Theory

When an acid of generic formula \(H_nA\), with \(n\geq1\), is dissolved in an aquous solution, a series of dissociations equilibrium are immediately estabished according to the 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 written reactions is an equilibrium characterized by a acidic dissociation constant \(k_a^{(i)}\) defined according to:

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

where \(1\leq i\leq n\) represents the index of the dissociation reaction numbered in order and starting from \(1\).

Given an initial concentration \(C_a\) of the acid substance, 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 more compact form, according to the summation:

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

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

To show how this can be done we can start by adopting a simple observation: each dissociation constant \(k_a^{(i)}\) can be used to write the concentration of two subsequent dissociation product one as a function of the other.

As an example the concentraion of the deprotonation product \(H_{n-i}A^{i-}\) can be use to express the concentration of its precursor \(H_{n-i+1}A^{(i-1)-}\) according to:

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

while, at the same time, it can also be used to express the concentration of its deprotonation product \(H_{k-1}A^{(n-k+1)-}\) of the following one 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 allowing to express the concentration of each species in the dissociation sequence as a function of any other.

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

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

while all the intermediates having lower protonation states \(n-m < n-1\) can be rewritten according to:

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

By introducting the cumulative dissociation contants:

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

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 of \(m=0\), \(\beta_0 := 1\). By observing that, when moving from \(m<i\) to the case of \(i>m\), the change in sign of the exponent \(i-m\) automatically takes care of bringing the \([H_3O^+]\) term to the denominator one can easily see how the two relations can be rewritten in a single expression that takes care of all the possible scenarios:

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

By substituting this relation in the acid mass balance the following result can be obtained:

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

From which:

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