BROWNIAN MOTION
The erratic random movement of microscopic particles in a fluid, as a result of continuous bombardment from molecules of the surrounding medium.
Brief History:
It was first observed in 1827 by a botanist Brownian.
The term “classical Brownian motion” describes the random movement of
microscopic particles suspended in a liquid or gas. Brown was investigating the
fertilization process in Clarkia
pulchella, then a newly discovered species of flowering plant, when he noticed
a “rapid oscillatory motion” of the microscopic particles within the pollen grains
suspended in water under the microscope. In
1827 the biologist Robert Brown noticed that if you looked at pollen grains in
water through a microscope, the pollen jiggles about. He called this jiggling
'Brownian motion', but Brown couldn't work out what was causing it. The first
of the three papers that Einstein published in 1905 finally came up with an
explanation.
Everything around us is made up of atoms and
molecules: the chair you're sitting on, the food you eat, the air you're
breathing. The idea of atoms has been around since the time of the ancient
Greeks, and a century before Einstein, the great chemist John Dalton had
suggested that all chemicals were made of tiny invisible molecules, which in
turn were made of even tinier atoms. The problem was that there was no proof of
their existence, until Einstein looked into the problem of Brownian motion.
Einstein explanation of
Brownian motion:
Einstein realized that the jiggling of the pollen
grains seen in Brownian motion was due to molecules of water hitting the tiny
pollen grains, like players kicking the ball in a game of football. The pollen
grains were visible but the water molecules weren't, so it looked like the
grains were bouncing around on their own.
Einstein also showed that it was possible to work
out how many molecules were hitting a single pollen grain and how fast the
water molecules were moving - all by looking at the pollen grains.
Importantly, Einstein's paper also made predictions
about the properties of atoms that could be tested. The French physicist Jean
Perrin used Einstein's predictions to work out the size of atoms and remove any
remaining doubts about the existence of atoms.
Explanation:
Brownian motion, also called Brownian movement,
any of various physical phenomena in which some quantity is constantly
undergoing small, random fluctuations. It was named for the Scottish botanist Robert Brown, the first
to study such fluctuations (1827).
If a number of particles subject to Brownian motion are
present in a given medium and there is no preferred direction for the random
oscillations, then over a period of time the particles will tend to be spread
evenly throughout the medium. Thus, if A and B are two adjacent regions and, at time t, A contains twice as many particles as B, at that instant
the probability of a particle’s leaving A to
enter B is twice as great as the probability
that a particle will leave B to
enter A.
The physical process in which a substance tends to spread steadily from regions
of high concentration to regions of lower concentration is called diffusion. Diffusion
can therefore be considered a macroscopic manifestation of Brownian motion on
the microscopic level. Thus, it is possible to study diffusion by simulating
the motion of a Brownian particle and computing its average behavior.
The first point we must understand here is that the
problem of Brownian motion, that is of randomly moving particles spurred by an
infinite thermal energy reservoir, is nothing different to the problem of a
random walker who takes either a step right or to the left with two different
probabilities. The question as to what the velocity trajectory of a Brownian
particle is at
position r at time t now maps over to a question of how many steps t, each of
length ∆,would a random walker( a drunk person) need to reach a point r to the
right( or left) of his starting position. Now if the system is sufficiently
randomized, that is the random walker can step to the right or left at any time
with an equal probability ½, it is not too difficult to see that his mean
displacement will be zero.
Zeamenskey.
(2012).Heat and Thermodynamics.Tata McGraw-Hill Education
The phenomenon of Brownian motion can be easily observed
in a laboratory if a colloidal solution is examined under an ultra-microscope.
As the direction of illumination is perpendicular to the axis of the
microscope. The suspended particles in the solution look like bright
illuminated spots. These illuminated particles continually move to and fro in a
random haphazard way. The particles spin, rise sink and rise again. The
movements of the particles is continuous and spontaneous. This nonstop random
and haphazard motion of the particles is called Brownian motion. The motion of
the particles become more conspicuous in a liquid of lower viscosity. The
phenomenon of Brownian movement gives a clear picture of the gaseous state of
matter. The motion of the molecules of a gas is similar in nature to the
Brownian motion of suspended particles in a colloidal solution.
Essential Feature:
1.
The motion of each particle is completely irregular and random. No two
particles are found to execute the same motion.
2.
The motion is continuous and takes place for ever.
3.
The smaller particles appear to be more agitated than the larger ones.
4.
The motion is independent of the nature of the suspended particles.
5.
The motion become more violent on increasing temperature.
6.
The motion is not modified due to the shaking of colloidal motion.
7.
The Brownian motion cannot be observed with particles of large size.
8.
The motion is more conspicuous in a liquid of lower viscosity.
9.
The laws of kinetic theory of gases are applicable to Brownian motion
too.
Lal.B, Subrahmanyam.N. (2008).Heat
Thermodynamics And Statistical Physics. S.
Chand, 2008
Langevin’s Theory of Brownian Motion:
According to Langevin the force acted on
suspended particle is of two types
1.
Frictional Force proportional to the velocity
f(
)=6
)=6
2.
Force due to all external influence of surrounding fluid.
According to this theory viscosity of liquid decreases as
temperature increases. Thus the temperature effect is negligible comparable to
the effect of viscosity. Diffusion, fluctuations in concentration and Brownian
motion represents a single phenomenon. Diffusion is a macroscopic phenomenon
while Brownian motion is a microscopic phenomenon.
Einstein’s theory of
Brownian motion:
According to Einstein’s theory of transitional Brownian motion
the particles lend to diffuse into the medium in course of time. Consequently
the diffusion coefficient must be related to the Brownian movement.
Conclusion:
Historically and conceptually, Brownian motion lies
intermediate between thermodynamics and statistical mechanics. On the one hand
it is a good quantitative model for thermodynamic fluctuations, and on the
other hand, as in Einstein’s original treatment, it can be usefully described
by a probability distribution and transition probability, which are the stuff
of statistical mechanics.
Differential Equation:
The term differential equation was coined by
Leibniz in 1676 for a relationship between the two differentials dx and dy for the two variables x and y. A differential equation is an equation which
contains a derivative of an unknown function. It tells something about a rate
of change, from which we hope to deduce facts about the function. Here is a differential
equation. A differential equation is basically a mathematical equation that
relates some function with its derivatives. In applications, the functions
usually represent physical quantities, the derivatives represent their rates of
change, and the equation defines a relationship between the two. A
differential equation contains one or more terms involving derivatives of one
variable (the dependent variable, y) with respect to another
variable (the independent variable, x).
For
example,
Unlike
algebraic equations, the solutions of differential equations are functions and
not just numbers. It represents the
relationship between a continuously varying quantity and its rate of change.
This is very essential in all scientific investigation. Partial differential equations (PDEs) are equations that
involve rates of change with respect to continuous variables. The position of a rigid body is specified by six
numbers, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature,
pressure, and so forth. The dynamics for the rigid body take place in a
finite-dimensional configuration space; the dynamics for the fluid occur in an
infinite-dimensional configuration space. This distinction usually makes PDEs
much harder to solve than ordinary
differential equations (ODEs), but
here again there will be simple solutions for linear problems. Classic domains
where PDEs are used include acoustics, fluid flow, electrodynamics, and heat transfer.
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