of the process are unaffected by pH. We're not talking about the degree of autoionization. $\endgroup$
    – Zhe
    Jul 25 '20 at 20:56
  • $\begingroup$ I don't think the kinetics of this reaction can be measured. $\endgroup$
    – Maurice
    Jul 26 '20 at 12:52
  • $\begingroup$ It’s complicated and hard to measure: ingentaconnect.com/content/scs/chimia/2012/00000066/00000004/… $\endgroup$ Jul 27 '20 at 3:26
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Ideal solutions

If we pretend for a moment that all the solutions under consideration are ideal, and that we have already reach equilibrium, we can consider the following reaction:

$$\ce{H2O(l) + H2O(l) <=> HO-(aq) + H3O+(aq)}$$

As Zhe said in the comments, the concentration of the reactants don't change much (even if you have $\pu{1 M}$ sodium hydroxide or $\pu{1 M}$ hydrochloric acid), so the forward rate at equilibrium is the same at any pH (again, assuming an ideal solution). That means the reverse rate is also the same at any pH. Because the product of the concentrations of hydroxide and hydronium ions is constant, the reverse rate constant does not depend on the pH.

Considering ionic strength and activity coefficients

The above argument does not work in a real solution with high concentrations of dissolved ions. If the pH is far from neutral, the concentration of ions (and counter-ions is high), and equilibrium concentrations (and with that the rates) will depend on the solutes present in solution.

Molecular level

At the molecular level, the auto-ionization of water is a special case because hydronium and hydroxide ions seem to diffuse faster than other ions (Grotthuss mechanism). At neutral pH, both species are rare. At acidic pH, there are more hydronium ions, but fewer hydroxide ions, so collisions are also rare (and reverse for basic). A high ionic strength would make it easier for ion pairs to form and separate, while a low ionic strength would be advantageous for the combining of hydronium and hydroxide.

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