Boltzmann’s derivation

The kinetic theory of dense gases began with the work of Enskog, who in 1922, generalized Boltzmann s derivation of the transport equation to apply it to a dense gas of hard spheres. Enskog showed that for dense gases there is a mechanism for the transport of momentum and energy by means of the intermolecular potential, which is not taken into account by the Boltzmann equation at low densities, and he derived expressions for the transport coeffi-... 

It is easy to extend Boltzmann s derivation to cover a mixture of monatomic gases whose particles all interact with pairwise additive central forces. To do this we denote the distribution function for gas molecules of species a by /a(ri,vi, 0- Then let Japifa,U) be the collision integral for collisions of particles of species a with particles of species /3, let be an external force acting on particles of species a, and let Tafa denote the collision term for collisions of particles of species a with the boundaries. Then/ satisfies... 

Boltzmann s derivations depended on the existence of matter being, ultimately, particulate. This is consistent with modern atomic theory. Boltzmanns ideas— including the idea that atoms behave statistically—have been accepted as a correct understanding of matter. 

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

Liouville s equation, derivation of Boltzmann s equation from, 41 Littlewood, J. E., 388 Lobachevskies method, 79,85 Local methods of solution of equations, 78... 

Other than an effect on backbone solvation, side chains could potentially modulate PPII helix-forming propensities in a number of ways. These include contributions due to side chain conformational entropy and, as discussed previously, side chain-to-backbone hydrogen bonds. Given the extended nature of the PPII conformation, one might expect the side chains to possess significant conformational entropy compared to more compact conformations. The side chain conformational entropy, Y.S ppn (T = 298°K), available to each of the residues simulated in the Ac-Ala-Xaa-Ala-NMe peptides above was estimated using methods outlined in Creamer (2000). In essence, conformational entropy Scan be derived from the distribution of side chain conformations using Boltzmann s equation... 

However, some problems remain unsolved. The three- and four-body results of Choh and Uhlenbeck and of Cohen, respectively, have to be compared with the corresponding expressions in the Prigogine s theory. Furthermore, for any concentration, one has to see how the systematic generalization of the Boltzmann equation derived by Cohen is related to the long-time evolution equation in Prigogine s theory. The aim of this work is to throw some light on these points. 

However, in above derivation it has been assumed that molecules of B are stationary, which is not correct. Because the molecules of gases are moving with different velocities and cannot be assumed to be stationary. Thus, the average velocity of molecules uA may be replaced by /8kb T/n i, where kh is Boltzmann s constant and p the reduced mass given by... 

The distribution of excess charge of hydrated ions in the diffuse layer can be derived by using Poisson s equation, d% dx = - o(jc)/e, and Boltzmann s distribution equation, Ci(x) = Cys) exp -Zie )/ 7 , to obtain the relationship in Eqn. 5-3 between the interfacial charge, om, and the diffuse layer potential, ohp ... 

The concentration profile of excess ions in the diffuse layer may be derived from Eqn. 5—3 and Boltzmann s distribution equation as a function of interfadal chai ge. Oh- Simple calculation gives the interfadal ionic concentration (at the OHP) to be c,.o = 1 M for an interfadal charge ay = 0.1 Cm" (corresponding to... 

In Eq. (1.36), Nj is the equilibrium number of point defects, N is the total number of atomic sites per volume or mole, Ej is the activation energy for formation of the defect, is Boltzmann s constant (1.38 x 10 J/atom K), and T is absolute temperature. Equation (1.36) is an Arrhenius-type expression of which we will see a great deal in subsequent chapters. Many of these Arrhenius expressions can be derived from the Gibbs free energy, AG. 

An alternative to Eq. 2.43 that is popular with modern kineticists is the Eyring equation (Eq. 2.44), which derives from the notion that the transition state is in (very unfavorable) equilibrium with the reactants but decays with a universal frequency given by kBT/h, where A b is Boltzmann s constant and h is Planck s constant. Then, by analogy with Eqs. 2.9 and 2.12, we have... 

Let q represent an observable quantity of a macroscopic system such as the circuit in Figure 1. Assuming that there are no other macroscopic observables, one can derive Eq. (15) from the equation of motion of all particles at the expense of a regrettable, but indispensable, repeated randomness assumption, similar to Boltzmann s Stosszahlansatz. 6 It then also follows that, provided q is an even variable, W has a symmetry property called detailed balancing. 6,7... 

In conclusion we must mention that a necessary condition for the validity of Eq, (3), and consequently of other formulas derived from Eq. (3) is that Ni < 1 for the state (or states) of lowest energy and a fortiori for all other states, When this inequality does not hold. Boltzmann s distribution law must be replaced by a more general and more precise distribution law, either that of Fermi and Dirac or that of Bose and Einstein according to the nature of the molecules. See also Statistical Mechanics. 

At temperatures where the potential minimum, e, is comparable to kT, equation (18) is not valid because the method of treating the thermal velocity distribution is no longer appropriate. When e becomes comparable to kT, orientation effects are believed to become important because in general the depth of the potential minimum is dependent on the configuration of both molecules. The probability of a particular orientation can be derived from Boltzmann s equation. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

GSS(Z Ri, R2) orGcc(Z Ri, R2) or Gim/2m(Z Ri, R2) h U Ti=—,h 2 x k kT kTIQQni P Used between spheres or cylinders of constant radii Ri, R2 of materials 1, 2. Planck s constant. Boltzmann s constant. Thermal energy. Thermal energy at room temperature. Pressure, negative spatial derivative of G per unit area between parallel planar surfaces negative pressure denotes attraction (a convention contrary to that in which pressure on a surface is defined in the direction of its outward normal... 

Here k is a constant, called Boltzmann,s constant, which will appear frequently in our statistical work. It has the same dimensions as entropy, or specific heat, that is. energy divided bv temperature. Jts value in absolute units is 1.379 X 10 16 erg per degree. This value is derived indirectly using Eq. (1.2), for the mifropy, derive tlir> porfftp.t... 

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