Theory

When an acid of generic formula \(H_nA\), with \(n\geq1\), is dissolved in an aqueous solution, a series of dissociation equilibria is immediately established according to the following sequence of reactions:

\[ \ce{H_nA + H_2O <=> H_{n-1}A^- + H_3O^+} \]

\[ \ce{H_{n-1}A^- + H_2O <=> H_{n-2}A^{2-} + H_3O^+} \]

\[ ... \]

\[ \ce{HA^{(n-1)-} + H_2O <=> A^{n-} + H_3O^+} \]

Each of the above reactions is an equilibrium characterized by an acid dissociation constant \(k_a^{(i)}\), defined as:

\[ k_a^{(i)}=\frac{[H_{n-i}A^{i-}][H_3O^+]}{[H_{n-i+1}A^{(i-1)-}]} \]

where \(1\leq i\leq n\) denotes the index of the dissociation reaction, numbered sequentially starting from \(1\).

Given an initial acid concentration \(C_a\), the following mass balance can be written:

\[ C_a = [H_nA]+[H_{n-1}A^-]+[H_{n-2}A^{2-}] + ... + [HA^{(n-1)-}] + [A^{n-}] \]

or, in a more compact form,

\[ C_a = \sum_{j=0}^{n} [H_{n-j}A^{j-}] \]

Using this relation, the concentration \([H_{n-j}A^{j-}]\) of each dissociation product can be expressed as a function of the total acid concentration \(C_a\) and the solution pH.

To show how this can be done, let us start with a simple observation: each dissociation constant \(k_a^{(i)}\) can be used to express the concentration of either of two consecutive dissociation products as a function of the other.

For example, the concentration of the deprotonated species \(H_{n-i}A^{i-}\) can be used to express the concentration of its precursor \(H_{n-i+1}A^{(i-1)-}\) according to:

\[ [H_{n-i+1}A^{(i-1)-}]=\frac{[H_3O^+]}{k_a^{(i)}}[H_{n-i}A^{i-}] \]

Likewise, it can also be used to express the concentration of the following deprotonation product according to:

\[ [H_{n-i-1}A^{(i+1)-}]=\frac{k_a^{(i+1)}}{[H_3O^+]}[H_{n-i}A^{i-}] \]

These relations can be chained together, allowing the concentration of each species in the dissociation sequence to be expressed as a function of any other species.

Using this observation, the acid mass balance can be rewritten as a function of the concentration of a generic intermediate species \([H_{n-i}A^{i-}]\). To do so, a two-step process can be employed. All species with higher protonation states (\(n-m>n-i\)) can be rewritten according to:

\[ [H_{n-m}A^{m-}]=\frac{[H_3O^+]^{i-m}}{\prod_{j=m+1}^{i} k_a^{(j)}}[H_{n-i}A^{i-}] \qquad \text{for} \qquad m < i \]

while all species with lower protonation states (\(n-m < n-1\)) can be rewritten according to:

\[ [H_{n-m}A^{m-}]=\frac{\prod_{j=i+1}^{m} k_a^{(j)}}{[H_3O^+]^{m-i}}[H_{n-i}A^{i-}] \qquad \text{for} \qquad m > i \]

By introducing the cumulative dissociation constants:

\[ \beta_j := \prod_{i=1}^{j} k_a^{(i)} \]

the previous relations can be rewritten as:

\[ [H_{n-m}A^{m-}]=\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}}[H_{n-i}A^{i-}] \qquad \text{for} \qquad m < i \]

\[ [H_{n-m}A^{m-}]=\frac{\beta_m}{\beta_{i}[H_3O^+]^{m-i}}[H_{n-i}A^{i-}] \qquad \text{for} \qquad m > i \]

where, for the special case \(m=0\), \(\beta_0 := 1\). By observing that, when moving from the case \(m<i\) to the case \(m>i\), the change in the sign of the exponent \(i-m\) automatically moves the \([H_3O^+]\) term to the denominator, it is easy to see that the two expressions can be combined into a single equation valid for all possible cases:

\[ [H_{n-m}A^{m-}]=\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}}[H_{n-i}A^{i-}] \]

Substituting this relation into the acid mass balance yields:

\[ C_a = [H_{n-i}A^{i-}]\sum_{m=0}^{n}\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}} \]

from which:

\[ [H_{n-i}A^{i-}] = C_a \bigg( \sum_{m=0}^{n}\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}} \bigg)^{-1} \]

