# 15.2: Metal-Ligand Association Constants

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## Introduction

The thermodynamic stability of a coordination complex is defined according the the equilibrium constant of its formation reaction. And since its formation is a many-step process, there are many reactions to consider. The absolute formation of a metal complex is defined by an equilibrium reaction that is independent of solution conditions and is expressed as if all ligands associate to the metal ion at once (this is practically impossible when there are multiple ligands). This absolute formation also accounts for all $$\ce{H+}$$ ions that may be released from the ligand when it binds to the metal ion. The absolute formation of a Cu complex is defined below.

$m \mathrm{Cu}+l \mathrm{L}+h \mathrm{H} \rightarrow \mathrm{Cu}_{m} \mathrm{L}_{l} \mathrm{H}_{h}$

$\beta_{l m h}=\frac{\left[\mathrm{Cu}_{m} \mathrm{L}_{l} \mathrm{H}_{h}\right]}{[\mathrm{Cu}]^{m}[\mathrm{L}]^{l}[\mathrm{H}]^{h}} \label{betalmh}$

Notice that, according to the equation above, $$\ce{H+}$$ ions are involved in this reaction, thus the reaction is pH-dependent. Most metal-ligand formations are dependent on pH because by its nature, a ligand is a Lewis Bases that can react with $$\ce{H+}$$. If we are interested in the thermodynamics in the context of aqueous solution, the $$\beta$$ value is not very useful; what we would actually want is a pH-dependent conditional affinity constant, ($$K_{cond}$$), that can be calculated from the $$\beta$$ and the $$pK_a$$ values of the ligands. Alternatively, an apparent binding constant ($$K_{app}$$, also called $$K_{eff}$$ for effective affinity constant) can be measured directly as the equilibrium constant under the specified solution conditions of pH and buffer components. An in-depth explanation of the relationship between $$\beta$$, $$K_{cond}$$, and $$K_{app}$$ is given below.

## Ligand Protonation Constants

General treatment of ligand protonation constants for a ligand with four dissociable protons is shown below:

### Stepwise Ligand Protonation Constants

$\begin{array}{ll} \mathrm{L}^{(\mathrm{n})}+\mathrm{H}^{+} \stackrel{\beta_{1}=K_{a1}}{\rightleftharpoons} \mathrm{LH}^{(\mathrm{n}+1)} &\quad K_{a1}=\frac{[\mathrm{LH}]}{[\mathrm{L}][\mathrm{H}]}=\beta_{1}\\ \mathrm{LH}^{(\mathrm{n}+1)}+\mathrm{H}^{+} \stackrel{K_{a2}}{\rightleftharpoons} \mathrm{LH}_{2}^{(\mathrm{n}+2)} &\quad K_{a2}=\frac{\left[\mathrm{LH}_{2}\right]}{[\mathrm{LH}][\mathrm{H}]}\\ \mathrm{LH}_{2}^{(\mathrm{n}+2)}+\mathrm{H}^{+} \stackrel{K_{a3}}{\rightleftharpoons} \mathrm{LH}_{3}^{(\mathrm{n}+3)} &\quad K_{a3}=\frac{\left[\mathrm{LH}_{3}\right]}{\left[\mathrm{LH}_{2}\right][\mathrm{H}]}\\ \mathrm{LH}_{3}{ }^{(\mathrm{n}+3)}+\mathrm{H}^{+} \stackrel{K_{a4}}{\rightleftharpoons} \mathrm{LH}_{4}^{(\mathrm{n}+4)} &\quad K_{a4}=\frac{\left[\mathrm{LH}_{4}\right]}{\left[\mathrm{LH}_{3}\right][\mathrm{H}]} \end{array} \label{stepwiseligand}$

### Absolute Ligand Protonation Constants

$\begin{array}{ll} \mathrm{L}^{(\mathrm{n})}+\mathrm{H}^{+} \stackrel{\beta_{1}=K_{a1}}{\rightleftharpoons} \mathrm{LH}^{(\mathrm{n}+1)} &\quad \beta_{1}=\frac{[\mathrm{LH}]}{[\mathrm{L}][\mathrm{H}]}=K_{a1}\\ \mathrm{L}^{(\mathrm{n})}+2 \mathrm{H}^{+} \stackrel{\beta_{2}}{\rightleftharpoons} \mathrm{LH}_{2}^{(\mathrm{n}+2)} &\quad \beta_{2}=\frac{\left[\mathrm{LH}_{2}\right]}{[\mathrm{L}][\mathrm{H}]^{2}}=K_{a1} K_{a2}\\ \mathrm{L}^{(\mathrm{n})}+2 \mathrm{H}^{+} \stackrel{\beta_{3}}{\rightleftharpoons} \mathrm{LH}_{3}^{(\mathrm{n}+3)} &\quad \beta_{3}=\frac{\left[\mathrm{LH}_{3}\right]}{[\mathrm{L}][\mathrm{H}]^{3}}=K_{a1} K_{a2} K_{a3}\\ \mathrm{L}^{(\mathrm{n})}+2 \mathrm{H}^{+} \stackrel{\beta_{4}}{\rightleftharpoons} \mathrm{LH}_{4}^{(\mathrm{n}+4)} &\quad \beta_{4}=\frac{\left[\mathrm{LH}_{4}\right]}{[\mathrm{L}][\mathrm{H}]^{4}}=K_{a1} K_{a2} K_{a3} K_{a4} \end{array} \label{absoluteligand}$

For the purpose of defining protonation constants in terms of absolute values and stepwise values, it is useful to start with a specific example. Here we will use glycine.   Many of us are familiar with protonation constants as pKa values, or stepwise protonation constants.  For glycine, the stepwise protonation events can be defined as dissociation events or association events.  The stepwise association equilibria for glycine are below.  The dissociation equilibrium would be written in the apposite direction and their equilibria constants would be defined as the inverse of the association constant expression.

$\begin{array}{ll} \mathrm{Gly}^{2-}+\mathrm{H}^{+} \stackrel{\beta_{1}=K_1}{\rightleftharpoons} \mathrm{GlyH}^{1-} &\quad \beta_{1}=K_1=\frac{\left[\mathrm{GlyH}^{-1}\right]}{\left[\mathrm{Gly}^{-2}\right]\left[\mathrm{H}^{+}\right]} \label{gly1}\end{array}$
$\begin{array}{ll} \mathrm{GlyH}^{1-}+\mathrm{H}^{+} \stackrel{K_2}{\rightleftharpoons} \mathrm{GlyH}_{2} &\quad K_2=\frac{\left[\mathrm{GlyH}_{2}\right]}{\left[\mathrm{GlyH}^{-1}\right]\left[\mathrm{H}^{+}\right]}\end{array}$

Take the log of $$K_1$$ or $$K_2$$ to find p$$K_1$$ or p$$K_2$$.

Many people are less familiar with the protonation constants that are defined as absolute association constants, or $$\beta$$ values.  For the first protonation event of glycine (equ. \ref{gly1}), the $$\beta_1$$ is defined by the same equilibrium constant expression as the more familiar $$K_1$$.  However, for the second protonation event, the $$\beta$$ value is defined as two protons associating with the unprotonated ligand (equ. \ref{glybeta} below).

$\mathrm{Gly}^{2-}+2 \mathrm{H}^{+} \stackrel{\beta_{2}=\mathrm{K}_{1} K_2}{\rightleftharpoons} \mathrm{GlyH}_{2} \quad \beta_{2}=\frac{\left[\mathrm{Gly} \mathrm{H}_{2}\right]}{\left[\mathrm{Gly}^{-2}\right]\left[\mathrm{H}^{+}\right]^{2}}=K_1 K_2 \label{glybeta}$

By multiplying the expression for K1 and K2 (Equations \ref{gly1} & \ref{glybeta}), we arrive at the equilibrium constant express ion for $$\beta_2$$.  Notice that $$\beta$$ values are absolute and are defined by association of protons to ligand in a single step.  On the other hand, $$K$$ values are stepwise protonation events.

