Boltzmann distribution vibrational states

How can the vibrational temperature of a polyatomic molecule in a molecular beam be assessed A traditional approach is through the measurement of vibrational hot bands. If it is assumed that the vibrations are in a Boltzmann distribution of states, the temperature of the molecule can be determined from the ratio of the hot band (1 — 0) intensity to the (0- 1) intensity according to 7(1 0)/I(0 1) =... 

Now we need to solve for K. This is the point at which TST turns into a statistical mechanics analysis. We invoke statistical mechanics because a transition state does not have a Boltzmann distribution of states (see the next section), because its lifetime is so fleeting. Using statistical mechanics, it is found that K is proportional to a new equilibrium constant K ) that can be viewed in the same manner as the simple equilibrium constants given in Chapter 2. Thus, this K is equal to exp(-AG /RT). The exact expression found for K is Eq. 7.13, where k, h, v, and T are the Boltzmann constant, Planck s constant, vibrational frequency, and absolute temperature, respectively (see Appendix 1 for values). 

The intensity distribution among rotational transitions in a vibration-rotation band is governed principally by the Boltzmann distribution of population among the initial states, giving... 

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 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Fig. 5.2 Radial distribution curves, Pv

Just as above, we can derive expressions for any fluorescence lifetime for any number of pathways. In this chapter we limit our discussion to cases where the excited molecules have relaxed to their lowest excited-state vibrational level by internal conversion (ic) before pursuing any other de-excitation pathway (see the Perrin-Jablonski diagram in Fig. 1.4). This means we do not consider coherent effects whereby the molecule decays, or transfers energy, from a higher excited state, or from a non-Boltzmann distribution of vibrational levels, before coming to steady-state equilibrium in its ground electronic state (see Section 1.2.2). Internal conversion only takes a few picoseconds, or less [82-84, 106]. In the case of incoherent decay, the method of excitation does not play a role in the decay by any of the pathways from the excited state the excitation scheme is only peculiar to the method we choose to measure the fluorescence (Sections 1.7-1.11). 

The frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

The reactant molecule BC is specified to be in an initial vibrational v and rotational state 7, which determines p and allows R to be set to the maximum bond extension compatible with total vibrational energy. The initial relative velocity uR may be varied systematically or it may be chosen at random from Boltzmann distribution function. The orientation angle, which specify rotational phase and impact parameter b are selected at random. 

Calculations based on the Boltzmann distribution law show that, at room temperature, most molecules will be in the v = 0 vibrational state of the electronic ground state and so absorption almost always occurs from So(v = 0) (Figure 2.2). 

At room temperature, most of the molecules are in the lowest vibrational level of the ground state (according to the Boltzmann distribution see Chapter 3, Box 3.1). In addition to the pure electronic transition called the 0-0 transition, there are several vibronic transitions whose intensities depend on the relative position and shape of the potential energy curves (Figure 2.4). 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

Figure 4.15 shows the Boltzmann distribution for several values of kT/E for a system where the states have evenly spaced energies. At low temperatures, most of the molecules can be found at the lowest energy states, with energy level equal to zero. When the temperature is increased, more and more molecules are promoted to higher energy states. When a molecule has several degrees of freedom, such as translations, rotations, and vibrations, each has its own quantum states and partition functions, and then the overall partition function is a product of all these separate partition functions ... 

Fig. 9.5. Schematic representation of acceptor (empty) and donor (filled) electronic states of ions in solution. The states are distributed in solution according to the Maxwell-Boltzmann law. Fluctuations of all states (i.e., ground and other higher energy states) are considered to give rise to a continuum distribution (vibrational model). (Reprinted with permission from J. O M. Bockris and S. U. M. Khan, J. Phys. Chem. 87 2599 copyright 1983 American Chemical Society.)...

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