Las constantes de equilibrio dan la medida de la estabilidad del complejo. Cuanto mayor es el valor de la constante más estable es el complejo.
Hay dos tipos de constantes de complejación:
- Sucesivas:
$ \begin{array}{l} \ce{M + L <=> ML} \quad k_1 = \dfrac{[\ce{ML}]}{[\ce{M}] [\ce{L}]} \\[1ex] \ce{ML + L <=> ML2} \quad k_2 = \dfrac{[\ce{ML2}]}{[\ce{ML}] [\ce{L}]} \\[1ex] \vdots \\[1ex] \ce{ML_{\normalsize \textit{n} \small - 1} + L <=> ML_{\normalsize \textit{n}}} \quad k_{\smash{\normalsize \textit{n}}} = \dfrac{ [\ce{ML_{\smash{\normalsize \textit{n}}}}] }{ [\ce{ML_{\normalsize \textit{n} \small - 1}}] [\ce{L}] } \end{array} $
Estas constantes sucesivas son las reales, estos equilibrios son los que se dan en el proceso de complejación, pero no se utilizan en los cálculos.
- Globales:
$ \begin{array}{l} \ce{M + L <=> ML} \quad \beta_1 = \dfrac{[\ce{ML}]}{[\ce{M}] [\ce{L}]} \\[1ex] \ce{M + 2 L <=> ML2} \quad \beta_2 = \dfrac{[\ce{ML2}]}{[\ce{M}][\ce{L}]^2} \\[1ex] \vdots \\[1ex] \ce{M + \textit{n} L <=> ML_{\normalsize \textit{n}}} \quad \beta_{\smash{\normalsize \textit{n}}} = \dfrac{ [\ce{ML_{\smash{\normalsize \textit{n}}}}] }{ [\ce{M}] [\ce{L}]^{\smash{\normalsize \textit{n}}} } \end{array} $
Estas constantes globales son las que se utilizan en los cálculos.
Relación entre constantes:
$ \begin{array}{l} \beta_1 = k_1 \\[1ex] \beta_2 = k_1 k_2 \\[1ex] \vdots \\[1ex] \beta_{\smash{\normalsize \textit{n}}} = k_1 k_2 \dotsm k_{\smash{\normalsize \textit{n}}} \end{array} $
Comprobación:
$ k_1 k_2 = \dfrac{\smash{\cancel{[\ce{ML}]}}}{[\ce{M}][\ce{L}]} \dfrac{[\ce{ML2}]}{\smash{\cancel{[\ce{ML}]}\!} [\ce{L}]} = \dfrac{[\ce{ML2}]}{[\ce{M}][\ce{L}]^2} = \beta_2 $
Los ligandos, al tener pares de electrones libres, pueden actuar además como bases. Así pues, también pueden definirse las constantes de protonación sucesivas y globales. Por ejemplo, para el carbonato:
$ \begin{array}{l} \left. \begin{alignedat}{2} &\ce{H2CO3 <=> H+ \! + HCO3^- &\quad&K_{a1}} \\[1ex] &\ce{HCO3^- <=> H+ \! + CO3^{2-} &&K_{a2}} \end{alignedat} \right\} \text{Constantes de acidez} \\[1em] \left. \begin{alignedat}{2} &\ce{H+ \! + CO3^{2-} <=> HCO3^-} &&k_1 = \dfrac{1}{\ce{K_{a2}}} \\[1ex] &\ce{H+ \! + HCO3^- <=> H2CO3} \, &\quad&k_2 = \dfrac{1}{\ce{K_{a1}}} \end{alignedat} \right\} \begin{split} & \\ &\smash{\text{Constantes de protonación}} \\ &\smash{\text{sucesivas}} \end{split} \\[1em] \left. \begin{alignedat}{2} &\ce{H+ \! + CO3^{2-} <=> HCO3^-} &&\beta_1 = k_1 = \dfrac{1}{\ce{K_{a2}}} \\[1ex] &\ce{2 H+ \! + CO3^{2-} <=> H2CO3} \, &\quad&\beta_2 = k_1 k_2 = \dfrac{1}{\ce{K_{a2}} \ce{K_{a1}}} \end{alignedat} \right\} \begin{split} & \\ &\smash{\text{Constantes de}} \\ &\smash{\text{protonación globales}} \end{split} \end{array} $
Por tanto, un ligando, además de participar en el proceso de complejación, también, al aceptar protones, puede hacerlo en reacciones del tipo ácido/base.
