A weak base is a base that does not accept all the available H+ ions, and forms an equilibrium between the base and its conjugate acid. Upon dissolution in water, a weak base does not dissociate completely, so that the resulting aqueous solution contains only a small proportion of hydroxide ions and the concerned basic radical, and a large proportion of undissociated molecules of the base.
pH, Kb, and Kw Bases yield solutions in which the hydrogen ion activity is lower than it is in pure water, i.e., the solution is said to have a pH greater than 7.0 at standard conditions, potentially as high as 14 (and even greater than 14 for some bases). The formula for pH is:
pH = − log 10 [ H + ] {\displaystyle {\mbox{pH}}=-\log _{10}\left[{\mbox{H}}^{+}\right]}
Bases are proton acceptors; a base will receive a hydrogen ion from water, H2O, and the remaining H+ concentration in the solution determines pH. A weak base will have a higher H+ concentration than a stronger base because it is less completely protonated than a stronger base and, therefore, more hydrogen ions remain in its solution. Given its greater H+ concentration, the formula yields a lower pH value for the weak base. However, pH of bases is usually calculated in terms of the OH− concentration. This is done because the H+ concentration is not a part of the reaction, whereas the OH− concentration is. The pOH is defined as:
pOH = − log 10 [ OH − ] {\displaystyle {\mbox{pOH}}=-\log _{10}\left[{\mbox{OH}}^{-}\right]}
If we multiply the equilibrium constants of a conjugate acid (such as NH4+) and a conjugate base (such as NH3) we obtain:
K a × K b = [ H 3 O + ] [ N H 3 ] [ N H 4 + ] × [ N H 4 + ] [ O H − ] [ N H 3 ] = [ H 3 O + ] [ O H − ] {\displaystyle K_{a}\times K_{b}={[H_{3}O^{+}][NH_{3}] \over [NH_{4}^{+}]}\times {[NH_{4}^{+}][OH^{-}] \over [NH_{3}]}=[H_{3}O^{+}][OH^{-}]}
As K w = [ H 3 O + ] [ O H − ] {\displaystyle {K_{w}}=[H_{3}O^{+}][OH^{-}]} is just the self-ionization constant of water, we have K a × K b = K w {\displaystyle K_{a}\times K_{b}=K_{w}}
Taking the logarithm of both sides of the equation yields:
l o g K a + l o g K b = l o g K w {\displaystyle logK_{a}+logK_{b}=logK_{w}}
Finally, multiplying both sides by -1, we obtain:
p K a + p K b = p K w = 14.00 {\displaystyle pK_{a}+pK_{b}=pK_{w}=14.00}
With pOH obtained from the pOH formula given above, the pH of the base can then be calculated from p H = p K w − p O H {\displaystyle pH=pK_{w}-pOH} , where pKw = 14.00. A weak base persists in chemical equilibrium in much the same way as a weak acid does, with a base dissociation constant (Kb) indicating the strength of the base. For example, when ammonia is put in water, the following equilibrium is set up:
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