Boltzmann ratio

Unfortunately, it is necessary to consider a more complex spin system in order to explain relaxation in the presence of other spins, the behaviour of a carbon atom in a methine group being a good example. This is because the dominant relaxation process in this case arises from the magnetic dipole of the bonded proton, so that the possibility of proton spin transitions must also be included. Figure 4.3 shows the simplest such system. The proton spin, conventionally labelled S, has its projection along Bq represented by the longer arrow. In this case, one must expect not only the one-spin transitions of probabilities Wi and but also concerted spin transitions of probabilities Wi ( flip-flip ) and Wq ( flip-flop ). In each case, will imply separate upwards and downwards rates of flow, in the appropriate Boltzmann ratio, and may also include an added transition rate due to irradiation. 

The magnetogyric ratios and y, arise in this equation from the different Boltzmann ratios appropriate to the different transitions. The relaxation remains exponential, if more rapid than before. 

For a sample at 293 K in a 4.69 T magnetic field, the ratio N /Nq = 0.99997. There are almost as many nuclei in the excited state as in the ground state because the difference between the two energy levels is very small. Typically, for every 100,000 nuclei in the excited state, there may be 100,003 in the ground state, as in this case. This is always the case in NMR the Boltzmann ratio is always very close to 1.00. For this reason, NMR is inherently a low-sensitivity technique. 

As can be seen from Fig. 2, the formation of excited P pP products will depend on the importance of the adiabatic pathway on the 2 A surface between spin-orbit-excited reactants and products. Whilst there is considerable variation in the observed product spin-orbit distribution, in general it is found that the lower P pP state is preferentially produced, in some cases almost exclusively. In the F + HX reaction, the product atom P p to P3/2° population ratio ranges from 0.10 [91] and approximately 0.07 [72,73,79] for X=C1 and Br, respectively, to <0.01 [69,73,87] for X=I. For the latter reaction, the reported ratio of ca. 0.5 by Burak and Eyal [78] appears to be in error since it disagrees with the results of these other experiments [69,73,87]. The observed spin-orbit ratio for the HQ and HBr reactions is close to the room-temperature Boltzmann ratio of F atom spin-orbit states. This similarity has been assumed to imply that the excited P pP products were formed on the 2 A surface from excited P p reactants [76,91]. However, this interpretation disagrees with the observations of Hepburn et al [79] on the F + HBr reaction. They find no dependence of the Br product spin-orbit population ratio on that of the F atom reactant. They conclude that Br P f ) products are formed by nonadiabatic transitions from the l A to the 2 A surface in the exit channel. 

In Equation 7.1, n+/n is the ratio of the number of positive ions to the number of neutrals evaporated at the same time from a hot surface at temperature T (K), where k is the Boltzmann constant and A is another constant (often taken to be 0.5 see below). By inserting a value for k and adjusting Equation 7.1 to common units (electronvolts) and putting A = 0.5, the simpler Equation 7.2 is obtained. 

The Boltzmann equation (Equation 18.2) shows that, under equilibrium conditions, the ratio of the number (n) of ground-state molecules (A ) to those in an excited state (A ) depends on the energy gap E between the states, the Boltzmann constant k (1.38 x 10" J-K" ), and the absolute temperature T(K). 

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

This is die fonn diat chemists and physicists are most accustomed to. The probabilities are calculated from the Boltzmann equation and the energy difference between state t and state it — 1. Because we are using a ratio of probabilities, the normalization factor, i.e., the partition function, drops out of the equation. Another variant when 6 is multidimensional (which it usually is) is to update one component at a time. We define 6, = 6, i,... 

Since the equilibrium probability Ed.s, t) contains the Boltzmann factor with an energy Tid.s, ), the condition (12) leads to the ratio of transition probabilities of the forward and backward processes as... 

The first factor on the right is equal to the ratio of the free volume of a if molecule in the clathrate to its molecular volume in the gas phase, whilst the second is the appropriate Boltzmann factor. 

By measuring the relative intensities of satellite and main lines, the population ratio is obtained, if it can be assumed that the dipole moment and line strength is not appreciably different in the two cases. From the population ratio R, the energy interval AE is obtained from the Boltzmann law i.e.,... 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

The outer layer (beyond the compact layer), referred to as the diffuse layer (or Gouy layer), is a three-dimensional region of scattered ions, which extends from the OHP into the bulk solution. Such an ionic distribution reflects the counterbalance between ordering forces of the electrical field and the disorder caused by a random thermal motion. Based on the equilibrium between these two opposing effects, the concentration of ionic species at a given distance from the surface, C(x), decays exponentially with the ratio between the electro static energy (zF) and the thermal energy (R 7). in accordance with the Boltzmann equation ... 

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