A continuación, ejemplo de cálculo empleando las constantes de complejación globales, estando presentes las especies:
$ \ce{M \quad L \quad ML \quad ML2} $
Equilibrios y constantes:
$ \begin{array}{l} \ce{M + L <=> ML} \quad k_1 = \dfrac{[\ce{ML}]}{[\ce{M}][\ce{L}]} \\[1ex] \ce{ML + L <=> ML2} \quad k_2 = \dfrac{[\ce{ML2}]}{[\ce{ML}][\ce{L}]} \\[1ex] \beta_1 = \dfrac{[\ce{ML}]}{[\ce{M}][\ce{L}]} \qquad \beta_2 = \dfrac{[\ce{ML2}]}{[\ce{M}][\ce{L}]^2} \end{array} $
Balance de masas:
$ c_{\ce{M}} = [\ce{M}] + [\ce{ML}] + [\ce{ML2}] $
Utilizando las constantes globales en el balance de masas se hallan las concentraciones de las distintas especies en función de $[\ce{L}]$. Esto es:
$\bullet \ [\ce{M}]$
$ \begin{array}{l} c_{\ce{M}} = [\ce{M}] + \beta_1 [\ce{M}] [\ce{L}] + \beta_2 [\ce{M}] [\ce{L}]^2 = [\ce{M}] (1 + \beta_1 [L] + \beta_2 [\ce{L}]^2) \\[1ex] \Rightarrow \, [\ce{M}] = \dfrac{c_{\ce{M}}}{1 + \beta_1 [\ce{L}] + \beta_2 [\ce{L}]^2} \end{array} $
$\bullet \ [\ce{ML}]$
$ \begin{array}{l} \begin{aligned} c_{\ce{M}} &= \dfrac{[\ce{ML}]}{\beta_1 [\ce{L}]} + [\ce{ML}] + \beta_2 [\ce{M}] [\ce{L}]^2 = \dfrac{[\ce{ML}]}{\beta_1 [\ce{L}]} + [\ce{ML}] + \beta_2 \dfrac{[\ce{ML}]}{\beta_1 [\ce{L}]} [\ce{L}]^2 \! = \\[1ex] &= [\ce{ML}] \! \left( \dfrac{1}{\beta_1 [\ce{L}]} + \dfrac{\beta_1 [\ce{L}]}{\beta_1 [\ce{L}]} + \dfrac{\beta_2 [\ce{L}]^2}{\beta_1 [\ce{L}]} \right) \end{aligned} \\[1ex] \begin{aligned} \Rightarrow \, [\ce{ML}] = \dfrac{c_{\ce{M}} \beta_1 [\ce{L}]}{1 + \beta_1 [\ce{L}] + \beta_2 [\ce{L}]^2} \end{aligned} \end{array} $
O, más simple, usando $\beta_1$ y aprovechando directamente la expresión encontrada anteriormente para $[\ce{M}]$:
$ \begin{array}{l} [\ce{ML}] = \beta_1 [\ce{M}] [\ce{L}] \\[1ex] \llap{\Rightarrow \, {}} [\ce{ML}] = \dfrac{c_{\ce{M}} \beta_1 [\ce{L}]}{1 + \beta_1 [\ce{L}] + \beta_2 [\ce{L}]^2} \end{array} $
$\bullet \ [\ce{ML2}]$
$ \begin{array}{l} \begin{aligned} c_{\ce{M}} &= \dfrac{[\ce{ML2}]}{\beta_2 [\ce{L}]^2} + \beta_1 [\ce{M}] [\ce{L}] + [\ce{ML2}] = \dfrac{[\ce{ML2}]}{\beta_2 [\ce{L}]^2} + \beta_1 \dfrac{[\ce{ML2}]}{\beta_2 [\ce{L}]^2} [\ce{L}] + [\ce{ML2}] = \\[1ex] &= [\ce{ML2}] \! \left( \dfrac{1}{\beta_2 [\ce{L}]^2} + \dfrac{\beta_1 [\ce{L}]}{\beta_2 [\ce{L}]^2} + \dfrac{\beta_2 [\ce{L}]^2}{\beta_2 [\ce{L}]^2} \right) \end{aligned} \\[1ex] \begin{aligned} \Rightarrow \, [\ce{ML2}] = \dfrac{c_{\ce{M}} \beta_2 [\ce{L}]^2}{1 + \beta_1 [\ce{L}] + \beta_2 [\ce{L}]^2} \end{aligned} \end{array} $
O de forma más directa y sencilla:
$ \begin{array}{l} [\ce{ML2}] = \beta_2 [\ce{M}] [\ce{L}]^2 \\[1ex] \llap{\Rightarrow \, {}} [\ce{ML2}] = \dfrac{c_{\ce{M}} \beta_2 [\ce{L}]^2}{1 + \beta_1 [\ce{L}] + \beta_2 [\ce{L}]^2} \end{array} $