Statistical mechanics

Phase space, $$\nu$$-space and $$\gamma$$-space

The space occupies by individual molecule can be represented by cartetian co-ordinate, to specify the position of particle, we have to mention x, y and z axis. If the particle is in motion then x, y and z axis only cannot justify the properties and its momenta are required. Let P_x, P_y, and P_zbe the momenta of the particle along x,y and z axis respectively then to give complete information of paticle six dimensions are required. The state of a paticle at one instant can be represented by a point in two dimensional space which is called phase space. The co-ordinate in phase space along x-axis is (x, p_x).

For a molecule in space we require six dimenstion which are x, y, z, P_x, P_y, P_z and the space passes by a molecule in a system with three degree of freedom is called $$\nu$$-space (molecular space).

The collective form of position space and momentum space for a position sapce and momentum space for a particle under consideration is called phase space. There are three co-ordinate in position space and three co-ordinate in momentum space (P_x, P_y, P_z).

$$\gamma$$-space

In a consideration of n number of molecule of a system. There are 6N degree of freedom. This space is called $$\gamma$$-space. Here the term $$\gamma$$ is for gas.

$$\gamma$$-space is also called Euclidian space.

The concept of ensemble

Assembly:

Assembly is a group of particular number of particle having identical properties in a space.

When various independent assembly are considered in a large system then it is called ensemble. These ensemble are made macroscopically as identical means, each assembly is characterised by the same value of macroscopic parameter.

Assembly are independent means for the calculation of possible energy state of an ensemble. The interaction among the assembly are not considered.

Types of ensemble

1. Micro-canonical ensemble (V, E, N)

It is a collection of large number of independent assemblies, each of which posses same volume, same energy and same number of molecular particle.

The assemblies are separated by rigid and well insulated wall. So that there would be no any interaction among the assembly and these become independent.

2.Canonical ensemble (N, T, V)

It is a collection of large number of independent assemblies each of which posses same number of molecule, same temperature and same volume.

Each assembly in canonical assemble is separated by rigid wall however the wall may be conducting. Hence, each assembly in canonical ensemble can be separated by rigid impermeable but conducting wall.

3. Grand canonical ensemble ($$\nu$$, T, V)

It is a collection of large number of independent assemblies which possess same $$\nu$$, T, V

Thermodynamic probability (W)

W = $$\frac{N\factorial

Total number of molecule = N

Number of compartment = n

Number of molecule in 1^stcompartment = n₁

Number of molecule in 2^ndcompartment = n₂

Number of molecule in 3^rdcompartment = n₃

Number of molecule in n^thcompartment = n_n

Macrostate

Each compartment wise distribution of a system of particle is called macrostate. Total number of macrostate for n particle is 1.

Microstate

Each distinct arrangement of particle within macrostate is called microstate. For n mumber of particle system, total number of microstate = 2ⁿ

Thermodynamic probability (W)

A macrostate corresponds to many microstate at any time a particular system is equally likely to be in any microstate.

The number of microstates corresponding to a particular macrostate is called thermodynamic probability. It is denoted by W.

It may also be defined as the number of quantum states lying between the energy states of E and E+dE, for a given volume(V) and number of molecule(N).

Thermodynamic probability is different from the ordinary concept of probabilty.The thermodynamic probability is 1 or greater than 1. However the ordinary probability for a particle is always less than one.

Let n₁, n₂, n₃and n_inumber of particles are distributed in the energy level of E₁, E₂, E₃....E_i

Then, N = En_i

E = n_iE_i

Mathematically, thermodynamic probability for n number of particle is given by

W =

Thermodynamic probability and entropy

According to Boltzman, the thermodynamic probability and entropy are related as,

s = k$$\ln$$w

where, s = entropy

k = Boltzmann constant. It is universal gas constant per molecule

i.e, k = $$\frac{R}{N_A}$$ N_Ais Avogadro,s number

= 1.38×10^-23

W= thermodynamic probability

Let us consider two system, each having entropy s₁and s₂. Since, entropy is extensive property of a system.

Total entropy of a system (s) = s₁+ s₂........(i)

Thermodynamic probability is multiplicative properties. The thermodynamic probability of whole system is the product of thermodynamic probability of each system.

Let w₁and w₂be the thermodynamic probability of individual system and w be the thermodynamic probabilty of whole system then w = w₁.w₂......(2)

Entropy is the measure of randomness or disorder of a system. Greater the randomness or disorder of a system, greater the entropy. So, the entropy is the function of probalbilty or configuration.

S = F(w)......(3)a

S₁= F(w₁)......(3)b

S₂= F(w₂)......(3)c

Putting the value of s, s₁and s₂in equation (i)

F(w) = F(w₁) + F(w₂)

F(w₁.w₂) = F(w₁) + F(w₂).....(4) [$$\because$$ w =w₁.w₂]

Equation (4) is similar to $$\ln$$xy = $$\ln$$x + $$\ln$$y

In general, F(w) = k$$\ln$$w

where, k is a constant and is called Boltzmann constant.

or, s = k$$\ln$$w.......(5)

References