The Maxwell-Boltzmann distribution law

Two types of distribution functions have been discussed so far the spatial distribution of molecules in a gravitational field, the Boltzmann distribution, and the speed distribution in a gas (the Maxwell distribution). These can be written in a combined form, the Maxwell-Boltzmann distribution law. 

The barometric formula governs the spatial distribution of the molecules in a gravity field according to the equation 

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Consider, as an example, the calculation of the mean-square speed of an ensemble of molecules which obey the Maxwell-Boltzmann distribution law. This quantity is given by... 

Figure 10 A graphical illustration of the Maxwell-Boltzmann distribution laws. Normalized speed---is vlvp, and normalized energy - is EUcT.

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

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... 

A comparison with Eq. (1) shows that, as far as the velocity distribution is concerned, Eq. (3") agrees with the Maxwell assumption. Hence we call Eq. (3) the Maxwell-Boltzmann distribution law.40... 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

Equation (1.4) expresses what is called the Maxwell-Boltzmann distribution law. If Eq. (1.4) gives the probability of finding any particular molecule in the fctii state, it is clear lhat it also gives the fraction of all molecules to be found in that state, averaged through the assembly. 

The barometer formula can be derived by elementary methods, thus checking this part of the Maxwell-Boltzmann distribution law. Consider a column of atmosphere 1 sq. cm. in cross section, and take a section of this column bounded by horizontal planes at heights ft and ft + dh. Let the pressure in this section be P we are interested in the variation of P with ft. Now it is just the fact that the pressure is greater on the lower face of the section than on the upper one which holds the gas up against gravity. That is, if P is the upward pressure on the lower face, P + dP the downward pressure on the upper face, the net downward force is dP,... 

We can show, as we did with the Fermi-Dirac statistics, that the distribution (6.5) approaches the Maxwell-Boltzmann distribution law at high temperatures. It is no easier to make detailed calculations with the Einstein-Bose law than with the Fermi-Dirac distribution, and on account of its smaller practical importance we shall not carry through a detailed... 

The Maxwell-Boltzmann distribution law shows that the ratio of the number of atoms in an excited upper state see Excited State) to the number in the ground state increases exponentially with temperatme. For example, the ratio N /No for Cs (resonance line at 852.1 mn), Ca (422.7 nm), and Zn (213.8 nm) increases from 4.44 x 10 , 1.21 x 10 , and 7.29 X 10- at 2000 K to 2.98 x lO, 6.04 x 10 and 1.48 X 10 at 4000 K. Even so, this means that for many transition metals only a small fraction of the vaporized atoms are in an excited state capable of emitting a line spectrum, even at the temperatme of an ICP. 

The distribution of velocities of the particles in an ideal gas is described by the Maxwell-Boltzmann distribution law ... 

The selection rule allows any transitions corresponding to Au = 1 if the molecule is assumed to be aharmonic oscillator (Sec. 1.3), Under ordinary conditions, however, only the fundamentals that originate in the transition from u = 0 to i = 1 in the electronic ground state can be observed. This is because the Maxwell-Boltzmann distribution law requires that the ratio of population at i = 0 and v = 1 states is given by... 

At room temperature, most of the scattering molecules are in the u = 0 state, but some are in higher vibrational states. Using the Maxwell-Boltzmann distribution law, we find that the fraction of molecules 4 with vibrational quantum number v is given by... 

The excess contribution can be calculated from the crystal field energies according to the following equations where Q is the partitioning function described by the Maxwell-Boltzmann distribution law in equation (4), T the temperature, R the universal gas constant, the energy of the level i, and g, its degeneracy ... 

In deriving Eq. (8), it is assumed that molecules A and B collide with a single relative velocity g. In a real gaseous sample containing both A and B molecules at thermal equilibrium, the distribution of relative velocities is described by the Maxwell-Boltzmann Distribution Law ... 

These equations express the quantal analogue of the Maxwell-Boltzmann distribution law, and they are more general in their scope than the Maxwell expression for the distribution of velocities (the latter will be derived from (12 72) in 12 11). 

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