## Metal-Ligand Association Constants

The metal-ligand association constant, $$\beta$$, is defined as the pH-independent absolute association constant and is often denoted $$\beta_{lm}$$ according to Equation \ref{betaml} below. M is metal, L is ligand, and $$l$$ and $$m$$ denote the complex $$\mathrm{M}_{m} \mathrm{L}_{l}$$. When $$m=1$$ the second subscript is omitted. Sometimes $$\ce{H+}$$ ion is incorporated into this expression, as seen with Equation \ref{betalmh}. If $$\ce{H+}$$ ion is incorporated, then the resulting $$\beta_{lmh}$$ is pH-dependent because it depends on the hydrogen ion concentration [H]. For the purposes here, $$\ce{H+}$$ ion is left out of the equation and we use $$\beta_{lm}$$ as defined here.

$m \mathrm{M}+l \mathrm{L} \stackrel{\beta_{l m}}{\rightleftharpoons} \mathrm{M}_{m} \mathrm{L}_{l} \quad \beta_{l m}=\frac{\left[\mathrm{M}_{m} \mathrm{L}_{l}\right]}{[M]^{m}[L]^{l}} \label{betaml}$

The association of metal and ligand can involve exchange of protons, which can effect the affinity of ligand for metal depending on the pH of solution.  The conditional $$\beta$$ value is denoted $$\beta '$$, or $$K$$.  The $$\beta '$$ is the pH-dependent constant, and is also referred to as the conditional constant or effective association constant.  The absolute $$\beta$$ value can be converted into a pH-dependant conditional value as follows:

$\beta_{l m}^{\prime}=\frac{\beta_{l m}}{\left(\alpha_{M}\right)^{\mathrm{m}}\left(\alpha_{L}\right)^{\mathrm{n}}}$
$\alpha_{M}=\frac{\left([\mathrm{M}]+[\mathrm{MOH}]+\left[\mathrm{M}(\mathrm{OH})_{2}\right]+\ldots\right)}{[M]} \label{m}$
$\alpha_{L}=\frac{\left([\mathrm{L}]+[\mathrm{LH}]+\left[\mathrm{LH}_{2}\right]+\ldots\right)}{[L]} \label{n}$

The expressions for $$\alpha_M$$ and $$\alpha_L$$ correspond to the fractions of metal and ligand in solution that do not exist as complexed species with OH or L, respectively.  These expressions can be reduced to the following equations.

$\alpha_{M}=1+10^{\left(\mathrm{pH}-\mathrm{pK}_{\mathrm{al}}\right)}+10^{\left(2 \mathrm{pH}-\mathrm{pK}_{\mathrm{al}}-\mathrm{pK}_{\mathrm{a} 2}\right)}+\ldots \label{o}$
$\alpha_{L}=1+10^{\left(\mathrm{pK}_{\mathrm{al}}-\mathrm{pH}\right)}+10^{\left(\mathrm{pK}_{\mathrm{al}}+\mathrm{pK}_{\mathrm{a} 2}-2 \mathrm{pH}\right)}+\ldots \label{p}$

Proof for Simplification of $$\alpha_M$$ (Conversion of Equation \ref{m} to Equation \ref{o})

Equation \ref{m} can be expressed as the following:
$\alpha_{M}=\frac{[M]}{[M]}+\frac{[M O H]}{[M]}+\frac{\left[M(O H)_{2}\right]}{[M]}+\ldots \label{q}$
The expression for metal hydroxide formation is
$M+O H \rightleftharpoons \mathrm{MOH} \quad \mathrm{K}_{1}=\frac{[\mathrm{MOH}]}{[\mathrm{M}][\mathrm{OH}]} \label{r}$
This expression can put in terms of [H] by multiplying by the equilibrium constant expression for the dissociation of water:
$\mathrm{K}_{\mathrm{w}}=[O H][H] \label{s}$
$\mathrm{K}_{1}=K_{1} K_{W}=\frac{[\mathrm{MOH}]}{[\mathrm{M}][\mathrm{OH}]} \times[O H][H]=\frac{[M O H][H]}{[M]} \label{t}$
Rearrange the expression and put [H] in terms of pH and $$K_1$$ in terms of p$$K_1$$.
$\frac{[\mathrm{MOH}]}{[\mathrm{M}]}=\frac{\mathrm{K}_{1}}{[\mathrm{H}]}=\frac{10^{-\mathrm{pK}_{1}}}{10^{-\mathrm{pH}}}=10^{\mathrm{pH}-\mathrm{pK}_{1}} \label{u}$
Treat the second hydrolysis component similarly:
$M+2(O H) \rightleftharpoons \quad M(O H)_{2} \label{v}$
$\beta_{2}=\frac{\left[M(O H)_{2}\right]}{[M][O H]^{2}}=\beta_{2}\left(K_{W}\right)^{2}=\frac{\left[M(O H)_{2}\right][H]^{2}}{[M]} \label{w}$
$\frac{\left[M(O H)_{2}\right]}{[M]}=\frac{\beta_{2}}{[H]^{2}}=\frac{K_{1} K_{2}}{[H]^{2}}=\frac{10^{-p K_{1}-p K_{2}}}{10^{-2 p H}}=10^{2 p H-p K_{1}-p K_{2}} \label{x}$
Substitute (\ref{u}) and (\ref{x}) into (\ref{q}) to get Equation \ref{o}.

