Partition Funtion and Maxwell-Boltzmann distribution


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Definition: It is the science that deals with average properties of the molecules, atoms, or elementary particles in random motion in a system of many such particles and relates these properties to the thermodynamic and other macroscopic properties of the system. STATISTICAL MECHANICS

The partition function:: 

The partition function is given by the equation: where sum is taken over all the different quantum states of the system The partition function:

Isolated system: 

A physical system whose boundaries are not linked with any other system is called isolated system. It doesn‘t interact with its surroundings Isolated system

Equilibrium Configuration : 

Equilibrium Configuration For large populations of particles (N), the most probable configuration will be the dominant configuration. That configuration is the only one we need to consider when calculating average thermodynamic properties. Equilibrium Configuration


Isolated System where N,V, U = constant … …. Є 0 Є 1 Є 2 Є i We must simultaneously satisfy 3 conditions: CONT…


Boltzmann Distribution: is the number of particle in energy levels, Continued…


What is the probability of a particle being in state i? Now we have Probability


Results Partition Function for non-degenerate states Partition Function for non-degenerate states

From 1 st Law of Thermodynamics dU = dQ + dW Result to: Derivation of β

Temperature Dependence of the Partition Function: 

Consider T→0 Temperature Dependence of the Partition Function This is the number of molecular states in the lowest energy level.

Consider T→∞ q=g 0 +g 1 +g 2 +…g i This is the total number of molecular states. q, the molecular partition function tells us the average number of states that are thermally accessible to a molecule at the temperature of the system.

Graphical Representation:: 

Graphical Representation: Low Temperature Moderate Temperature High Temperature i P i ( ℰ i ) All molecular states become accessible as T goes toward ∞

Maxwell Boltzmann distribution : 

Maxwell Boltzmann distribution


Scottish physicist james clerk Maxwell developed his kinetic theory of gases in 1859. Maxwell determined the distribution of velocities among the molecules of a gas.Maxwell’s finding was later generalized in 1871 by a German physicist,ludwig Boltzmann to express the distribution of energy among the molecules. History:

Definition : 

Definition A law expressing the distribution of energy among the molecules of a gas in thermal equilibrium by statistical mechanics .

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Points to note. No particles have no energy. Most particles have average energy. There is no upper limit. Area under curve equals number of particles. Energy or speed no. of molecules

Maxwell’s four assumptions: : 

Maxwell’s four assumptions: X The diameter of molecules are much smaller than the distance between them


The collision between molecules conserve energy The molecules move between collisions without interacting as a constant speed in a straight line The positions and velocities of the molecules are initially at random. CONT…


We have assumed that gas molecules can only have the translational kinetic energy. Therefore, it applies only to Monatomic Gas. Maxwell distribution deals only with slower speed particles A calculation of the evolution of a system of gravitating masses (galaxies) in an expanding universe surprisingly suggests that Maxwell's velocity distribution is not an acceptable end point. LIMITATIONS OF MAXWELL DISTRIBUTION


In any system, the particles present will have a very wide range of energies. For gases , this can be shown on a graph called the Maxwell-Boltzmann Distribution which is a plot of the number of particles having each particular energy . LINK OF PARTITION FUNCTION AND MAXWELL BOLTZMAN DISTRIBUTION

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Among the particles of the system maxwell distribution law for particles is: For energy distribution: