The 'bite size' effect in chelating ligands

My question is in relation to the following data, which describes the log K for the equilibrium of Mn+ + Ligand $\ce{<=>}$ [ML]n+


[![enter image description here](https://i.stack.imgur.com/jiRZF.png)](https://i.stack.imgur.com/jiRZF.png)


As far as my understanding goes, this is an example of the effect of changing the 'bite size' of the ligand. In TRIEN, the bite-size is greater than in 2,3,2-tet, as shown on the image below, which means that it prefers to bind to metals of larger radius. However, this data suggests that both ligands bind better to Cu2+, which is to be expected for 2,3,2-tet, but not for TRIEN.


[![enter image description here](https://i.stack.imgur.com/HFowq.jpg)](https://i.stack.imgur.com/HFowq.jpg)


Granted, the logK reflect this 'bite-size' effect somewhat as log K increases for
Cu2+ and decreases for Pb2+. However, my question is why this effect is not more pronounced in the TRIEN ligand because I would expect logK(Cu2+)>logK(Pb2+) based on the arguments above?




For James Gaidis information, the OP is correct and the data given in OP's table is accurate, which is from a peer-reviewed paper (Ref.1). In that paper, binding affinity of $\ce{Ni^2+}$ was compared with various other metal ions with different diameters:


[![binding affinity of Ni(II) compared with various other metal ions-1](https://i.stack.imgur.com/mbrzo.png)](https://i.stack.imgur.com/mbrzo.png)


Here, the ligand $\ce{L^6}$ is TRIEN (10-member chain) and the ligand $\ce{L^7}$ is 2,3,2-tet (11-member chain). As explained in the above chart, TRIEN is preferred when by the ions when atomic radius increased over larger 2,3,2-tet (larger the metal ion, the more strongly it prefers the smaller ligand, TRIEN). Similarly, the smaller the metal ion, the more it prefers to complex with the 13-membered chain, 2,3,2-tet For example, for $\ce{Cu^2+}$ (smaller cation), $\Delta K = \log K\_\ce{L^7} - \log K\_\ce{L^6} = 3$. For same two ligands, for $\ce{Pb^2+}$ (significantly larger cation), $\Delta K = \log K\_\ce{L^7} - \log K\_\ce{L^6} = -2.5$ (assumed from the plot). When they have changed the 10-member chain of TRIEN to 12-member macrocycle $\ce{L^1}$, and 11-member chain of 2,3,2-tet to 14-member macrocycle $\ce{L^3}$, the same trend continued:


[![Macrocycle ligands](https://i.stack.imgur.com/bvJhu.png)](https://i.stack.imgur.com/bvJhu.png)
[![binding affinity of Ni(II) compared with various other metal ions-2](https://i.stack.imgur.com/TkM6K.png)](https://i.stack.imgur.com/TkM6K.png)



> 
> This behavior is simply explained by using macrocycles. It is the fact that the hole size in $\ce{L^1}$ is actually larger than that in $\ce{L^3}$. This rather surprising result comes about because the most stable form of $\ce{L^1}$ as indicated by molecular mechanics calculations is the *trans*-I(++++) form (as shown in following diagram; Ref.2), which has all its nitrogens oriented so that their hydrogens are on the same side of the macrocycle. The hole size in this conformer of $\ce{L^1}$ is such that metal ions showing $\ce{M-N}$ lengths of $\pu{2.11 Å}$ fit best. On the other hand, the hole size in $\ce{L^3}$ in its most stable conformer, the *trans*-III(++--) form, is such that metal ions with $\ce{M-N}$ bond lengths of $\pu{2.05 Å}$ fit best:
> 
> 
> 


[![Most stable configurations](https://i.stack.imgur.com/VeRoA.png)](https://i.stack.imgur.com/VeRoA.png)


This contraction-expansion ability can be shown by the drawing of the $\ce{[Cu(L^1)Br]+}$ cation, showing how the copper atom is raised above the plane of the donor atoms. The ligand has the trans-I(++++) configuration. In addition, the macrocyclic ring of $\ce{L^1}$ is much more flexible with respect to expansion and contraction, and is, in particular, able to accommodate large metal ions much more easily than can $\ce{L^3}$:


[![Crystal structure of Cu complex](https://i.stack.imgur.com/kdlWC.png)](https://i.stack.imgur.com/kdlWC.png)


All of these arguments (changing confirmation, etc.) are applied to $\ce{L^6}$ (TRIEN) and $\ce{L^7}$ (2,3,2-tet) ligands as well such that changing the so-called "bite distance" to adjust the size of metal ions. A valuable discussion of other factors affecting chelating including solvent can be find in Ref.3.


**References:**


1. Vivienne J. Thöm and Robert D. Hancock, "The stability of nickel(II) complexes of tetra-aza macrocycles," *J. Chem. Soc., Dalton Trans.* **1985**, *(9)*, 1877-1880 (DOI: <https://doi.org/10.1039/DT9850001877>).
2. Vivienne J. Thöm, Christine C. Fox, Jan C. A. Boeyens, and Robert D. Hancock, "Molecular mechanics and crystallographic study of hole sizes in nitrogen-donor tetraaza macrocycles," *J. Am. Chem. Soc.* **1984**, *106(20)*, 5947–5955 (DOI: <https://doi.org/10.1021/ja00332a032>).
3. Robert D. Hancock and Arthur E. Martell, "Ligand design for selective complexation of metal ions in aqueous solution," *Chem. Rev.* **1989**, *89(8)*, 1875–1914 (DOI: <https://doi.org/10.1021/cr00098a011>).


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