Proof for Simplification of $$a_L$$ (Conversion of Equation \ref{n} to Equation \ref{p}):

Equation \ref{n} can be expressed as the following:
$\alpha_{M}=\frac{[M]}{[M]}+\frac{[M O H]}{[M]}+\frac{\left[M(O H)_{2}\right]}{[M]}+\ldots \label{y}$
The stepwise equilibria for protonation of L are shown in (\ref{stepwiseligand}).  For the first protonation, The Expression for $$K_{a1}$$ in (\ref{stepwiseligand}) can be rearranged to Equation \ref{z} and put in terms of pH and pKa1 instead of [H] and Ka1:
$\frac{[H L]}{L}=\frac{[\mathrm{H}]}{\left[\mathrm{K}_{\mathrm{a} 1}\right]}=\frac{10^{-\mathrm{pH}}}{10^{-\mathrm{pK}_{\mathrm{al}}}}=10^{\mathrm{pK}_{\mathrm{al}}-\mathrm{pH}} \label{z}$
Treat the term for the second protonation similarly using the expression of $$\beta_2$$ from the asbolute protonation equilibria shown in \ref{absoluteligand}.  The expression for $$\beta_2$$ can be rearranged and put in terms of pH and pKa’s (recall that $$\beta_2=K_{a1} K_{a2}$$) .
$\frac{\left[H_{2} L\right]}{[\mathrm{L}]}=\frac{[\mathrm{H}]^{2}}{\left[\beta_{2}\right]}=\frac{[\mathrm{H}]^{2}}{\left[\mathrm{K}_{\mathrm{a} 1} \mathrm{K}_{\mathrm{a} 2}\right]}=\frac{10^{-2 \mathrm{pH}}}{10^{-\mathrm{p} \mathrm{K}_{\mathrm{al}}-\mathrm{p} \mathrm{K}_{\mathrm{a} 2}}}=10^{\mathrm{pK}_{\mathrm{al}}+\mathrm{pK}_{\mathrm{a} 2}-2 \mathrm{pH}} \label{aa}$
Substitute (\ref{z}) and (\ref{aa}) into (\ref{y}) to get Equation \ref{p}.

Please note that for the equations above, the p$$K_{a1}$$ value is the highest p$$K_a$$ value for any given ligand.  We define p$$K_a$$ as an ASSOCIATION constant.  In many sources, the $$K_a$$ or p$$K_a$$ values given are actually ACID DISSOCIATION constants defined by the dissociation rather than association of ligand and proton.  It is very important to use the association constant value for the least acidic proton for p$$K_{a1}$$ to get the correct values using the $$\alpha_M$$ and $$\alpha_L$$ equations above.

Other components in solution, such as buffer or competing ligand can affect the measured stability constant.  The apparent stability constant ($$K_{app}$$) is defined as the constant measured in the presence of other species (salt, buffer components) that may interfere with determination of the pure pH-dependent conditional association constant.

Curated or created by Kathryn Haas

This page titled 15.2: Metal-Ligand Association Constants is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Kathryn Haas.