Boltzmann Distribution: In most cases of interest of chemistry the particles adopt the Boltzmann distribution.
Qualitative considerations: the general expression for W given by eq. can be used to compare the number of ways that different distributions of molecules throughout allowed energies can be achieved.
We want to investigate the values of W for various distributions, subject to these constrains:
1. The total number of particles is fixed i.e. we are dealing with a particular sample.
2. The system of particles has some fixed energy.
The way in which W varies with the type of distribution can be seen by dealing with specific simple systems.
Consider an energy pattern in which there is only one state at each of the allowed energies. Then all theg_{1 }values equal 1. This simplifies the experience for the number of ways in which the distribution can be achieved to:
W = N!/N_{1}! N_{2}! N_{3}!......
To begin, we deal with a manageable number of particles instead of Avogadro's number. Then the N! Values are small enough that the value of W can be calculated directly. From the W values for various distributions- subject to these two constraints- the distribution that corresponds to the target is, most probable distribution can be discovered.
Then notice that the more the particles are spread out through the available states, subject to the two constrains, the larger the value of W. the largest value of W is obtained with the final distribution, the most probable distribution is then in which the particles are spread out as much as possible. The bunching up in the lower energy states is more than compensated for by a spreading out through the higher energy states.
The way most probable distribution varies with the total energy of the system, or the average energy of the system, is illustrated by the examples. All the distributions have the same energy levels. The higher the average energy, the more the particles are spread out into the higher levels.
From the preceding qualitative considerations, you see that the most probable distribution of molecules throughout the energies allowed to them depends on the average energy. From the kinetic molecular studies of the average energy, or the energy per mole of particles, depends on the absolute temperature. It follows that the most probable distribution depends on the Boltzmann distribution shows this dependence for the most systems that are of interest in chemistry.
The Boltzmann distribution shows the number of particles per state of energy is compared to the other. The Boltzmann distribution expresses two of the allowed energies and the numbers of particles in the number of states at these energies, is
N_{i}/g_{i}/N_{j}/g_{j} = e^{-}(E_{i}^{-} E_{j})/ (kT)
The distribution implied by these exponential expressions can seen to be like the most probable distributions arrived at the population of the states decrease rapidly as we go to the higher energies.
Boltzmann distribution: from the preceding qualitative considerations, the most probable distribution of molecules throughout the energies allowed to them depends on the average energy, or the energy per molecule of particles, depends on the absolute temperature. The Boltzmann distribution shows this dependence for most systems that are of interest in chemistry.
At lower temperatures the exponential factor makes the population particles are crowded rapidly with increasing energy. The particles are crowded into the states of lower energy. At higher temperatures the exponential fall-off is less rapid. The particles can spread out into the higher energy states.
The Boltzmann distribution shows the number of particles per state at one energy compared to that at energy. The Boltzmann distribution expression, with I and j indicating two of the allowed energies, Ni and Nj the number of particles with these energies, and g_{i} and g_{j} the number of states at these energies.
Derivation of the Boltzmann distribution: the development that leads to the Boltzmann distribution expression is now given. This development is an optional section that can be bypassed without loss of continuity.
The goal is to find values of N, N_{1}, N_{2},.... That maximize W. it is mathematically more convenient to look for a maximum in In W, which occurs also where W is a maximum. (the logarithmic expression allows Sterling's approximation In x! = x In x - x for large numbers, which is derived to be applied.) when a Sterling's approximation is used for N_{i} terms we obtain:
In W = N_{0} In g) + N_{1} In g_{1} +.... + N! - N_{0} In N_{0} - N_{1} In N_{1} - ....+ N_{0} + N_{1} +....
= N_{0} (1 + In g_{0}/N_{0}) + N_{1} (1+ In g_{1}/N_{1}) +.... + In N!
We cannot now proceed directly to the desired maximum by setting this derivative equal to zero. We must recognize that there are limitations on the values the Ni's may take. [They cannot, for example, all go to zero, or infinity, might suggest.]
The two constrains are;
1. The total number of molecules is fixed. This total number is given by:
N_{0} + N_{1} + N_{2} + ... = Σ N_{i}
2. The total energy is also some fixed quantity. We can ensure that this is so by requiring that the energy in excess which the system would have if all the molecules were in the lowest energy levels be fixed. This "thermal energy" is:
3. N_{0} (0) + N_{1} (ε_{1} - ε_{0}) + N_{2} (ε_{2} - ε_{0}) +..... = Σ (ε_{i} - ε_{0})N