BROWNIAN MOTION:

A random movement of macroscopic particles suspended in liquid or

gases resulting from the impact of molecules of the surrounding medium
is called Brownian motion.

Mean free path:

The mean free path is the average of all path lengths between
collisions.

In physics, mean free path is the average distance over which a moving
particle travels before substantially changing its direction or energy,
typically as a result of one or more successive collisions with other
particles

Zeroth law of thermodynamics:

When a body, ‘A’, is in thermal equilibrium with another body, ‘b’, and
also separately in thermal equilibrium with a body ‘, C’, then body, ‘B’ and
‘C’, will also be in thermal equilibrium with each other. This statement
defines the zeroth law of thermodynamics. The law is based on
temperature measurement.

First law of thermodynamics:

1
A thermodynamic system in an equilibrium state possesses a state
variable known as internal energy(E). Between the two systems, the
change in the internal energy is equal to the difference of the heat
transfer into the system and the work done by the system.

The first law of thermodynamics states that the energy of the universe
remains the same. Though it may be exchanged between the system and
the surroundings, it can’t be created or destroyed. The law basically
relates to the changes in energy states due to work and heat transfer. It
redefines the conservation of energy concept.

ΔU = Q + W
 ΔU = Change in internal energy of the system
 q = Algebraic sum of heat transfer between system and
surroundings
 W = Work interaction of the system with its surroundings

Second law thermodynamics:

The second law of thermodynamics states that. any spontaneously
occurring process will always lead to an escalation in the entropy (S) of
the universe. In simple words, the law explains that an isolated system's
entropy will never decrease over time

Real life Example of second law of thermodynamics is that: When

we put an ice cube in a cup with water at room temperature. The
water releases off heat and the ice cube melts. Hence, the
entropy of water decreases

2
Third law of thermodynamics:
The third law of thermodynamics states that the entropy of a system
approaches a constant value as the temperature approaches absolute zero.
The entropy of a system at absolute zero is typically zero, and in all cases is
determined only by the number of different ground states it has.

The value of entropy of a completely pure crystalline substance is zero at

absolute zero temperature”

Internal energy:
Internal energy U of a system or a body with well defined
boundaries is the total of the kinetic energy due to the
motion of molecules and the potential energy ...
ΔU=q+w
Quasistatic Process:
A quasistatic process is an ideal thermodynamic
process that occurs slowly to maintain thermodynamic equilibrium
throughout the process. It is a process that happens at an infinitesimally
slow rate. Slow heat transfer between two bodies at two finitely different
temperatures is an example of a quasi-static process

Heat engine:
A heat engine is a device that converts heat to work. It

takes heat from a reservoir, then does some work like moving a piston,
lifting weight etc. and finally discharges some heat energy into the sink

3
Maxwell demons:
Maxwell's demon is a thought experiment that would
hypothetically violate the second law of thermodynamics..it is
hypothetical, intelligent being (or a functionally equivalent device)
capable of detecting and reacting to the motions of individual
molecules.

Joule Thomson process:

In thermodynamics, the Joule–Thomson effect
describes the temperature change of a real gas or liquid
when it is forced through a valve or porous plug while
keeping it insulated so that no heat is exchanged with the
environment. This procedure is called a throttling process or
Joule–Thomson process.

Joule Thomson process Result:

 The gases involved must have very low saturation
temperatures and pressures.

 In order to bring the gas into a state characterized by a

positive Joule–Thomson effect, a foregoing preparation is
necessary: the gas has to be precooled in auxiliary
classical refrigeration systems.

 To obtain very low temperatures, very high pressure

drops are necessary.

4
  At sufficiently low temperatures, all gases show a cooling
effect.

 At ordinary temperatures, all gases except hydrogen and

helium show cooling effect. Hydrogen and helium show
heating instead of cooling at room temperature.

 The fall in temperature is directly proportional to the

difference in pressure on the two sides of the porous
plug.

 The fall in temperature for a given difference of pressure

decreases with rise in the initial temperature of the gas. It
was found that the cooling effect decreased with the
increase of initial temperature and became zero at a
certain temperature and at a temperature higher than
this temperature. instead of cooling, heating was
observed. This particular temperature at which the Joule-
Thomson effect changes sign is called the temperature of
inversion.

Macroscopic properties of particles:

 Macroscopic properties of matter are the properties

in bulk matter

 Visible to naked eye

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 In a scale that is visible to naked eye, which includes
centi-, kilo-. mega-, etc.

 Volume, pressure, temperature, density. etc.

