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Example 13.2: The Inverse Proportionality of [H3O+] and [OH−]

A solution of carbon dioxide in water has a hydronium ion concentration of \(2.0\times 10^{-6 }\)M. What is the concentration of hydroxide ion at 25 °C?

Solution

\(c_{\mathrm{\ce{H3O+}}}\) \(= 2.0\times 10^{-6}\ \mathrm{M}\)


\([\ce{H3O+}]\) \(= \dfrac{c_{\mathrm{\ce{H3O+}}}}{1\ \mathrm{M}}\)    (in PQcalc, \(c_{\mathrm{solute}}\) has units and [solute] does not)

\(\ \ \ =\dfrac{2.0\times 10^{-6}\ \mathrm{M}}{1\ \mathrm{M}}\)

\(\ \ \ =2.0\times 10^{-6}\)


\([\ce{OH-}]\) = ?


We know the value of the ion-product constant for water at 25 °C:


\(K_{\mathrm{w}}\) \(= 1.0\times 10^{-14}\)


Thus, we can calculate the missing equilibrium concentration.
Rearrangement of the \(K_{\mathrm{w}}\) expression yields that [OH-] is directly proportional to the inverse of [H3O+]:

\([\ce{OH-}]\) \(= \dfrac{K_{\mathrm{w}}}{[\ce{H3O+}]}\)

\(\ \ \ =\dfrac{1.0\times 10^{-14}}{2.0\times 10^{-6}}\)

\(\ \ \ =5.0\times 10^{-9}\)


\(c_{\mathrm{\ce{OH-}}}\) \(= 1\ \mathrm{M} \cdot [\ce{OH-}]\)

\(\ \ \ =1\ \mathrm{M} \cdot 5.0\times 10^{-9}\)

\(\ \ \ =5.0\times 10^{-9}\ \mathrm{M}\)


The hydroxide ion concentration in water is reduced to 5.0 × 10^-9 M as the hydrogen ion concentration increases to 2.0 × 10^-6 M. This is expected from Le Châtelier’s principle; the autoionization reaction shifts to the left to reduce the stress of the increased hydronium ion concentration and the [OH^-] is reduced relative to that in pure water.
A check of these concentrations confirms that our arithmetic is correct:

\(K_{\mathrm{w,check}}\) \(= [\ce{H3O+}] \cdot [\ce{OH-}]\)

\(\ \ \ =2.0\times 10^{-6} \cdot 5.0\times 10^{-9}\)

\(\ \ \ =1.0\times 10^{-14}\)


Think about it: For pure \(Failed to interpret math QQQwater, [H3O+] QQQ = [\ce{OH-}]\) \( = 1.0\times 10^{-7}\). The concentration of hydroxide for the solution considered here is:
a) larger than \(1.0\times 10^{-7 }\)M because [H3O+] is larger than \(1.0\times 10^{-7}\)
b) larger than \(1.0\times 10^{-7 }\)M because [H3O+] is smaller than \(1.0\times 10^{-7}\)
c) smaller than \(1.0\times 10^{-7 }\)M because [H3O+] is larger than \(1.0\times 10^{-7}\)
d) smaller than \(1.0\times 10^{-7 }\)M because [H3O+] is smaller than \(1.0\times 10^{-7}\)