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Sunday, February 18, 2018

GIBBS CANONICAL ENSEMBLE

(SYSTEM IN CONTACT WITH A HEAT RESERVOIR)

Canonical ensemble describes those systems that are not isolated but are in contact with a heat reservoir.

The system under consideration together with a heat reservoir forms a closed system and then system of interest is taken as sub system of this closed system.

Any sub system of an isolated system in statistical equilibrium can be represented by a canonical ensemble.

The probability density of a canonical ensemble depends both on Energy 'E' and temperature 'T'

It is given by

ρ(E,T) = Ae-E/𝜏

Partition function for canonical ensemble

z=(1/h³ⁿ.n!)∫exp(-E(q,p)/KTdΓ

The statistical energy of a system in a canonical ensemble is given by


𝜎 = logZ  +(E/𝜏);  𝜎 = logZ  +T[∂(log Z)/∂T]

Thermodynamical entropy 'S' of a system in a canonical ensemble is given by

S=K𝜎

S=KlogZ + (E/T); U=KT2[∂(log Z)/∂T]

The entropy at absolute zero in a canonical ensemble can be expressed as

 S=K.log g

gₒ is statistical weight of ground state

Saturday, February 17, 2018

What is Chemical Potential?

For a chemical system, molar free energy is known as Chemical Potential.

A chemical substance that is free to move from one place to another place, will move spontaneously from a state of higher chemical potential to a state of lower chemical potential.

In the position of equilibrium, the chemical potential is constant through the entire system.

Let us consider a general heterogeneous system consisting of an independent components in several coexisting phases.

To start with, it is convenient to describe a given phase by its chemical composition, which is specified by the no. of mole 'Ni' of each species i, its volume V and its entropy 'S'.

If we consider internal energy (U)

U=U(S,V,N₁,N₂,.....Nᵢ,.....Nn)
μi=❴∂U/∂Nᵢ❵S,V,Nj ; j= except 'i'

'μi' is chemical potential of component 'i' in given phase.

dU=TdS-PdV+Σμᵢi.dNᵢ for i=1...n

We can also consider chemical potential 'μ' in terms of Helmoltz free energy 'F'.

F = F(T,V,N₁,N₂......Nn)

μ1=❴∂F/∂N1❵T,V,N₂,....

μ2=❴∂F/∂N2❵T,V,N₁,N₃,....

The chemical potentials are thus the rate of change of free energy per mole, at constant volume and temperature.

μ can also be expressed as

μi=❴∂G/∂Nᵢ❵T,P,Nj

A System in external field will be in equilibrium if the temperature and chemical potential of each component of the system is constant through out, i.e.

dT₁=0 and dμᵢ=0




Monday, December 11, 2017

Plot of Binding Energy per Nucleon against Mass Number - Important Conclusions

What is Binding Energy?

Binding Energy (BE) is the energy required to break a Nucleus into free neutrons and free protons.

According to Einstein's relative theory, mass of a system bound by energy 'B' is less than mass of its constituents by B/c².

BE/Nucleon(B/A) vs Mass Number (A) Plot



Important Conclusions

a) Approximately for most of Nuclei B/A ~ Constant.
b) B/A falls off at small values of A

Reason: For very light Nuclei a large fraction of their nucleons resides on the surface rather than inside. This reduces the B/A value as a surface nucleon is surrounded by fewer nucleons compared to a nucleon residing in interior and consequently is not so strongly bound.

c) B/A falls off at large values of A. This is clearly a Coulomb effect. Between every pair of Protons, there is a Coulomb repulsion which increases as Z². Notice that for naturally occurring nuclei, Z² increases faster than A and so Coulomb effect cannot adequately compensated by an increase in A.

d) B/A against A plot is peaked about A~50.
 Binding Energy can be increased by either breaking a heavy nucleus into parts or fusing light nuclei together.  It is easy to see that when binding energy is increased, energy in other forms can be released , since a decrease in 'M' corresponds to conversion of mass into energy.

e) The peak of the plot corresponds to iron. This explains large abundance of Fe(iron) in nature.

f) The plot indicates that binding becomes strong for a grouping of four particles. This unit is 𝛂 particle (2 neutrons + 2 protons).

The peaks in figure at mass numbers 4,8,12,16,20 & 24 are clear evidence of this effect. This effect is due to a pairing  force which exists  between a pair of neutrons and pair of protons.

g)  On closer inspection, it is found that B/A against A plot shows discontinuities  at neutron or proton number values 2,4,8,20,50,82 & 126. At these values of neutron or proton numbers, the BE is found to be unusually large. Large BE means high stability.

Monday, December 4, 2017

What is a Nuclear Reactor?



