Autoprotolysis Constant of Water

My chemistry textbook states that $[\ce{H3O+}][\ce{OH-}] = K\_w$ in all aqueous solutions, which we know of as $K\_w$. However, I was wondering why this was true. The textbook states that it is true because $K\_w$ is an equilibrium constant, but why are we allowed to assume that the other ions take no part in this reaction?


To clarify further, consider the following reaction
$$\ce{HCl + H2O (l) <=> H3O+ + OH- + Cl- }$$
Of course this isn't balanced, but why doesn't the hydrogen ion on the left affect the equilibrium constant?




Your textbook is correct, the $K\_\mathrm{w}$ expression is as follows:
$$K\_\mathrm{w} = [\ce{H3O+}][\ce{OH-}]=1.0\times10^{-14}$$


Since for every $\ce{OH-}$ there must be one $\ce{H+}$ ($\ce{H+}$ in water is synonomous with $\ce{H3O+}$) the mol ratio is 1:1, if moles are equal the molarities are equal. I'm going to substitute $x$ in for both
$$K\_\mathrm{w}=[x][x]=[x]^2=1.0\times10^{-14}; x= 1.0\times10^{-7}.$$


So the molarity of $[\ce{OH-}]$ and $[\ce{H3O+}]$ are both $1.0\times10^{-7}$ That's a really really low amount, small enough to ignore.


However, a lot of reactions (especially in Organic Chemistry) you'll see a negative ion, for example, $\ce{Cl-}$ pulling the $\ce{H+}$ off of a water. So when $[\ce{OH-}]$ or $[\ce{H3O+}]$ are needed, water can be torn apart to make it. 


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!!Bing!! !!Bing!! !!Bing!! Congratulations - You're thinking like chemist!!


Your textbook has stated an oversimplification which isn't really true. Let's define $\ce{K\_{w}}$ based on concentrations as: $$\ce{K\_{w} = [H^+][OH^{-}]}$$


(1) The value of $\ce{K\_{w}}$ changes with temperature. 


(2) $\ce{K\_{w}}$ changes with pressure.


(3) $\ce{K\_{w}}$ depends on the total ionic strength of the solution, which includes spectator ions. 


See Wikipedia article "Self-ionization of water" section ["Dependence on temperature, pressure and ionic strength."](https://en.wikipedia.org/wiki/Self-ionization_of_water#Dependence_on_temperature.2C_pressure_and_ionic_strength)


The "real" relationship at any temperature and pressure is given in the [goldbook](http://goldbook.iupac.org/A00532.html) as noted by Martin: $$\mathrm{K^{\*}\_{w} = 1.00 \times 10^{-14} = a(H^+) a(OH^-)}$$


where $\mathrm{a(H^+)}$ and $\mathrm{a(OH^-)}$ are the activities of $\ce{H^+}$ and $\ce{OH^-}$. In dilute solution the activity of an ion can be approximated by its concentration. A "dilute solution" is dilute enough if a linear relationship holds between concentration and activity. In general this would be 0.1 N or less. 


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