A buffer solution contains a mixture of a weak acid and its conjugate base or a weak base and its conjugate acid. The important aspect of the buffer solution is that the pH of the solution is minimally changed when small amounts of acid or base are added or when the solution is slightly diluted.
Calculation of Buffer Capacity
The magnitude of a buffer’s resistance to pH changes is referred to as its buffer capacity, denoted as β. The extent of pH change caused by the addition of a strong acid or strong base depends on the buffer’s concentration. A low buffer concentration will not be effective in buffering against a comparable or higher concentration of a strong acid or base. Therefore, it is important to know the buffer capacity at a given concentration and pH. The buffer capacity, β, is defined as:
$$ β=-\frac{dC_A}{dpH}=\frac{dC_B}{dpH} $$
where CA and CB are the concentrations of a strong acid and a strong base added to a buffer, respectively. When a strong base (NaOH) is added to a buffer solution consisting of a weak acid and its conjugate base, the electroneutrality equation will be:
$$ C_B+[H^+]=[OH^-]+[A-] $$
or
$$ C_B=[OH^-]+[A-]-[H^+] $$
Differentiating CB with pH yields:
$$ β=\frac{dC_B}{dpH}=\frac{d[A^-]}{dpH}+\frac{d[OH^-]}{dpH}-\frac{d[H^+]}{dpH} $$
The terms on the right-hand side of this equation can be rewritten as follows:
$$ \frac{d[A^-]}{dpH}=2.303(C_a+C_b)\frac{K_a[H^+]}{(K_a+[H^+])^2} $$
$$ \frac{d[H^+]}{dpH}=-\frac{2.303d[H^+]}{dln[H^+]}=-2.303[H^+] $$
$$ \frac{d[OH^-]}{dpH}=\frac{2.303d[OH^-]}{dln[OH^-]}=2.303[OH^-] $$
where Ca and Cb are the concentrations of the weak acid and its conjugate base initially present, respectively. Then, the equation will be:
$$ β=2.303([OH^-]+[H^+]+\frac{1}{2.303}\frac{d[A^-]}{dpH}) $$
Now, if we assume [H+] >> [OH−],
$$ β=2.303([H^+]+\frac{(C_a+C_b)K_a[H^+]}{(K_a+[H^+])^2}) $$
If we assume Ca >> [H+] and Cb >> [OH−] ,
$$ β=2.303\frac{(C_a+C_b)K_a[H^+]}{(K_a+[H^+])^2} $$
Maximum Buffer Capacity
The maximum buffer capacity can be derived at dβ/dpH=0:
$$ \frac{dβ}{dpH}=\frac{2.303(C_a+C_b)K_a(K_a-[H^+])}{(K_a+[H^+])^3} $$
and
$$ \frac{2.303(C_a+C_b)K_a(K_a-[H^+])}{(K_a+[H^+])^3}=0 $$
Therefore, the maximum buffer capacity occurs at pH=pKa . The maximum buffer capacity is given by:
$$ β=0.576(C_a+C_b) $$
Reference:
- Kim, C. (2004). Advanced pharmaceutics : physicochemical principles. London: CRC Press LLC.