Buffer capacity

The buffer capacity \(\beta\) indicates the ability of an acid-base system to resist changes in \(\mathrm{pH}\) induced by the addition of a strong acid or base. The buffer capacity is defined as:

\[ \beta := \frac{d C_b}{d \mathrm{pH}} \]

where the quantity \(C_b\) represents the amount of strong base required to increase the \(\mathrm{pH}\) of the solution. The definition of the buffer capacity can also be rewritten as a function of the \([H_3O^+]\) concentration. To do so, one can consider the following sequence of equalities:

\[ \beta = \frac{d C_b}{d \mathrm{pH}} = \frac{d C_b}{d [H_3O^+]}\frac{[H_3O^+]}{d \mathrm{pH}} = \frac{d C_b}{d [H_3O^+]}\left(\frac{d \mathrm{pH}}{d[H_3O^+]}\right)^{-1}\]

Recalling the definition \(\mathrm{pH}:= -\log{[H_3O^+]}\), the last term can be computed as:

\[ \frac{d \mathrm{pH}}{d[H_3O^+]} = - \frac{d}{d[H_3O^+]} \log{[H_3O^+]} = - \frac{1}{[H_3O^+] \log{10}} \]

Substituting this expression into the previous equation yields:

\[ \beta = -\log{10}[H_3O^+] \frac{d C_b}{d [H_3O^+]} \]

Using this form of the buffer capacity, an explicit expression for a generic acid-base system can be obtained by considering the following charge balance equation:

\[ [OH^-] + \sum_A \sum_{i=1}^{n_A} i [H_{n_A - i}A^{i-}] = [H_3O^+] + C_b \]

where the index \(A\) runs over all weak acid-base species in the system, while the index \(i\) runs over all possible deprotonation states, from \(1\) to \(n_A\). Note that an increase in the deprotonation state directly corresponds to a higher charge state, which is taken into account by multiplying the concentration of each species by the index \(i\). Strong acids and bases are not explicitly included in the balance because their neutralization produces water, whose contribution is already represented by the \([H_3O^+]\) and \([OH^-]\) terms. Finally, the term \(C_b\) represents the amount of strong base hypothetically added to the acid-base system to evaluate its buffer capacity and can be interpreted as the concentration of a hypothetical counter-ion (e.g. \(Na^+\) during the addition of the strong base \(NaOH\)).

From the above equation one can easily obtain:

\[ C_b = \frac{k_w}{[H_3O^+]} + \sum_A \sum_{i=1}^{n_A} i [H_{n_A - i}A^{i-}] - [H_3O^+]\]

whose derivative with respect to the \([H_3O^+]\) concentration can be computed as:

\[ \frac{d C_b}{d [H_3O^+]} = -\frac{k_w}{[H_3O^+]^2} + \sum_A \sum_{i=1}^{n_A} i \frac{d[H_{n_A - i}A^{i-}]}{d[H_3O^+]} - 1 \]

Considering the previously derived definition of \(\beta\), it follows that:

\[ \beta = \log{10}\left[\frac{k_w}{[H_3O^+]} - \sum_A \sum_{i=1}^{n_A} i [H_3O^+]\frac{d[H_{n_A - i}A^{i-}]}{d[H_3O^+]} + [H_3O^+]\right] \]

The only remaining term to evaluate is the derivative of \([H_{n-i}A^{i-}]\) with respect to \([H_3O^+]\). Recalling the expression derived in the previous section,

\[ [H_{n-i}A^{i-}] = C_A \bigg( \sum_{m=0}^{n_A}\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}} \bigg)^{-1} \]

its derivative can be computed as:

\[ \frac{d[H_{n_A - i}A^{i-}]}{d[H_3O^+]} = -C_A \bigg( \sum_{m=0}^{n_A}\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}} \bigg)^{-2} \sum_{m=0}^{n_A}\frac{(i-m)\beta_{m}[H_3O^+]^{i-m-1}}{\beta_{i}} \]

Substituting this result into the previous equation finally yields:

\[\beta = \log{10}\left[ \frac{k_w}{[H_3O^+]} + \sum_A C_A \sum_{i=1}^{n_A} i \bigg( \sum_{m=0}^{n_A}\frac{\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}} \bigg)^{-2} \sum_{m=0}^{n_A}\frac{(i-m)\beta_{m}[H_3O^+]^{i-m}}{\beta_{i}} + [H_3O^+] \right] \]