Microscopic properties of particles:

 Microscopic properties are the properties of the

constituents of bulk matter

 Invisible to naked eye

 In a scale that is invisible to naked eye, which includes

milli-, micro-, nano-, pico-, etc.

 Intermolecular forces, chemical bonding. atomicity, e

Fermi-Dirac Distribution
The total number of different and distinguishable ways is
k
gi !
W=∏ n ! g ! (i)
i i ( i−n ) i

Taking natural logarithm on both sides of equation (i),

k

ln W =∑ [ln g i ¿−ln ni−ln ( gi−n i) ! ]¿

i=1

As ni & giare very large numbers, we can use Sterling approximation

ln n! = (n ln n) –n
Applying Sterling’s approximation, we have

6
 k

ln W= ∑ ¿¿
i=1

k
=∑ ¿¿ -ni ln ni -( gi−ni ) ln ¿ ¿)]
i=1

Here gi is not subject to variation whereas ni varies continuously. Differentiating both

sides, we have
k
1
δ (ln W)=∑ ¿¿ -ln ni δ n i+( gi -ni ) δ ni
( gi−ni ) + ln( i- i
g n ¿ δ ni ¿
i=1
k

=∑ [ln ( gi−n i )−ln n i¿ ]δ ni ¿

i=1

To get the state of maximum thermodynamic probability

δ ¿=0
k

∑ [ln ( gi−n i )−ln n i¿ ]δ ni ¿ =0

i=1

k
Or ∑ ¿¿]δ ni=0
i=1
k
Or ∑ ¿¿ =0
i=1

In addition, our system must satisfy the two auxiliary conditions:

i) Conservation of total number of particles, i.e., n=a constant
 n=∑
i
ni=constant

i.e., δE=∑
i
δ ni =0

ii) Conservation of total energy of the system i.e., E= a constant.

E=∑
i
ni E i constant

δ E= ∑ E i δ ni= 0
i

For this, let us multiply eqn. by α and eqn by β and add the resulting expression to eqn
∑ ¿¿ )+α + β E i ¿ δ ni ¿ δ ni=0
i=1

AS the variations δ ni are independent of each other, we get

7
 i n
ln( g −n ¿ +α + β Ei =0
i i

ni
or g i−ni
=e−(α+β E ) i

g i−ni (α + β E )
or ni
=e i

g i (α + β E )
or ni
=e +1 i

g i
or ni = α+β Ei
e +1
This equation represents the most probable distribution of the particles among
various energy levels for a system obeying Fermi- Dirac statistics and is therefore
known as Fermi-Dirac distribution law, for an assembly of fermions.

See beck effect:

The See beck effect is a phenomenon in which a temperature difference between

two dissimilar electrical conductors or semiconductors produces a voltage difference
between the two substances.

When heat is applied to one of the two conductors or semiconductors, heated electrons
flow toward the cooler conductor or semiconductor. If the pair is connected through an
electrical circuit, direct current flows through that circuit.

Peltier effect:
The Peltier effect is the reverse phenomenon of the See beck effect; the
electrical current flowing through the junction connecting two materials will emit or
absorb heat per unit time at the junction to balance the difference in the chemical
potential of the two materi

8
9
MAXWELL-BOLTZMANN
DISTRIBUTION FUNCTION
FOR IDEAL GAS
Energy distribution law in General Form

For Ni represents particles , Gi possible quantum states and

total number of ways of arranging all the particles in the given
distribution is W.
The most probable distribution of the particles among the
energy states in equilibrium is that for which the probability of
occurrence is maximum, so, W is Maximum.

k dlog W
∑ n dni = 0 & d(logW)=0 i=1 d

Deduce Maxwell-Boltz man Distribution Function for An

Ideal Gas
We have,
𝑔𝑖𝑛𝑖 log
𝑊 = log 𝑁! + log[ 𝜋 ( )]
𝑛𝑖 !

