2. Brownian motion
Describes the random movement of
microscopic particles suspended in a
liquid or gas
Robert Brown saw particle of pollen
dance around in fluid
under microscope
Atoms/molecules move randomly as
they collide with each other(have
kinetic energy)
Randomly moving
atoms/particles collide with larger
atom
3. Einstein theory of Brownian motion
• In 1905 proposed a quantitative explanation for Brownian motion
• Conducted a theoretical analysis of Brownian motion
• Found that Brownian motion properties and diffusion constant
are related.
• Einstein determined the diffusion coefficient in two ways
• From the irregular random motion of the suspended particles
• From the difference in osmotic pressure
4. Main result of Einstein's paper on Brownian motion
• The mean square displacement ⟨𝑥⟩^2 suffered by a spherical
Brownian Particle, of radius a, in time ‘t’ is given by
Where E is the viscosity of the fluid, R is the gas constant and Nav is the
Avogadro number.
5. • Einstein found a special case of fluctuation- dissipation theorem in
this paper as
Where γ - coefficient viscous dray force, D - diffusion constant, T -
temperature.
6. • Einstein also found a diffusion equation
• He established a link between random walk of single particle the
diffusion of many particles. For the initial condition P(x,0) =δ (x), the
solution of the diffusion equation is
7. Langevin's Theory of Brownian Motion
• In 1908, Langevin presented a simpler approach of Brownian motion
• This was based on the following assumptions
i. Brownian motion is entirely irregular and there is no effect of gravity on it
ii. Each Brownian particle In liquid sufferers a large no of collisions by
the molecules of the liquid (approx. 1022 collisions per second)
iii. Each collision is expected to produce a deflection in the path of the particle
8. • According to Langevin, the force experienced by a Brownian particle
in liquid is of 2 types
1. Viscous Force (Fv) : Assuming a Spherical Brownian particle, the viscous force
experienced by the particle in the given fluid is given by the equation
Where, is the coefficient of viscosity of the liquid
r is the radius of Brownian particle
V is the velocity of Brownian particle
2. Fluctuating Force (Fl) : Force due to all external influences of surrounding
medium
Net Force (F) = Viscous Force (Fv) + Fluctuating Force (Fl)
9. • Taking the average values in equation
• According to law of equipartition of energy
10. • Hence we get
• Or, if ,
• With further simplification, U can be written as
• As m is very small, is very large and hence where
• Therefore
