Maxwell-Boltzman distribution

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

Top typical saturation curve and variation of mean electron energy with applied field. Middle fraction of the electron swarm exceeding the specific energy at each field strength. Calculated assuming constant collision cross-section and Maxwell-Boltzman distribution. Bottom variation of products typical of involvement of ionic precursors (methane) and excited intermediates (ethane) with applied field strength... 

The Maxwell-Boltzman distribution function permits calculation of the effect of temperature on each electronic transition. Designated by /V0 and /Ve, the number of atoms in the ground and excited states, one obtains ... 

This difference between fermions and bosons is reflected in how they occupy a set of states, especially as a function of temperature. Consider the system shown in Figure E.10. At zero temperature (T = 0), the bosons will try to occupy the lowest energy state (a Bose-Einstein condensate) while for the fermions the occupancy will be one per quantum state. At high temperatures the distributions are similar and approach the Maxwell Boltzman distribution. 

Fig. 23.—Illustration of the energy supplied from an outside source in photochemical reactions. The normal thermal energy supplied from collisions by the Maxwell-Boltzman distribution is insufficient to produce the chemical reaction.

Other distribution functions such as the Maxwell-Boltzman distribution, (%2) distribution, etc. are used as well [99], A discrete law of bubble size distribution (Poisson distribution) is presented in [10]... 

This distribution equation is known as the Maxwell-Boltzman distribution. 

In classical molecular dynamics simulations, atoms are generally considered to be points which interact with other atoms by some predehned potential form. The forms of the potential can be, for example, Lennard-Jones potentials or Coulomb potentials. The atoms are given velocities in random directions with magnitudes selected from a Maxwell-Boltzman distribution, and then they are allowed to propagate via Newton s equations of motion according to a finite-difference approximation. See the following references for much more detailed discussions Allen and Tildesley (1987) and Frenkel... 

In spite of the difference in the underlying concepts and the forms of equations, Eqs. (3.3) and (3.4), both descriptions reflect the statistical sense of the rate constant. The latter statement is crucially important for better understanding of the problem existing in heterogeneous kinetics. Indeed, the above-mentioned theories are based on gas statistics and the given equations assume an equilibrium Maxwell-Boltzman distribution for gas species, which in the absence of reaction interact only via elastic collisions. If this can be considered as a satisfactory approximation for gas reactions at moderate temperatures and pressures discussed here (with some exceptions—see Section III.D), its applicability to the processes involving surface sites (i.e., elements of solid lattice) or adsorbed species is not so obvious. 

Rate constants of heterogeneous reactions are usually represented in the three-parameter form discussed above. However, this form has a clear physical sense only for reactions in ideal gases at a strictly kept equilibrium Maxwell-Boltzman distribution. Despite several attempts to adopt transition-state theory to heterogeneous reactions, and to those proceeding in adsorbed layers in particular (e.g., Krylov et al., 1972 Zhdanov et al., 1988), its applicability in these cases is doubtful. 

No special equilibrium between activated complexes and reactants is assumed. It is supposed however, that within the space between qi and qi+Aqi, q2 and q2+Aq2, configurations (activated complexes) have impulses (motions) between pi and pi+Api,p2 and P2+AP2 respectively. These configurations are computed in accordance with Maxwell-Boltzmann distribution. Recall from the gas laws that an energy profile for molecules can be describe by the Maxwell-Boltzman distribution diagram (Figure 3.3). As the temperature goes up, the population of molecules with more energy also increases. 

Velocities are allocated randomly to atoms to produce a Maxwell-Boltzman distribution (Figure 9.4). If a low temperature is chosen (for instance 100 K, blue curve) then most atoms will have roughly the same velocity. In terms of the model at the... 

Figure 9.4 Maxwell-Boltzman distributions at various temperature...

The relative intensities of rotational lines in a vibrational band of an electronic absorption spectrum are almost entirely dependent on the relative populations on the initial rotational levels (Maxwell-Boltzman distribution). Use this fact to make a sketch of fhe Q-branches in Example 10.1 assuming a sample temperature of 300K. How would fhey appear under low resolution ... 

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