Boltzmann expression

In Chapter 1 we saw that the Boltzmann equation S = k log W gives the same qualitative relationship between entropy and disorder and suggested that a fundamental property of entropy is a measure of the disorder in a system. In Chapter 10 we will explore this relationship in more detail on the molecular level, and use the Boltzmann expression to develop quantitative relationships between entropy and disorder. 

The Boltzmann expression can be used to calculate the relative populations of molecules in any rotational state 7 compared to the lowest rotational state 7 = 0 at temperature T (K) ... 

Because the population ratio is determined by the appropriate Boltzmann expression where A is the energy difference between states and k is the... 

How much more positive than E does the electron energy have to be for us to relax the Fermi-Dirac law and use the simpler Boltzmann expression From... 

Before deriving the expressions for gas-phase collision frequency, we need to discuss the relative velocity that is important in collisions. The reduced mass mn is also obtained from this analysis, and will be the appropriate mass to use in the Maxwell-Boltzmann expression for collision velocities. 

From the Boltzmann expression (Sidebar 5.11) for the statistical mechanical entropy in this lowest-possible (T = 0) macrostate, we therefore obtain... 

Substitution of the Boltzmann expression (S8.6-3) for p into (S8.6-1) leads to the Poisson-Boltzmann (PB) equation... 

We can estimate the factor by which the thermal noise will be reduced with the Boltzmann expression ... 

The form of distribution (17) recalls a Boltzmann expression with modulus of distribution 7. Attempts at a direct physical explanation of this result are thwarted by the obvious dependence of 7, not only on the state of the surface, but also on the nature of the gas whose adsorption proceeds according to equation (1). Nevertheless, formula (17) makes very plausible the experimentally observed constancy of the functional dependence A(Q) itself which leads to equation (1). It seems natural that with training or sintering of the surface, the liberation or destruction of points with different heats of adsorption may proceed in such a way as to preserve the exponential relation between A and Q, changing only the constants D, Q0, and especially 7. 

This is called an energy in two squared terms. The Maxwell-Boltzmann expression for such a situation is... 

Let us examine now the effect of the excluded volume at low surface potentials. In the linear approximation of the Poisson—Boltzmann expression, the increase in the number of counterions in the vicinity of the interface equals the decrease in the number of co-ions. If the co-ions have a larger size, one expects the available volume near the surface to be larger than that in the bulk. As a result, a concentration of ions in excess to that predicted by the Poisson—Boltzmann equation is expected to occur in the vicinity of the surface, when the volume exclusion is taken into account. 

The traditional theory of the double layer is based on a combination of the Poisson equation and Boltzmann distribution. While this involves the approximation that the potential of mean force used in the Boltzmann expression equals the mean value of the electrical potential [9], the results thus obtained are satisfactory at least for 1 1 electrolytes. The equations proposed in the present paper use the approximations inherent in the Poisson—Boltzmann equation, but also include the effect of the polarization field of the solvent which is caused by a polarization source assumed uniformly distributed on the surface and by the double layer itself. 

The ionic concentrations at the surface can be related to their bulk concentrations via the Boltzmann expressions... 

Using Boltzmann expressions for the ion distributions, the Poisson—Boltzmann equation becomes... 

At equilibrium the distribution of molecules among the various energy states E is given by the Maxwell-Boltzmann expression. Thus for a molecule with n classical internal, harmonic oscillators, the fraction of molecules with energy J i, E2,. , En present in these oscillators is... 

The entropy in a solid arises first from corrfignrational terms that for a perfect solid are zero. However, for a solid showing orientational or translational disorder, corrfignrational expressions based on the Boltzmann expression S = khi(W)may be used. In this section, we shall pay more attention to the second term, which is arises from the population of the vibrational degrees of freedom of the solid. Thus the entropy of a solid may be written as ... 

The Boltzmann expression for one mole of SOjFj molecules having six possible orientations is... 

Allison el al. (1991) state that the activity difference between ions near the surface and those far away is the result of electrical work in moving the ions across the potential gradient between the charged surface and the bulk solution, The activity change of an ion moved from the surface to the bulk solution is described by EDL theory with an exponential Boltzmann expression... 

Since AH is the energy required to form the activated state (AB) from A and B, is the Boltzmann expression for the fraction of... 

This result is yet another example of an equation (like the Boltzmann expression for the entropy) which connects macroscopic and microscopic quantities. The origins of this equation can be obtained on the basis of thermodynamic reasoning and to probe these details the reader is encouraged to examine chap. 16 of Callen (1985) or the first sections of Feynman (1972). In the following section we will examine how this machinery may be brought to bear on some model problems. 

The problem is to get some device which would substitute the average distribution of the discrete ions in the ionic atmosphere around the centralj-ion, given by n, in the Maxwell-Boltzmann expression, by a continuous charge density which could be taken to be equivalent to pj in the Poisson equation. This would enable Poisson s equation to be combined with a Maxwell-Boltzmann distribution. 

The relative populations of ground-state (Nq) and excited-state (iVJ populations at a given flame temperature can be estimated from the Maxwell-Boltzmann expression ... 

Structural information about melting processes is growing quite rapidly in bulk and in its significance for the chemical physics of condensed states of matter. General statistical considerations about the numbers of ways of constructing the solid phase Wg and the liquid phase and the application of the Boltzmann expression... 

The exponentially decreasing term in the Boltzmann expression would seem to favour the very lowest energy states. However, this would lead to the paradoxical situation in which everything in the Universe should be at zero enthalpy. This can be resolved as follows. 

The total energy of the Universe is not zero. At the instant of the Big Bang (or whatever) the world was endowed with a large amount of energy. This energy will never go away, no matter how hard we try. So it must be distributed somehow. But how does that fit with the Boltzmann expression It is all to do with w . 

It is of interest to know the number of thermally excited atoms relative to the number of ground state atoms at a given flame temperature. In a quantity of atoms, under the same external conditions, the electrons are not all in the same energy level but are statistically distributed among the levels. At a flame temperature T (in K), the ratio of the number of atoms in an excited (upper) state u to the number of atoms Ao in the ground state is given by the Maxwell-Boltzmann expression... 

Long et al. (1973) took this analogy with vapour condensation one step further. At equilibrium, they related the dispersed and flocculated particle number concentrations (in actual fact, the particle volume fractions) through a simple Boltzmann expression ... 

For a continuous strain history, the Boltzmann expression becomes... 

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