The Boltzmann Distribution Law

To calculate the most probable configuration, we start with equation (10.15) and take the logarithm of both sides to get 

We next eliminate the factorial expressions by substituting Stirling s approx-imationj so that 

Stirling s approximation is a mathematical relationship that becomes more accurate as N becomes larger, and becomes very precise for the large number of units we will consider as we work with moles of ideal gas. Substituting equations (10.17) and (10.18) into equation (10.16) gives 

To find the most probable distribution (largest W) we make an infinitesimal displacement in n, (with N held constant) and set the displacement equal to zero. That is 

Equation (10.21) can be simplified further by noting that since N = then 

Some systems that can be considered as states of different energy. 

Liquid AEis positive Solid AEis positive Solution AEis negative  

States of different energy populated according to a Boltzmann distribution where n, and w2 are the populations of states having energies and E2, respectively. 

An explanation of how the population of states varies with the energy separating them is provided by a principle known as the Boltzmann distribution law, which is named after Ludwig Boltzmann (1844-1906). A simplified form of the relationship can be written as 

It can be seen that the relative population of the higher energy state decreases as the energy difference increases. However, because T occurs in the denominator of the exponential function, one way to increase the population of the upper state is to increase the temperature. Going from the lower state to the upper one is an endothermic process, and increasing the temperature favors the endothermic direction of the process. 

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The intensity distribution among the rotational transitions is governed by the population distribution among the rotational levels of the initial electronic or vibronic state of the transition. For absorption, the relative populations at a temperature T are given by the Boltzmann distribution law (Equation 5.15) and intensities show a characteristic rise and fall, along each branch, as J increases. 

We need one more equation before we can derive the Boltzmann distribution law. We note that the total energy U - C0 is given by... 

A second equation linking parameters / and Qy is needed so that they may be calculated individually. To derive such an equation it was assumed that the concentration of ions Cj is determined by the Boltzmann distribution law. 

If the origin of the coordinate system is located in the centre of the central kth ion, the number of particles dNt in the volume dV is given by a distribution function, expressed by Debye and Hiickel in terms of the Boltzmann distribution law (cf. p. 215)... 

Quantitatively, the effect of temperature on solubility can be explained by considering the dissolution process as represented by the population of states of unequal energy. The Boltzmann distribution law relating the populations nx and n2 for states Et and E2 which are of unequal energy can be written as... 

At thermal equilibrium the population of any series of energy levels is described by the Boltzmann distribution law. If N0 molecules are in the ground state then the number Ni in any higher energy level is given by the equation ... 

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). 

Since our solvents are permanent dipoles that develop an orientation under the influence of an outside electric field activity in opposition to the disordering influence of thermal agitation, such an orientation process is governed by the Boltzmann distribution law and results in the dielectric constant being strongly dependent on the absolute temperature. Thus, as the systems become cooler, the random motion of their molecules decreases and the electric field becomes very effective in orienting them the dielectric constant increases markedly with reduction in temperature at constant volume. 

In this article we use transition state theory (TST) to analyze rate data. But TST is by no means universally accepted as valid for the purpose of answering the questions we ask about catalytic systems. For example, Simonyi and Mayer (5) criticize TST mainly because the usual derivation depends upon applying the Boltzmann distribution law where they think it should not be applied, and because thermodynamic concepts are used improperly. Sometimes general doubts that TST can be used reliably are expressed (6). But TST has also been used with considerable success. Horiuti, Miyahara, and Toyoshima (7) successfully used theory almost the same as TST in 66 sets of reported kinetic data for metal-catalyzed reactions. The site densities they calculated were usually what was expected. (Their method is discussed further in Section II,B,7.)... 

Introducing the ni(r) value as given in Eq. (2.10) into Eq. (2.12) one obtains the Boltzmann distribution law in terms of charge densities... 

When a system is in thermodynamic equilibrium the level population, i.e. the number of atoms A in the excited state, is given by the Boltzmann distribution law ... 

Adsorption. In the simple theory of the space charge inside a semiconductor, it was assumed that all the electrons and holes are free to move up to the surface. Being susceptible to thermal motion, their concentrations from a- = 0 to x — °° were said to be given by the interplay of electrical and thermal forces only, as expressed by the Boltzmann distribution law and Poisson s equation. 

In the discussion of many properties of substances it is necessary to know the distribution of atoms or molecules among their various quantum states. An example is the theory of the dielectric constant of a gas of molecules with permanent electric dipole moments, as discussed in Appendix IX. The theory of this distribution constitutes the subject of statistical mechanics, which is presented in many good books.1 In the following paragraphs a brief statement is made about the Boltzmann distribution law, which is a basic theorem in statistical mechanics. 

It is convenient and useful to express the Boltzmann distribution law in two forms a quantum form and a classical form. The quantum form of the law, in its application to atoms and molecules, may be expressed as follows The relative probabilities of various quantum states of a system in equilibrium with its environment at absolute temperature T, each state being represented by a complete set of values of the quantum numbers, are proportional to the Boltzmann factor e Wn/kT, in which n represents the set of quantum numbers, Wn is the energy of the quantized state, and k is the Boltzmann constant, with value 1.3804 X 10 16 erg deg 1. The Boltzmann constant k is the gas-law constant R divided by Avogadro s number that is, it is the gas-law constant per molecule. 

In case that the energy can be expressed as the sum of two terms, one depending only on the coordinates (the potential energy) and the other only on the momenta (the kinetic energy), the Boltzmann distribution law for coordinates can be discussed separately from that for momenta, because the Boltzmann factor can be split into the product of two exponential terms, one involving only the coordinates and the other only the momenta. 

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... 

Many interesting and useful concepts follow from classical statistical con side rations (eg, the Boltzmann distribution law) and their later modifications to take into account quantum mechanical effects (Bose-Einstein and Fermi-Dirac statistics). These concepts are quite beyond the scope of the present article, and the reader should consult Refs 14 16. A brief excursion into this area is appropriate, however. A very useful concept is the so-called partition function, Z, which is defined as ... 

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