The symbol W in the Boltzmann formula engraved into the Boltzmann tombstone in the 1930s. [2] |

In simpler terms, W is the number of different ways P particles can be distributed in a system composed of N different, but connected, compartments. [5]

To note, there is not a good English translation equivalent of the German word

Formulations

There are a number of similar but different formulas (and accompanying explanations) given for this "multiplicity", or probability, as it was called by Boltzmann, each of which seems to differ details, depending on the components of the system, e.g. atoms, molecules, bosons, etc. [1]

In 1969, American chemical engineer

where

In 2003, Danish chemical-physicist John Avery gave the following expression for W: [6]

where

In 1999, American physicist Ralph Baierlein gives the following definition of entropy:

and defines the change in entropy ΔS = Sf – Si in a process, such as between the entropy of ice as compared to water, as being a function of the difference between the two multiplicities of the two states:

or

and then defines the

Two-container/eight-particle example

To go through a simple example, given by Belgian thermodynamicist Ilya Prigogine, suppose we have eight particles, N = 8, contained in a two-compartment system, such as two spherical gas bulbs connected via an open stopcock. The problem, then, is to find the probability of the various possible distributions of particles between the two compartments. There is, for instance, only way of placing the eight particles in a single half. If we assume the particles to be distinguishable, there are eight different ways of putting one particle in one half and seven in the other. Equal distribution of the eight distinguishable particles between the two halves can be done in:

different ways. Prigogine, however, stops here and does not go on to calculate the total number of complexions or multiplicity for this system [5]

Orbital description

In molecular orbital theory, multiplicity is a quantity used in atomic spectra to describe the energy levels of man-electron atoms characterized by Russell-Saunders coupling given by 2S+1, where S is the total electron spin quantum number. The multiplicity of an energy level is indicated by a left superscript to the value of L, where L is the resultant electron orbital momentum of the individual electron orbital angular momenta l. [3]

References

1. Pauling, Linus. (1969).

2. Muller, Ingo. (2007).

3. Daintith, John. (2004).

4. Planck, Max. (1901). "On the Law of Distribution of Energy in the Normal Spectrum".

5. Prigogine, Ilya. (1984).

6. Avery, John (2003).

7. Baierlein, Ralph. (1999).

External links

● Wahrscheinlichkeit (German → English) – Wikipedia.