Boltzmann probability law

Figure 1.1 Illustration of the Boltzmann probability law of Equations (1.6) and (1.7). The state probability distribution is plotted at two different temperatures for a system with ten possible microstates with energy ranging from 10 21 to 10-20 joules. At the lower temperature (T = 273 K), the lower-energy states are significantly more probable than the higher-energy states. At the higher temperature (T = 1000 K), the energy distribution becomes more uniform than at the lower temperature.

AB, corresponding to the left and right sides of Equation (10.14). If the probability of complex existing (the relative amount of time the system is found in the AB state) is p, then the Boltzmann probability law says that the energy difference between the two possible states is —ksT In As we shall see, the probability p is a function of V, the volume of the system. We define Ka = which will be shown to be related to (but not equal to) the equilibrium constant for the reaction. 

As is apparent in the figure, a minimum in energy occurs where the molecules A and B are separated by tq. Conformations in or near this energy correspond well to the AB complex in Equation (10.14). Values of r that are sufficiently greater than ro correspond to the A + B uncomplexed state. To make this definition precise, we note that the probability of the distance between A and B, RAB, is directly related to U(r) following the Boltzmann probability law ... 

All these details of energy and interactions are dependent on Boltzmann and the distribution law. The probability of maintaining the molecular configurations implied in the diagrams (particularly the summit) can be calculated by determining the probability of their existence by means of Boltzmann s law. 

It is convenient and useful to express the Boltzmann distribution law in two forms a quantum form and a classical form. The quantum form of the law, in its application to atoms and molecules, may be expressed as follows The relative probabilities of various quantum states of a system in equilibrium with its environment at absolute temperature T, each state being represented by a complete set of values of the quantum numbers, are proportional to the Boltzmann factor e Wn/kT, in which n represents the set of quantum numbers, Wn is the energy of the quantized state, and k is the Boltzmann constant, with value 1.3804 X 10 16 erg deg 1. The Boltzmann constant k is the gas-law constant R divided by Avogadro s number that is, it is the gas-law constant per molecule. 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

According to Maxwell and Boltzmann, the law of equipartition of energy can be defined as, the energy given to a gas is equally distributed amongst each degree of freedom. The degree of freedom is the system to represent the probable velocity of a molecule. 

If a molecule comprised of s classical oscillators is in contact with a thermostat at temperature T, the probability P( C)d that it will carry vibrational energy in the range e, C + d is obtained from the Boltzmann distribution law by substituting S2( )d for the degeneracy, viz. 

Equation (1.4) expresses what is called the Maxwell-Boltzmann distribution law. If Eq. (1.4) gives the probability of finding any particular molecule in the fctii state, it is clear lhat it also gives the fraction of all molecules to be found in that state, averaged through the assembly. 

In this method a random number generator is used to move and rotate molecules in a random fashion. If the system is held under specified conditions of temperature, volume and number of molecules, the probability of a particular arrangement of molecules is proportional to exp(-U/kT), where U is the total intermolecular energy of the assembly of molecules and k is the Boltzmann constant. Thus, within the MC scheme the movement of individual molecules is accepted or rejected in accordance with a probability determined by the Boltzmann distribution law. After the generation of a long sequence of moves, the results are averaged to give the equilibrium properties of the model system. 

The average displacement, x = R — Rq), would be zero. But the term in x. Equation (6.17), makes a positive value of x more probable than a negative value. This is based on the Boltzmann distribution law, and a negative value for g. A calculation gives... 

Because the energy differences 8E between the sublevel states of each multiplet of mA and nB are very small, 8E ambient temperature and above, they are nearly equally populated under equilibrium conditions (Boltzmann s law, Equation 2.9) and the probability of the formation of any given encounter spin state will be equal to all the others as there are mn choices, it will be equal to the spin-statistical factor a = (mn) ... 

According to Boltzmann distribution law, the fraction of molecules which in the most probable state at temperature T possesses the energy Et is given by... 

Although we have obtained the Boltzmann distribution law for systems in which the values of nt are large, its validity is Irnore general and it can be shown to apply equally well to systems in which the probability of any particular energy state being occupied is very small. 

Fiu. 49-1.—The probability values Pn for system-part a in a system of five coupled harmonic oscillators with total quantum number n 10 (closed circles), and values calculated by the Boltzmann distribution law (open circles). 

To answer this question, the Boltzmann distribution law is invoked, which states that the probability F of a particle having an energy AH, or greater is given by ... 

In the three-dimensional network of long-chain molecules, the chain ends of each single coil are separately fixed at (0,0,0) and x,y,z). All the possible conformations Q of this coil with fixed end locations should be proportional to the probability W(x,y,z) associated with this end-to-end distance. According to the Boltzmann s law, S = khvQ, where k is the Boltzmann constant, as well as to the relationship in (3.3), we obtain... 

---