Relationship between the water ionization constant and ionic strength

I read in a textbook that 



> 
> the water ionization constant ($K\_\mathrm{w}$) increases as the ionic strength of the solution increases
> 
> 
> 


and it confused me. If the ionic strength increased, then the activity of each of the ions $\ce{H3O+}$ and $\ce{OH-}$ would decrease, leading to a decrease in the value of $K\_\mathrm{w}$. 


Can someone please explain the mistake in my way of thinking?




The [equilibrium constant](http://goldbook.iupac.org/html/E/E02177.html) is given though $$K\_x = \prod\_{\ce{B}} x\_{\ce{B}}^{\nu\_{\ce{B}}},$$
where $x$ is a specific quantity. We can choose that to be the activity $a$.


For the auto-ionisation
$$\ce{H2O (aq) <=>[H2O] H+ (aq) + OH- (aq)}\tag{1}\label{water}$$ 
we find therefore
$$K\_a = \frac{a(\ce{H+})\cdot a(\ce{OH-})}{a(\ce{H2O})}.$$


Assuming the equilibrium $\eqref{water}$ is largely on the left side and therefore there is an excess of water, a reasonable simplification is to treat $a(\ce{H2O})$ as constant.
$$K\_a' = K\_a\cdot a(\ce{H2O}) = a(\ce{H+})\cdot a(\ce{OH-})$$


In ideally diluted solutions we also find
$$\lim\_{c\to0}\gamma\_i = \frac{a\_i}{c\_i}=1,$$
and therefore 
$$K\_c = c(\ce{H+})\cdot c(\ce{OH-}) \overset{\mathrm{def}}{=} K\_\mathrm{w}.$$


The [ionic strength](http://goldbook.iupac.org/html/I/I03180.html) is defined as
$$I\_c = \frac12\sum\_{\ce{B}}c\_{\ce{B}}z\_{\ce{B}}^2,$$
or for our case
$$I\_{c,\mathrm{w}} = \frac12 \bigg[c(\ce{H+}) + c(\ce{OH-})\bigg].$$


Because of $\eqref{water}$ we can write
$$I\_{c,\mathrm{w}} = c(\ce{H+})$$
and
$$K\_\mathrm{w} = c(\ce{H+})^2, $$
therefore
$$I\_{c,\mathrm{w}} = \sqrt{K\_\mathrm{w}}.$$


From this relation you can see, that when $K\_\mathrm{w}$ increases, the ionic strength also increases, vice versa.



> 
> If the ionic strength increased, then the activity of each of the ions $\ce{H3O+}$ and $\ce{OH−}$ would decrease, leading to a decrease in the value of $K\_\mathrm{w}$.
> 
> 
> 


The first part is correct, the activity of *each* ion would decrease. However, there are more ions, so the activity of the ensemble increases more (given ideal dilution), therefore also leading to an increase in $K\_\mathrm{w}$.


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