Boltzmann distribution, population ratio

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. 

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.

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

Fig. 1.1 Schematic representation of the population difference of spins at different magnetic field strengths. The two different spin quantum number values of the ]H spin, +34 and -34, are indicated by arrows. Spins assume the lower energy state preferentially, the ratio bet-ween upper and lower energy level being given by the Boltzmann distribution.

The ratio of populations at equilibrium is given by the Boltzmann distribution ... 

As stated earlier, in the state of thermal equilibrium at room temperature, dihydrogen (H2) contains 25.1% parahydrogen (nuclear singlet state) and 74.9% orthohydrogen (nuclear triplet state) [19]. This behavior reflects the three-fold degeneracy of the triplet state and the almost equal population of the energy levels, as demanded by the Maxwell-Boltzmann distribution. At lower temperatures, different ratios prevail (Fig. 12.5) due to the different symmetry of the singlet and the triplet state [19]. 

The rotational population distributions were Boltzmann in nature, characterized by 7Ji = 640 35 K. This seems substantially lower than yet somewhat larger than the temperature associated with the translational degree of freedom. The lambda doublet species were statistically populated. The population ratio of i =l/t =0 was roughly 0.09, consistent with a vibrational temperature Ty— 1120 35K. The same rotational and spin-orbit distributions were obtained for molecules desorbed in t = 1 as for f = 0 levels. Finally, there was no dependence in the J-state distributions on desorption angle. 

The relative population ratio FJFi was slightly higher than expected from a 300 K thermal distribution (e.g. 2.1 vs 1.8). Of particular note, in comparison to a simple Boltzmann distribution, there was a substantial absence of population in the F2(J < S.S) levels from that expected based on a thermal (300 K) distribution. Approximately 1% of the desorbed molecules were vibrationally excited. 

The population of nuclei in energy level E2 is slightly less than that in energy level Ei, which is a little more stable. Population ratio (Boltzmann distribution) calculations that can be conducted using equation (9.5) for T = 300 K and B0 = 5.3 T lead to R = 0.999 964 (where k = 8.314/6.022 x 10"23 J KT1). 

We see from (4.104) that, although the vibrational quantum number is not changing, the frequency of a pure-rotational transition depends on the vibrational quantum number of the molecule undergoing the transition. (Recall that vibration changes the effective moment of inertia, and thus affects the rotational energies.) For a collection of diatomic molecules at temperature T, the relative populations of the energy levels are given by the Boltzmann distribution law the ratio of the number of molecules with vibrational quantum number v to the number with vibrational quantum number zero is... 

We described the nuclear Overhauser effect (NOE) among protons in Section 3.16 we now discuss the het-eronuclear NOE, which results from broadband proton decoupling in 13C NMR spectra (see Figure 4.1b). The net effect of NOE on 13C spectra is the enhancement of peaks whose carbon atoms have attached protons. This enhancement is due to the reversal of spin populations from the predicted Boltzmann distribution. The total amount of enhancement depends on the theoretical maximum and the mode of relaxation. The maximum possible enhancement is equal to one-half the ratio of the nuclei s magnetogyric ratios (y s) while the... 

At thermal equilibrium, the Boltzmann distribution determines the populations in various energy levels. For any two quantum states, the ratio of populations between the higher energy state and the lower energy state at equilibrium will always be ... 

According to quantum mechanics, only those transitions involving Ad = 1 are allowed for a harmonic oscillator. If the vibration is anhar-monic, however, transitions involving Au = 2, 3,. .. (overtones) are also weakly allowed by selection rules. Among many Au = 1 transitions, that of u = 0 <-> 1 (fundamental) appears most strongly both in IR and Raman spectra. This is expected from the Maxwell-Boltzmann distribution law, which states that the population ratio of the u = 1 and u = 0 states is given by... 

The rotational energy distribution of desorbed NO from on-top species on Pt(l 1 1) at A. = 192 nm is observed to have a non-Boltzmann form, as shown in Fig. 14. Furthermore, the population in the two spin-orbit states is substantially inverted, since the population ratio of 2 = 1/2 and 3/2 is 1 2.2 in low J region. For desorption of hep hollow species at = 193 nm, on the other hand, the population ratio... 

The relative population of each spin state is determined by the Boltzmann distribution, Eq. (2.8). Under conditions of a typical NMR experiment the ratio of spin state populations is near unity, differing only by a few parts per million. 

Saturation leads to equalization in the populations of the energy levels, contrary to the Boltzmann distribution. On the other hand, a number of NMR techniques can be employed to increase the population difference well beyond that given by the Boltzmann distribution. In some instances it is convenient to retain the formalism of the Boltzmann relation by defining a spin temperature Ts that satisfies Eq. 2.19 for a given ratio n /na. For times much less than T, it is meaningful to have Ts = T, the macroscopic temperature of the sample, because the spin system and lattice do not interact in this time frame.Viewed in this way, saturation... 

The Boltzmann distribution law allows us to determine the fraction of molecules existing within two energy states e, and ey. If the number of molecules in each is , and nj respectively, the ratio of these two populations is given by the equation... 

The nuclei, which can be likened to small magnets (or magnetic dipoles) arc distributed over different levels in accordance with the Boltzmann distribution low energy levels are more highly populated and the population ratio of 2 successive levels is given by ... 

The transition probability for the upward transition (absorption) is equal to that for the downward transition (stimulated emission). The contribution of spontaneous emission is neglible at radiofrequencies. Thus, if there were equal populations of nuclei in the a and f spin states, there would be zero net absorption by a macroscopic sample. The possibility of observable NMR absorption depends on the lower state having at least a slight excess in population. At thermal equlibrium, the ratio of populations follows a Boltzmann distribution... 

The intensity of an absorption line depends on the strength of the transition and on the population ratio of the initial and final states. The probability that the energy level E is occupied is N(E ), the number of molecules in that level, divided by Na, the total number of molecules in the system. This is given by the Boltzmann distribution... 

The vibrational temperature of a molecule prepared in a supersonic jet can be estimated from the observed populations of its vibrational levels, assuming a Boltzmann distribution. The vibrational frequency of HgBr is 5.58 X 10 s , and the ratio of the number of molecules in the n = 1 state to the number in the n = 0 state is 0.127. Estimate the vibrational temperature under these conditions. 

The population of nuclei located in energy state E2, is slightly less numerous than that in the more stable state E. Expression 15.5 calculates the ratio of these two populations (Boltzmann distribution equilibrium). 

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