Since, the number of particles is very large

log 𝑁! = 𝑁 log 𝑁 − 𝑁

log 𝑛𝑖! = 𝑛𝑖 log 𝑛𝑖 − 𝑛𝑖

Now differentiating equation, Thus,

𝑑 log 𝑊 𝑛𝑖

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 +
𝑑𝑛𝑖 = log 𝑔𝑖 − ( log 𝑛𝑖 𝑛𝑖 )
+1

=log 𝑔𝑖 − log 𝑛𝑖
𝑖
𝑛
= - log( )
𝑔𝑖

Substituting this equation we have,

𝑖
𝑛
-log (𝑔𝑖) − 𝛼 − 𝛽𝐸𝑖 = 0

𝑖
𝑛
or, log(𝑔 𝑖) = −𝛼 − 𝛽𝐸𝑖

or, 𝑛𝑖 = 𝑔𝑖𝑒(−𝛼−𝛽𝐸𝑖) or, 𝑛𝑖 = 𝑔𝑖𝑒−𝛼𝑒−𝛽𝐸𝑖 (where i

= 1,2,3------k)

This equation is known as the Maxwell-Boltzmann energy distribution law in general form.
1
Now, 𝛽 =
𝐾𝑇

Substitution,
−𝐸𝑖

𝑛𝑖 = 𝑔𝑖𝑒−𝛼𝑒 𝐾𝑇
𝐸𝑖
−
Where, K is Boltzmann constant and T it’s absolute temperature. The quantity 𝑒 𝐾𝑇 is known as the Boltzmann
factor.
We write, N the total number of particles as,
𝑁 = ∑𝑖 𝑛𝑖
𝐸
− 𝑖
= ∑𝑖 𝑔𝑖 𝑒 −𝛼 𝑒 𝐾𝑇

𝐾
−𝛼 − 𝑖
=𝑒 𝑔𝑖 𝑒 𝐾𝑇

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 MEAN SPEED INTEGRATION

Ni number of molecules
Vi velocity of each molecule
Then, mean speed will be,

∑i=1NiVi v¯ =

∑i=1NiVi

If ∑Ni = N, then
i

Ni V i
Now from MAXWELL-BOLTZMANN distribution law of SPEED ,
its found as,

m
N(V) = 4πN(2π kT)32v2e− m2kv2t dv ———-(1)

As the speed of molecules is approximately continuous ,

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 dv ————(2)
Substituting equation (1). At equation (2)
1 ∫∞ m 32 2 − m2vkt
= 4π N( ) v e
N 0 2π k T
dv
2

m 32 ∫ ∞ 2 − m2vkt
= 4π ( ) v . ve
2π k T 0
2
dv ————-(3)

If we take , m2kvt2 = x

Or, 2mkt x = v2 ————4

Now from, derivate the equation (4),

Or, kmtdx = vdv ———(5)

Now from equation (3) &(5),

= 4π(2πmkT)32 ∫0∞ 2(mkt2)2 xe−xdx

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 xe−xdx

8k t
mπ
=

This equation gives the mean or average speed of a gas

molecule.

Thomson effect:

The evolution or absorption of heat when electric current

passes through a circuit composed of a single material that has
a temperature difference along its length. This transfer of heat
is superimposed on the common production of heat associated
with the electrical resistance to currents in conductors is known
as Thomson effect.

Phase transition:
A phase transition (or phase change) is the physical process of
transition between one state of a medium and another. Commonly the
term is used to refer to changes among the basic states of matter: solid,
liquid, and gas, and in rare cases, plasma.

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 First Order Phase Transition:
First-order phase transitions are those
that involve a latent heat During such a transition, a system either absorbs or
releases a fixed (and typically large) amount of energy per volume. During this
process, the temperature of the system will stay constant as heat is added: the
system is in a "mixed-phase regime" in which some parts of the system have
completed the transition and others have not

Second order phase transition:

Second order phase transition is
defined as the phenomenon that takes place with no change in entropy
and volume at constant temperature and pressure.

Ultraviolet Catastrophe:
The ultraviolet (UV) catastrophe, also called the Rayleigh–Jeans catastrophe, is the
prediction of classical electromagnetism that the intensity of the radiation emitted
by an ideal black body at thermal equilibrium goes to infinity as
wavelength decreases.

15
 Figure : The ultraviolet catastrophe is the error at short wavelengths in the Rayleigh–Jeans law for
the energy emitted by an ideal black body. The error, much more pronounced for short
wavelengths, is the difference between the Rayleigh–Jeans law —black—and Planck’s law—blue.

A black body is an idealized object that absorbs and emits all frequencies.
Classical physics can be used to derive an approximated equation describing
the intensity of a black body radiation as a function of frequency for a fixed
temperature. The result is known as the Rayleigh-Jeans law, which for
wavelength λ, is:

2 c k BT
Bλ(T)= λ
4

where Bλ is the intensity of the radiation —expressed as the power emitted per unit
emitting area, per steradian, per unit wavelength (spectral radiance)
c is the speed of light,
kB is the Boltzmann constant,
and T is the temperature in kelvins.

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