Molecular states equilibrium populations

Under equilibrium conditions excited molecular states are populated according to the familiar Boltzmann equation, lV(excited) =N ground) exp-AE/ T, where AE is the excitation energy. For a laser to be possible, the equilibrium has to be disturbed in such a way that a population inversion AN arises as defined by Eq. (1) . 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

Figure 5.21 Energy distribution of the Ms states for S = 10 and D = —0.5 cm in zero (left) and +3 T (middle) applied field along the molecular z axis, with illustration of equilibrium population distributions assuming only the lowest states are populated. (Right) Population distribution immediately after saturation at +3 T then removal of the field. The arrows indicate the path to relaxation via absorption then emission of thermal energy (phonons) from/to the lattice...

The assumption so far is that once the ion transfer has taken place from the probe to the block sites, displacing the original block ion 2, subsequent transitions occur to sustain the current. If this is not the case, if the configuration B(l)R(2) is sufficiently long lived, then the current-determining step may be the subsequent transfer to the transmembrane channel. The formulation of that case is similar to that already considered. One can assume an equilibrium population of B(1)B(2) states leading to the final state / (1,2) because of the multiple site character of the final state in the membrane channel, B(l,2), not previously mentioned as a final state, is indeed possible. Finally, with respect to the current-generating molecular mechanisms, there is the possibility that B(1)B(2) is a virtual state with a lifetime effectively... 

The origins of symmetry induced nuclear polarization can be summarized as follows as mentioned above molecular dihydrogen is composed of two species, para-H2, which is characterized by the product of a symmetric rotational wave-function and an antisymmetric nuclear spin wave function and ortho-H2, which is characterized by an antisymmetric rotational and one of the symmetric nuclear spin wavefunctions. In thermal equilibrium at room temperature each of the three ortho-states and the single para-state have practically all equal probability. In contrast, at temperatures below liquid nitrogen mainly the energetically lower para-state is populated. Therefore, an enrichment of the para-state and even the separation of the two species can be easily achieved at low temperatures as their interconversion is a rather slow process. Pure para-H2 is stable even in liquid solutions and para-H2 enriched hydrogen can be stored and used subsequently for hydrogenation reactions [54]. 

At infinite pressure, as we argued in Chapter 1, the rate of replenishment of the reactive molecular states is so fast that even allowing for the decay processes, the populations of these states remain at their Boltzmann equilibrium values. Hence, we can write... 

What may be different in the collisions of molecular ions with helium and argon atoms in drift tubes is the rate at which an equilibrium population of vibrational states is reached (determined, of course, by the relative rates of excitation and de-excitation in ion/buffer gas atom collisions) and this is also a function of E/N. This phenomenon is discussed by Kriegel et al (1987). Data for the N2O + NO charge transfer reaction are also given in Figure 5 which also indicate that the equilibrium population of the vibrational states of N20 ions is the same in collisions with helium and argon atoms (for further discussion... 

For a sample of many molecules, under the assumption of no interaction between them, the probabilities are directly proportional to the numbers of molecules in each state, the populations, which we designate as n. For a large collection of A molecules at some temperature T, the number found in the ith energy level of an A molecule is n = NPj, where N is the total number of A molecules. Is it possible for a large collection of molecules to have populations of molecular quantum states other than those dictated by the Maxwell-Boltzmann law Yes. But in that event, the system is not at equilibrium, which means that it is not stable and is undergoing change. The distribution law holds for equilibrium conditions, and under those conditions, it can be used to determine the number of molecules in particular energy level states. 

Reestablishment of the equilibrium population is associated with changes in the spin state, through the influence of the exclusion principle on the space-spin symmetry of the molecular states. As a consequence, the signals of methyl Cs may appear with different sign compared to the other resonances. As a consequence, the absolute intensity of the spectrum is not the same any more for microwave frequencies of constructive interference of quantum rotor and DNP effects. It should be mentioned that quantum rotor mechanisms constitute a separate polarization mechanism with great potential [13,15,16]. 

In this pressure range the actual populations of excited molecular states are close to the equilibrium populations given by f. The rate coefficient /c2 is therefore the thermal equilibrium average of the specific rate coefficient k(E) of the unimolecular reaction. The rate coefficient for the reverse recombination of a simple bond fission reaction... 

There will exist an equilibrium between the two states. If the ener between the two states, E, is of the order of kT, then the relative populations of the two states will vary with the temperature of the sample. In the Fe(III) dithiocarbamate series of complexes, [FeCRiRgdtcla], AE can be varied by suitable choice of substituents Ri and Rj. Although these are well removed from the FeSg molecular core, they can appreciably affect the electronic parameters of the central iron atom and of the surrounding crystal field of the sulfur atom by way of the conjugated system of the ligand (237). The results of the spin crossover are reflected in magnetic susceptibility data. 

One would prefer to be able to calculate aU of them by molecular dynamics simulations, exclusively. This is unfortunately not possible at present. In fact, some indices p, v of Eq. (6) refer to electronically excited molecules, which decay through population relaxation on the pico- and nanosecond time scales. The other indices p, v denote molecules that remain in their electronic ground state, and hydrodynamic time scales beyond microseconds intervene. The presence of these long times precludes the exclusive use of molecular dynamics, and a recourse to hydrodynamics of continuous media is inevitable. This concession has a high price. Macroscopic hydrodynamics assume a local thermodynamic equilibrium, which does not exist at times prior to 100 ps. These times are thus excluded from these studies. 

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

Fig. 8.8 The principle of the Dyson model. Each point in the phase diagram represents a possible composition of a molecular population. The horizontal axis is a, where (a+ 1) is the number of monomer types. On the vertical axis, b represents the quality factor of the polymeric catalysis. The transition region consists of populations which can have both an ordered and a disordered equilibrium state. In the death region there are only disordered states, while in the immortal region (in the Garden of Eden ), there is no disordered state (Dyson, 1988)...

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