Boltzmann potential energy relation

In more complex situations, where e.g. rotations and vibrations are involved, there will be squared terms relating to each rotational and each vibrational mode. Each rotational mode involves the square of the angular velocity and contributes one squared term. Each vibrational mode involves a term from the potential energy and a term for the kinetic energy of vibration, and contributes two squared terms if the vibration is harmonic. If there are a total of 2s squared terms, then this is called energy in 2s squared terms. The Maxwell-Boltzmann distribution is correspondingly more complex (Section 4.5.8). 

It is now clear what conditions we must have for equilibrium between metal a, metal 6, and the gas. The vapor pressure outside metal a must be the correct one for thermal equilibrium with that metal, the pressure outside b must be the correct one for equilibrium with it, and the pressures outside the two metals must be related according to the Boltzmann factor, to ensure equilibrium between the different parts of the gas. Thus let the potential energy of a mole of electrons outside the metal a bo Eaf and outside metal b be E. Then the Boltzmann factor leads to the relation... 

The situation of the barrier is different from what it would be with the Boltzmann statistics, as can be seen most clearly from Fig. XXIX-6. Here we have drawn energies, both inside and outside the metal, as in Fig. XXIX-4. The zero of energy is taken to be at the bottom of the picture. Then at the absolute zero there will be filled energy levels up to the energy Wi and empty levels above, the filled ones being shaded in Fig. XXIX-6. We can now see that the potential energy of an electron outside the metal, Wa, is related to Wi and to Lq, the heat of vaporization... 

The Boltzmann distribution describes the relative population of molecules at different altitudes in a column of air above the surface of the earth due to the differences in their gravitational potential energy at these altitudes. The most convenient form of this distribution—called the barometric equation—relates the pressure Fy in the column at altitude h to its value Pq t the surface of the earth as Fy = Pq exp[—TfgP/RT], where M, is the molar mass of the gas and g is the acceleration of gravity. Calculate the pressure at an altitude of 1 km for a gas with average molar mass 29 g mol when the temperature is 298 K and the pressure at the surface of the earth is 1 atm. 

However, the intermolecular force laws play a central role in the model determining the molecular interaction terms (i.e., related to the collision term on the RHS of the Boltzmann equation). Classical kinetic theory proceeds on the assumption that this law has been separately established, either empirically or from quantum theory. The force of interaction between two molecules is related to the potential energy as expressed by... 

To enable us to discuss the electrostatics of electrolyte solutions we need to introduce another fundamental principle - Boltzmann s distribution law — which relates the probability of particles being at a given point at which they have a potential energy, or free energy, A G, relative to some chosen reference state. This probability may be expressed in terms of the average concentration, c, at the point considered relative to that, r", at the reference level, taken as the zero of energy. If the temperature is T, then... 

Here, (r) is the force derived from the potential energy 7(r). The friction constant y and the random force are related through the fluctuation dissipation theorem < (t) (0)> = 2myk Td t), where T is the temperature and kg is Boltzmann s constant. The random thermal noise compensates for the energy dissipated by the frictional term —yp. Because we focus on finite segments of trajectories, the treatment of noise that is correlated in time is awkward within the specific methodology presented in this chapter. 

M makes an angle j8 with the field F, its potential energy is proportional to —(jlF ooa. By Boltzmann s relation (p. 79) the number of molecules at temperature T which possess this potential energy is proportional to e +i BOBplkT trigonometrical factors. If kT... 

To obtain Eqs. 11, 12, 13, and 14, it was assumed that the concentration of each ionic species within the electric double layer is related to the electric potential energy by a Boltzmann distribution. A comparison of Eq. 11 with the numerical results obtained by Prieve and Roman [2] shows that the thin-layer polarization model is quite good over a wide range of zeta potentials when Ka > 20. If 1(1 is small and Ka is large, the interaction between the diffuse counterions and the particle surface is weak and the polarization of the double layer is also weak. In the limit of... 

---