A Nuclear Reactor is a systematic arrangement to convert Nuclear Energy into thermal energy and then to Electrical energy .  Nuclear Reactor uses fissile material, heavy atomic nuclei, called as Nuclear Fuel. Fissile material leads to nuclear fission when the nuclei are hit by suitable energy Neutrons.
  
Example for Fissile Material is Uranium oxide.

Fission reaction of Uranium is as follows:



The energy evolved is distributed as kinetic energy of fission fragments  and heat.

This heat energy transmitted to a coolant which leads to generation of steam that could drive turbine system for conversion of thermal energy into electrical energy.

There are different types of Nuclear Reactors operating across the world.

a)  Boiling Water Reactors
b)  Pressurized Water Reactors
c)  Pressurized heavy water Reactors
d)  Fast Breeder Reactors

Sunday, September 17, 2017

How does Stern Gerlach Experiment proved spatial quantization and spin of electron

THE STERN-GERLACH EXPERIMENT



 Procedure implemented:


The silver atoms beam is produced by heating silver in a small electric oven. The beam is passed through an inhomogenous Magnetic Field.

Arrangement to produce inhomogenous Magnetic Field


We have one of pole pieces of the magnet flat with a cylindrical groove and the other in the form of a knife edge, parallel to groove.

The intensity of magnetic field increases as we go towards upper knife edge pole from center and it decreases as we go below towards lower pole.

A photographic plate is arranged to record the configuration of beam after its passage through the field.

The whole arrangement is placed in a vacuum. In absence of magnetic field, a trace of form of a narrow strip is obtained as shown in fig (a).

In presence of inhomogenous magnetic field the strip splits up into two components as shown in fig(b). 



The splitting of silver beam into two components in inhomogenous field verifies existence of electro spin and postulate of space quantization as shown below: 

Silver has an atomic number 47. According to Pauli's exclusion principle, all inner shells and sub shells are completely filled except outer most electron in 5S state.  Thus, it is a monovalent element.
The 5S electron is responsible for magnetic moment of atom.

When all silver atoms possessing a magnetic moment 'μᴊ' pass through inhomogenous magnetic field, they experience different amount of force in vertical direction depending on their orientation and alignment with magnetic field.

If magnetic moment  'μᴊ' can have all possible orientations then beam of Silver atoms consisting of Millions of atoms having all possible orientations will spread out into a broad continuous band on emerging from magnetic field. So a broad continuous patch should be observed on photographic plate.

Experimentally only two narrow strips are obtained on photographic plate. Therefore, predictions of classical physics are not correct in this case.

The two narrow strips show that 'μᴊ' cannot have all possible orientations, but only two possible orientations as shown in below figure



We know that 'μᴊ' is proportional to angular momentum 'J' and hence direction of 'J' relative to a well defined direction is given by

J = [√j(j+1)   ]* h/2Π

There are (2j+1) possible orientations of J. The Stern-Gerlach experiment shows that (2j+1)=2 or j=1/2.

Thus, J=(√3 /2)* (h/2Π)

It is known that angular momentum 'J' of Silver atoms entirely due to spin of its valence electrons. Thus, we conclude that the electron has a spin angular momentum

S = [√s(s+1)]*(h/2Π); where s=1/2.

Thus, Stern and Gerlach found that intial beam split into two distinct parts, corresponding to two opposite spin orientations in magnetic field that are permitted by space quantization.


Thursday, August 17, 2017

THE RAINBOW - EXPLANATION

Of all the optical phenomenon in every day life, the rainbow is loveliest.

Reflection of sun light by the rain drops is certainly an essential  element of an explanation but refraction-plays a role, too.

The following figure shows crucial path of light




The circle represents the cross section of a spherical rain drop. For the light ray, the sequence is Refraction, Reflection, and Refraction.

The angle (less than 90o) between the incident direction and the emergent direction is called as "Return Angle".

 A Ray from the sun strikes the spherical rain drop and some light is refracted into the water. Here we may ignore the portion that is reflected by the drops' surface. Next, the Ray proceeds to the far side of drop and is reflected there. Now we may ignore the portion that is refracted. Finally, the ray strikes the underside of drop and is refracted out into the air.

In precisely which direction does the emergent ray travel?

Two rules - i) Snell's Law for Refraction & ii) Equality of the angles of incidence and reflection

 suffice for answering that question. Once the initial point of contact between the rain drop and the ray from the sun has been specified. Also because the index of refraction depends on color, we must specify the color of light.

Lets start considering Red. Working out the complete path for many rays-i.e. for many different initial points of contact-reveals a surprising geometric property: The return angle for red light never exceeds 42.5 deg C, and most rays have a return angle 42 deg.



So return angle will decide the color of light reaching our vision of sight. Hence, different colors emerge from different sets of rain drops and produce a colored Rainbow.