Boltzmann Distribution

Boltzmann Distribution:

For an ideal gas, there are two key features of the Maxwell distribution:

  • There is an average energy (1/2)kT in each degree of freedom.
  • The probability of a molecule having energy E is proportional to e-E/kT.

The degree of freedom means a way in which a molecule is free to move and has energy- in this case the x, y, and z-directions- to give total average kinetic energy as 3/2kT.

Boltzmann generalized these features of Maxwell’s distribution to arbitrary large systems. He was the first to realize a deep connection between the thermodynamic concept of entropy and statistical analysis of possible states of a large system- that the increase in entropy of a system with time causes a change in macroscopic variables to those values which correspond to the largest possible number of microscopic arrangements. He showed that for a given energy, the number of available microscopic states is far greater for macroscopic values corresponding to thermal equilibrium. For example, for a given energy, there are far more microscopic arrangements of gas molecules possible (in which the gas is uniformly distributed in a box, than the corresponding microscopic value) of all the gas molecules being in the left-hand half of the box. Thus, if a litre of gas goes through all possible microscopic arrangements, over the course of time, there is in fact, a negligible probability of it being all in the left-hand half in time equal to the age of the universe. So, if we arrange for all the particles to be in the left-hand half by using a piston to push them there and then remove the piston, they would uniformly distribute throughout the box.

Boltzmann proved that the thermodynamic entropy S of a system (at a given energy E) is related to the number W of microscopic states available to it by:

S = k log W

Where k is Boltzmann’s constant. (There were some ambiguities in counting the number of possible microscopic arrangements which were rather troublesome. For example, how many different velocities can a particle in a container have?). Thus, for any system, large or small, in thermal equilibrium at temperature T, the probability of being in a particular state at energy E is proportional to e-E/kT. This is called Boltzmann’s distribution.

Nature of Radiations from Radioactive Substances
Alpha and Beta Particle Change
Group Displacement Law
Radioactive Disintegration Series
Rate of Radioactive Disintegration or Decay (Disintegration Rate)
Half-Life Period and Average-Life Period
Nuclear Reactions
Difference Between Nuclear and Chemical Reaction
Artificial Transmutation of Elements
Artificial or Induced Radioactivity
Nuclear Fission and Nuclear Fusion
Radio-Carbon Dating
Boltzmann distribution– Wikipedia

Comments (No)

Leave a Reply