NMR of water protons the enhancement factor

If one or more ligands L in laige excess interact with a metal ion in a metal complex CM in the presence of free metal ions M in solution, then the exchange of ligand L among three sites should be considered. A typical case is when L is a coordinating solvent molecule, e.g. water. The molar fraction of water nuclei is given by 

The two limiting cases of the metal completely free (K - 00) or completely bound (K - 0), where K is the dissociation constant of the CM complex, would give 

When the CM complex is fully formed, [M] = 0 and [CM] = Cm, therefore, from Eq. (4.40), e = So- w thus defined as the enhancement factor measured in a solution where all the metal is complexed. Since usually q p,eo should be smaller than unity if the intrinsic nuclear relaxation times are the same in the metal complex and in the aquaion. However, as often T cm T m owing to a longer correlation time xc in the complex (Chapter 3), q can be larger than unity. This is particularly true when C is a macromolecule (e.g. a protein) and M is a metal ion with long electronic relaxation times. 

As it appears in Eq. (4.40), eo is defined only in terms of the molar fraction of bound metal ion that is, independently of the actual concentrations. By using an equation for /b analogous to Eq. (4.32), the enhancement factor can be expressed as  

Both K and o can be obtained through a two-parameter fitting of the s data obtained at various Cc and/or Cm concentrations. 

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