Explain Ostwald’s Dilution Law.

Hint: The law relates the dissociation constant of a weak electrolyte with the degree of dissociation and the initial concentration of the electrolyte.

Complete step by step solution:
In words, Ostwald’s Dilution Law states that a weak electrolyte will undergo complete ionization only at infinite dilution.
To derive the law, first take into consideration, the dissociation constant of an acid.
\[{{K}_{d}}=\dfrac{[{{H}^{+}}][{{A}^{-}}]}{[HA]}\]
Where,
\[{{K}_{d}}\]= dissociation constant
\[[{{H}^{+}}]\] = concentration of cation
\[[{{A}^{-}}]\] = concentration of anion
\[[HA]\] = concentration of acid
From this point onwards, let us denote $[HA]$ by ${{c}_{0}}$.
A weak acid will not dissociate when dissolved in water and this will inhibit its ability to be a good electrolyte. Hence, a weak acid will be a weak electrolyte.
Let us take the degree of dissociation of this electrolyte to be $\alpha $.
Thus, the concentration of each ionic entity will be $\alpha {{c}_{0}}$. The fraction of undissociated acid that will be left can be denoted by $(1-\alpha )$; it follows that the concentration of the undissociated acid will be $(1-\alpha ){{c}_{0}}$. Thus, the newly formed equation will be:
\[{{K}_{d}}=\dfrac{(\alpha {{c}_{0}})(\alpha {{c}_{0}})}{(1-\alpha ){{c}_{0}}}\]
\[{{K}_{d}}=\dfrac{{{\alpha }^{2}}}{(1-\alpha )}\cdot {{c}_{0}}\]
For weak electrolytes, as a very small amount of dissociation is seen, $\alpha \ll 1$. Hence, it can be neglected in $(1-\alpha )$. The equation is modified to
\[{{K}_{d}}={{\alpha }^{2}}\cdot {{c}_{0}}\]
\[\alpha =\sqrt{\dfrac{{{K}_{d}}}{{{c}_{0}}}}\]
This is known as Ostwald’s dilution law for weak electrolytes.

Note: The same can be calculated for a weak base by considering the dissociation constant of a weak base and modifying it accordingly. Remember that the values given by this equation are approximate at best even for weak electrolytes and that this law fails miserably for strong electrolytes. Always verify the nature of an electrolyte before using this equation.