Protons equilibria relaxation

Let us first consider the nuclei //a, Hys, and as lying in a straight line and equidistant from one another (Fig. 4.5). The central proton, can relax by interactions with two neighbors, //a and He, while //a and He can relax by interaction with only one neighbor, so //b can relax twice as quickly as //a or Hq. If we assume that relaxation can occur only through dipole-dipole relaxation, and that an equilibrium steady state has been... 

The proton longitudinal relaxation rate of deoxygenated water is 0.3 s at 25°C, with about 25% of this value being attributed to intermolecular dipolar relaxation. In that case, of course, no dispersion occurs. However, for water in equilibrium with air, due to paramagnetic molecular oxygen, the relaxation rate increases by 0.1 at low fields and exhibits a dispersion around 40 MHz (47). 

These results may be interpreted in the following way (22) At room temperature the species exists mainly as dimer (or ion quadruplet) with rapid migration of MeaSi groups on the NMR time scale. As the sample is cooled to -20°C, trimethylsilyl migration becomes slow so that two peaks are observed for the dimer. The related compound LiN(SiMc3)2 shows similar monomer-dimer equilibria in solution in THF and hydrocarbons (72). Below —20°C, monomer (ion pair) is observable in equilibrium with dimer. Between -20°C and -60°C silyl migration is rapid in the monomer but slow in the dimer (compared to the proton spin relaxation time). Finally at -80°C anionic rearrangement for the monomer also becomes slow, so that two pairs of 2 1 peaks are observed. 

At this point we need to consider that there is another process operating in this system. When the populations of the spin states have been disturbed from their equilibrium values, as in this case by irradiation of the proton signal, relaxation processes will tend to restore the populations to their equilibrium values. Unlike excitation of a spin from a lower to a higher spin state, relaxation process are not subject to the same quantum mechanical selection rules. Relaxation involving changes of both spins simultaneously (called double-quantum transitions) are allowed in fact they are relatively important in magnitude. The relaxation pathway labeled W2 in Fig. 4.6 tends to restore equilibrium populations by relaxing spins from state N4 to Ni. We shall represent the number of spins that are relaxed by this pathway by the symbol d. The populations of the spin states thus become as follows ... 

The intensity of the signal—its relative lightness or darkness in the image—depends on the concentration and spin relaxation times of the various protons. Spin relaxation time is the time it takes for the perturbed magnetization associated with a proton to return to its equilibrium value. The relaxation time is quite sensitive to the environment and is different for water in blood and various tissues. 

Clearly, equation (6.4.1) has to be further expanded at pH significantly above neutrality when the dissociation of EH" to E + H has to be taken into account. This is a typical example of a relaxation linked to a protonic equilibrium. Such systems can be studied with pH jumps coupled to pressure or temperature dependent buflFers (see pp. 205-6). In the particular case of the reaction of alcohol dehydrogenase the cycle must be completed during the catalytic reaction when a proton is taken up by the enzyme during product dissociation. 

Most proton transfer reactions are fast they have been carefully studied by relaxation methods. A system consisting of a conjugate acid-base pair in water is a three-state cyclic equilibrium as shown in Scheme IV. [The symbolism is that used by Bemasconi. ... 

Scheme VIII has the form of Scheme II, so the relaxation time is given by Eq. (4-15)—appjirently. However, there is a difference between these two schemes in that L in Scheme VIII is also a participant in an acid-base equilibrium. The proton transfer is much more rapid than is the complex formation, so the acid-base system is considered to be at equilibrium throughout the complex formation. The experiment can be carried out by setting the total ligand concentration comparable to the total metal ion concentration, so that the solution is not buffered. As the base form L of the ligand undergoes coordination, the acid-base equilibrium shifts, thus changing the pH. This pH shift is detected by incorporating an acid-base indicator in the solution.

It is important to avoid saturation of the signal during pulse width calibration. The Bloch equations predict that a delay of 5 1] will be required for complete restoration to the equilibrium state. It is therefore advisable to determine the 1] values an approximate determination may be made quickly by using the inversion-recovery sequence (see next paragraph). The protons of the sample on which the pulse widths are being determined should have relaxation times of less than a second, to avoid unnecessary delays in pulse width calibration. If the sample has protons with longer relaxation times, then it may be advisable to add a small quantity of a relaxation reagent, such as Cr(acac) or Gkl(FOD)3, to induce the nuclei to relax more quickly. 

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

Unsually short NMR T, relaxation values were observed for the metal-bonded H-ligands in HCo(dppe)2, [Co(H2)(dppe)]+ (dppe = l,2-bis(diphenylphosphino)ethane), and CoH(CO) (PPh3)3.176 A theoretical analysis incorporating proton-meta) dipole-dipole interactions was able to reproduce these 7) values if an rCo H distance of 1.5 A was present, a value consistent with X-ray crystallographic experiments. A detailed structural and thermodynamic study of the complexes [H2Co(dppe)2]+, HCo(dppe)2, [HCo(dppe)2(MeCN)]+, and [Co(dppe)2(MeCN)]2+ has been reported.177 Equilibrium and electrochemical measurements enabled a thorough thermodynamic description of the system. Disproportionation of divalent [HCo(dppe)2]+ to [Co(dppe)2]+ and [H2Co(dppe)2]+ was examined as well as the reaction of [Co(dppe)2]+ with H2. 

The physical mechanism of entirely nonadiabatic and partially adiabatic transitions is as follows. Due to the fluctuation of the medium polarization, the matching of the zeroth-order energies of the quantum subsystem (electrons and proton) of the initial and final states occurs. In this transitional configuration, q, the subbarrier transition of the proton from the initial potential well to the final one takes place followed by the relaxation of the polarization to the final equilibrium configuration. 

Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

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