# Why macromolecule ligands such as Crown ethers form more stable complexes than open chain ligands?

This was in J. Huheey's book and no explanation was given

## 1 Answer

The reason for crown ethers being more stable than their equally long open chain derivatives is the difference in Gibbs Free energy. (If you are not used to these terms, the Wikipedia article will help I think, though it is quite, long and thorough.)

The main point to take from the use of the Gibbs free energy in this case is this: it is made up of two parts, enthalpy ($\Delta{}H$) and entropy ($\Delta{}S$).

$$\Delta{}G = \Delta{}H - T \cdot{} \Delta{}S$$

If only the binding affinity of the (crown) ether chain is considered, only the enthalpy is relevant. The strength of the individual ether moieties is as strong in a crown as it is in a chain. The difference then, lies with the entropy. To have a crown ether form a structure around a ligand 'costs some entropy'. The same entropy difference should be considered for the chain ether, with added to that the cost of pulling the ends of the chain together! That difference in entropy results in the difference in $\Delta{}G$ and therefore yields a more stable complex. (The ends of the crown ether are also pulled together, this cost of entropy however, has been paid in an earlier reaction when making the crown ether, and therefore has no influence here.)

You can see this also from the relation between the Gibbs free energy and the equilibrium constant of the complex (In the imaginary reaction $\ce{A + B -> AB})$:

$$\Delta{}G = - RT \cdot{} Ln (K)$$

where,

$$K = \frac{[\text{AB}]}{[\text{A}] \cdot{} [\text{B}]}$$

A higher $\Delta{}G$ leads to a higher equilibrium constant.

Hope this helped, and I'd be happy to clarify anything further.