# 9.3 Metals Complexes With Ligands Other Than Hydroxide

This page explains what is meant by a stability constant for a complex ion, and goes on to look at how the stability constant's magnitude is governed in part by the entropy change during a ligand exchange reaction.

## Sequential Reactions

If you add ammonia solution to a solution containing hexaaquacopper(II) ions, [Cu(H2O)6]2+, four of the water molecules are eventually replaced by ammonia molecules to give [Cu(NH3)4(H2O)2]2+. This can be written as a single equilibrium reaction to show the overall effect:

$\ce{[Cu(H2O)6]^{+2} + 4NH3 <=> [Cu(NH3)4(H2O)2]^{+2} + 4H2O} \label{all}$

However, in fact the water molecules get replaced one at a time, and so this is made up of a series of sequential reactions:

$\ce{[Cu(H2O)6]^{+2} + NH3 <=> [Cu(NH3)(H2O)5]^{+2} + H2O}\label{step1}$

$\ce{ [Cu(NH3)(H2O)5]^{+2} + NH3 <=> [Cu(NH3)2(H2O)4]^{+2} + H2O} \label{step2}$

$\ce{[Cu(NH3)2(H2O)4]^{+2} + NH3 <=> [Cu(NH3)3(H2O)3]^{+2} + H2O} \label{step3}$

$\ce{[Cu(NH3)3(H2O)3]^{+2} + NH3 <=> [Cu(NH3)4(H2O)2]^{+2} + H2O} \label{step4}$

Although this can look a bit daunting at first sight, all that is happening is that first you have one, then two, then three, then four water molecules in total replaced by four NH3 ligands. Using the approach from Section 9.2, in which we omit the water molecules, we could write these as:

$\ce{Cu^{+2} + NH3 <=> Cu(NH3)^{+2}} \label{step1a}$

$\ce{ Cu(NH3)^{+2} + NH3 <=> Cu(NH3)2^{+2}} \label{step2a}$

$\ce{Cu(NH3)2^{+2} + NH3 <=> Cu(NH3)3^{+2}} \label{step3a}$

$\ce{Cu(NH3)3^{+2} + NH3 <=> Cu(NH3)4^{+2}} \label{step4a}$

## Individual stability constants

Let's take a closer look at the first of these equilibria (Equation \ref{step1a}). Like any other equilibrium, this one has an equilibrium constant, $$K_{eq}$$ - except that in this case, we call it a stability constant. Because this is the first water molecule to be replaced, we call it $$K_1$$ and is given by this expression:

$K_1 = \dfrac{[\ce{Cu(NH3)^{+2}}]}{\ce{[Cu^{+2}}][\ce{NH3]}}$

Note that, at the top of this page, square brackets were used in the chemical reactions to illustrate that the charge on a molecule applied to the entire molecule. In the equation here for $$K_1$$, the square brackets [ ] are back to what we are used to them referring to:  molar concentrations in aqueous solution.

$$K_1$$ for this reaction is 1.78x10^4. Compared to many equilibrium constants we have seen for acid-base reactions, solubility product reactions, and Henry's Law gas-aqueous partitioning, the magnitude of this $$K_1$$ is much, much larger. This suggests that there is a strong tendency to form the ion containing an ammonia molecule. A high value of a stability constant shows that the ion is easily formed. Each of the other equilibria above also has its own stability constant, K2, K3 and K4. For example, K2 is given by:

$K_2 = \dfrac{[\ce{Cu(NH3)_2^{+2}}]}{\ce{[Cu(NH3)^{+2}}][\ce{NH3]}}$

And $$K_2$$ has a value of 4.07x10^3. Again, this value of K is fairly high compared to other $$K_{eq}$$ we have worked with previously. So, that suggests that this reaction also tends to be favorable.

The equations for $$K_3$$ and $$K_4$$ would be:

$K_3 = \dfrac{[\ce{Cu(NH3)_3^{+2}}]}{\ce{[Cu(NH3)_2^{+2}}][\ce{NH3]}}$

$K_4 = \dfrac{[\ce{Cu(NH3)_4^{+2}}]}{\ce{[Cu(NH3)_3^{+2}}][\ce{NH3]}}$

The following table shows all four of the stability constants. As with Ka and Ksp values, these are often expressed as log transformations. Because formation constants are generally greater than 10^0 (or 1), they are often expressed as logK values as opposed to pK values.

Metal-Ligand Complex n = _; Kn Value log Kn
Cu(NH3)+2 1; K1 1.78 x 104 4.25
Cu(NH3)2+2 2; K2 4.07 x 103 3.61
Cu(NH3)3+2 3; K3 9.55 x 102 2.98
Cu(NH3)4+2 4; K4 1.74 x 102 2.24

The fact that all the K values are greater than 1 generally reflects that the sequential metal-ligand complexes are each more favored than the prior complex. However, there are diminishing returns - notice that the incremental K values are declining as you replace more and more waters. This is common with individual stability constants.

## Cumulative Reactions and Beta Values

Instead of writing individual sequential reactions in which one ligand molecule is added at a time, sometimes these reactions are written as cumulative reactions. In this approach, the starting point for each reaction is the central metal ion, and the other reactant is n ligand molecules. The next four reactions illustrate this approach:

$\ce{ Cu^{+2} + NH3 <=> Cu(NH3)^{+2}} \label{step1b}$

$\ce{ Cu^{+2} + 2 NH3 <=> Cu(NH3)2^{+2}} \label{step2b}$

$\ce{ Cu^{+2} + 3 NH3 <=> Cu(NH3)3^{+2}} \label{step3b}$

$\ce{ Cu^{+2} + 4 NH3 <=> Cu(NH3)4^{+2}} \label{step4b}$

And, their equilibrium constant equations are:

$\beta_1 = \dfrac{[\ce{Cu(NH3)^{+2}}]}{\ce{[Cu^{+2}}][\ce{NH3]}}$

$\beta_2 = \dfrac{[\ce{Cu(NH3)_2^{+2}}]}{\ce{[Cu^{+2}}][\ce{NH3]^2}}$

$\beta_3 = \dfrac{[\ce{Cu(NH3)_3^{+2}}]}{\ce{[Cu^{+2}}][\ce{NH3]^3}}$

$\beta_4 = \dfrac{[\ce{Cu(NH3)_4^{+2}}]}{\ce{[Cu^{+2}}][\ce{NH3]^4}}$

The first reaction and Keq equation look the same as the approach above. However, for subsequent reactions, note the differences due to the stoichiometric coefficient on the NH3 molecule in the chemical reaction. Also, instead of calling these stability constants 'K', the greek letter beta is used.

This approach is chemically and mathematically identical to the sequential approach, but it can have advantages when applying these concepts to engineering problems. We will do some of these problems in class and in lab. The next section in this reading shows that the beta value for each complex is the mathematical product of the preceding K values leading up to the formation of that complex.

## Summary

Whether you are looking at the sequential replacement of individual water molecules with ligand molecules, or cumulative reactions producing the complex ions, a stability constant is simply the equilibrium constant for the reaction you are looking at. The larger the value of the stability constant, the further the reaction lies to the right. That implies that complex ions with large stability constants are more stable than ones with smaller ones. Stability constants tend to be very large numbers. In order to simplify the numbers a "log" scale is often used.