Of course you must have some means to measure the equilibrium. In aqueous solution, we address the equilibrium:
HA(aq) + H_2O(l) rarr A^(-) + H_3O^+
Given a "glass electrode", an electrode sensitive to H_3O^+, we can measure the concentration of H_3O^+, which given the stoichiometry of the reaction is equal to the concentration of A^-, and the starting concentration of HA should also be known. And thus:
K_a=([A^-][H_3O^+])/([HA])
And so if (say) the glass electrode gives a value for [H_3O^+], say [H_3O^+]=x.
And thus: K_a=(x^2)/([HA]-x).
This is quadratic in x, that may be solved exactly, given that we know the starting value of [HA]. Typically, simple approximations are used so we do not have to pfaff about with quadratics, i.e.
x_1~=sqrt(K_a[HA])
Successive approximations (x_1, x_2, x_3) rapidly give a value for x that is identical to the value that would be obtained by the quadratic equation.
