Maxwell-Boltzmann laws

The approach that we will follow is known as the Debye-Hiickel theory. The activity laws discussed in the following are derived from a knowledge of electrostatic considerations, and apply to ions in solution that have an energy distribution that follows the well-known Maxwell-Boltzmann law. Strong electrostatic forces affect the behaviour and the mean positions of all ions in solution. 

Fig. 9.5. Schematic representation of acceptor (empty) and donor (filled) electronic states of ions in solution. The states are distributed in solution according to the Maxwell-Boltzmann law. Fluctuations of all states (i.e., ground and other higher energy states) are considered to give rise to a continuum distribution (vibrational model). (Reprinted with permission from J. O M. Bockris and S. U. M. Khan, J. Phys. Chem. 87 2599 copyright 1983 American Chemical Society.)...

The apparently random stepwise or zig-zag movement of colloidal particles (Figure 6.1) was first observed by the botanist Robert Brown in 1827, and named after him. It provided early evidence for the molecular kinetic theory and was interpreted as arising from the random buffeting or jostling of the particles by molecules of the surrounding medium. The directions of movement of the molecules of the medium immediately adjacent to the particles are randomly oriented, while their speeds are distributed according to the Maxwell-Boltzmann law. The force acting upon the surface of a colloidal particle is proportional both to the frequency with which molecules collide with it and to the velocity of these molecules. The former is proportional to the local density of the molecules within one free path of the surface. Since the local... 

These qualitative remarks are rendered clearer by a simple calculation. Suppose that for a solid vibrating in one degree of freedom the energies of the atoms must correspond to 0, e, 2e, 3e,..., that is, the successive values are multiples of a standard quantum . Out of N atoms, the number which would normally possess energy greater than j is given by the Maxwell-Boltzmann law to be (see... 

In classical statistical mechanics, each particle is regarded as occupying a point in phase space, i.e. to have an exact position and momentum at any particular instant. The probability that this point will occupy any small volume of the phase space is taken to be proportional to the volume. The Maxwell-Boltzmann law gives the most probable distribution of the particles in phase space. 

The reactant molecules are distributed amongst their states according to the Maxwell-Boltzmann laws. 

Molecules of the same nature are distributed on these various energy levels according to the Maxwell-Boltzmann law which involves a temperature factor. Therefore, at room temperature most of them are at the lowest level (Eo, vo, ro), which is called the ground state. Changes in all three forms of internal energy... 

For a sample of many molecules, under the assumption of no interaction between them, the probabilities are directly proportional to the numbers of molecules in each state, the populations, which we designate as n. For a large collection of A molecules at some temperature T, the number found in the ith energy level of an A molecule is n = NPj, where N is the total number of A molecules. Is it possible for a large collection of molecules to have populations of molecular quantum states other than those dictated by the Maxwell-Boltzmann law Yes. But in that event, the system is not at equilibrium, which means that it is not stable and is undergoing change. The distribution law holds for equilibrium conditions, and under those conditions, it can be used to determine the number of molecules in particular energy level states. 

To calculate the mean velocity of displacement, we use the Maxwell-Boltzmann law, which indicates that the probability for a particle of mass m to have its velocity ranging between two values dx/df and dx/di +d(dx/dr) according to a degree of freedom is ... 

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