T r depends on the size of the molecule which can be

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'T r depends on the size of the molecule, which can be calculated rigorously if the molecule is spherical, or approximately if it is not. The appropriate expression is (2.13) * The nuclear longitudinal relaxation time, T" can be defined as the rate constant by which the populations of the M/ t and M/ = - t (for protons) levels reach their equilibrium value after an external perturbation (e.g., a radiofrequency pulse in an NMR experiment). The transverse relaxation time, T 2 , can be defined as the average lifetime of a hydrogen nucleus in a given spin state. The NMR linewidth is inversely proportional to T 2 . The relation T 2 .,; T, always holds.
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64 2 I THE REACTION PATHWAYS OF ZINC ENZYMES AND RELATED BIOLOGICAL CATALYSTS where 1] is the microviscosity of the solution, a is the radius (or approximate radius) of the molecule, k B is the Boltzmann constant, and T is the absolute temperature. For CA, T r can be safely calculated to be =10 -8 S at room tem- perature. Since the correlation time T c in high-spin cobalt proteins varies be- tween 10 -II and 10 12 s, it must therefore be determined by the electronic relaxation time. Water IH NMRO profiles are often analyzed by using the classical dipolar interaction approach, as first described by Solomon: 72 (2.14) where /La is the permeability of vacuum, 'YI is the nuclear magnetogyric ratio, ge is the electron g-factor, S is the electron spin quantum number, r is the electron-nucleus distance, and Ws and w[ are the electron and nuclear Larmor frequencies, respectively. This equation describes the dipolar interaction be- tween the magnetic moment of nucleus I (Ii 'YI y'I (I + 1)) and the magnetic moment of the electrons S (ge/LB y'S(S + I)) as a function of the correlation time (T c ) and of the magnetic field (expressed as w[ and ws). Neglect of the zero-field splitting of the S = i manifold may introduce an error in the quantitative esti- mates within a factor of two. 73 Fitting of the data for pseudotetrahedral complexes shows that they have T s of 10 II s, whereas five-coordinate complexes have a shorter T s , on the order of 10 -12 s. The latter derivatives also have exchangeable protons that could correspond to a water molecule in the coordination sphere, whereas the former do not. 25 The T s values are thus proposed as indicators of the coordination num- ber in low-symmetry, four- and five-coordinate cobalt complexes. The shorter electronic relaxation times are related to low-lying excited states, which, inde- pendently of the particular mechanism, favor electron relaxation. 74 Short electronic relaxation times in paramagnetic compounds cause only mi- nor broadening of IH NMR lines, whereas the isotropic shifts (i.e., the shifts due to the presence of unpaired electron(s), usually very large) are independent of the value of the electronic relaxation times. For cobalt-substituted carbonic anhydrase, the IH NMR spectra have been recorded for several derivatives, and the proton signals of histidines coordinated to the metal were found to be shifted well outside the diamagnetic region (Figure 2.14).75 Five-coordinate species give sharper signals than four-coordinate ones. The spectra
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