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Created by Byun, Jaeyoon on Dec 2024
Created by Byun, Jaeyoon on Dec 2024
==Foundation of Brownian motion==
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). It was named for the Scottish botanist Robert Brown, first person to study this motion. It was discovered during investigation of fertilization process in Clarkia pulchella, a flowering plant, as he noticed a "rapid oscillatory motion" of the microscopic particles within the pollen grains suspended in water under the microscope. Initially, he believed it was unusual activity by pollen, but it showed same movement even pollen was dead. Many scientists, such as Albert Einstein, analyzed this strange phenomenon quantitatively.
==Theory of Brownian Movement==
===Osmotic Pressure in Suspended Particles===
Consider '''z''' gram of molecules of non-electrolyte dissolve in a volume V* that is part of total volume '''V'''. If volume '''V*''' is separated from pure solvent by a partition permeable for solvent but not for the solute, "osmotic pressure", '''p''', is exerted on this partition, satisfying the following equation (when '''V*/z''' is great enough):
<math>pV^* = RTz</math>
On the other hand, if small suspended particles are present in fractional volume '''V*''', which particles can't pass to the solvent, no force (excluding gravity) will acting on this partition, according to classical Theory of Thermodynamics.
In molecular-kinetic theory of heat, however, provides different conception. According to this theory, a dissolved molecule is differentiated from a suspended body by its dimensions only. So the following assumption is made: suspended particles perform an irregular movement in the liquid on account of the molecular movement of the liquid. If particles are suspended from volume '''V*''' by the partition, there will be osmotic pressure '''p''' where:
<math>p = RTn/NV^* = RTv/N </math>
in which '''v''' = number of suspended particles present in volume '''V*''' and '''N''' stands for actual number of molecules.
====Osmotic Pressure in Molecular-Kinetic Theory of Heat====
If '''p_1''', '''p_2''',... '''p_i''' are the variables of state of system and if the complete system of equations of change of these variables of state is given as <math>\frac{\partial p_v}{\partial t}= &Phi_v(p_1,...,p_i)(v=1,2,..,l)</math> or ______, then the entropy of the system is given as '''S''' = _____ where T is absolute temperature, E is energy of system, and E is the energy as a function of p. Integration over all possible values, free energy, '''F''', is obtained:
F =
Consider, now, a quantity of liquid enclosed in a volume V with n solute molecules in V* by semi-permeable partition. In order to show that F is depended on the magnitude of the volume V* where all suspended particles are contained, variable B will be incorporated. Let x1, y1, z1 be the rectangular coordinates system of first particle, x2, y2, z2 for second particle, and xn, yn, zn for nth/last particle. Allocating in domains of parallelopiped form dx1, dy1, dz1, dx2, dy2, dz2, ...., and dxn, dyn, dzn within V*, the function can be written as:
dB = dx1dy1...dzn * J,
in which J is independent from x1,y1,z1,...,zn.
If there is a second system of same domain with dx1'dy1'dz1', dx2'dy2'dz2',..., dxn'dyn'dzn', the function can be written as:
dB' = dx1'dy1'...dzn'*J
If dx1dy1...dzn = dx1'dy1'...dzn', then equation becomes:
dB/dB' = J/J'
Based on this theory, dB/B or dB/B' is equal to probability that centers of gravity of particles at specific moment in domains. So, if movement of single particle is independent of one another and is homogenous with no force exerted to particle, probability will be equal, thus
dB/B = dB'/B'
which ultimately establishes the equation:
J = J'
This proves that J is independent of both V* and centres of gravity.
Integrating dB gives:
B = ______
Thus, F =
and Osmotic pressure, p =
This proves that existence of osmotic pressure by utilizing from molecular-kinetic theory of Heat.
==Example==
==References==
https://www.mit.edu/~kardar/research/seminars/motility/eins_brownian.pdf
https://www.britannica.com/science/Brownian-motion
https://en.wikipedia.org/wiki/Brownian_motion

Revision as of 18:48, 6 December 2024

Created by Byun, Jaeyoon on Dec 2024

Foundation of Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). It was named for the Scottish botanist Robert Brown, first person to study this motion. It was discovered during investigation of fertilization process in Clarkia pulchella, a flowering plant, as he noticed a "rapid oscillatory motion" of the microscopic particles within the pollen grains suspended in water under the microscope. Initially, he believed it was unusual activity by pollen, but it showed same movement even pollen was dead. Many scientists, such as Albert Einstein, analyzed this strange phenomenon quantitatively.

Theory of Brownian Movement

Osmotic Pressure in Suspended Particles

Consider z gram of molecules of non-electrolyte dissolve in a volume V* that is part of total volume V. If volume V* is separated from pure solvent by a partition permeable for solvent but not for the solute, "osmotic pressure", p, is exerted on this partition, satisfying the following equation (when V*/z is great enough):

[math]\displaystyle{ pV^* = RTz }[/math]

On the other hand, if small suspended particles are present in fractional volume V*, which particles can't pass to the solvent, no force (excluding gravity) will acting on this partition, according to classical Theory of Thermodynamics.

In molecular-kinetic theory of heat, however, provides different conception. According to this theory, a dissolved molecule is differentiated from a suspended body by its dimensions only. So the following assumption is made: suspended particles perform an irregular movement in the liquid on account of the molecular movement of the liquid. If particles are suspended from volume V* by the partition, there will be osmotic pressure p where:

[math]\displaystyle{ p = RTn/NV^* = RTv/N }[/math]

in which v = number of suspended particles present in volume V* and N stands for actual number of molecules.

Osmotic Pressure in Molecular-Kinetic Theory of Heat

If p_1, p_2,... p_i are the variables of state of system and if the complete system of equations of change of these variables of state is given as [math]\displaystyle{ \frac{\partial p_v}{\partial t}= &Phi_v(p_1,...,p_i)(v=1,2,..,l) }[/math] or ______, then the entropy of the system is given as S = _____ where T is absolute temperature, E is energy of system, and E is the energy as a function of p. Integration over all possible values, free energy, F, is obtained: F =

Consider, now, a quantity of liquid enclosed in a volume V with n solute molecules in V* by semi-permeable partition. In order to show that F is depended on the magnitude of the volume V* where all suspended particles are contained, variable B will be incorporated. Let x1, y1, z1 be the rectangular coordinates system of first particle, x2, y2, z2 for second particle, and xn, yn, zn for nth/last particle. Allocating in domains of parallelopiped form dx1, dy1, dz1, dx2, dy2, dz2, ...., and dxn, dyn, dzn within V*, the function can be written as: dB = dx1dy1...dzn * J, in which J is independent from x1,y1,z1,...,zn. If there is a second system of same domain with dx1'dy1'dz1', dx2'dy2'dz2',..., dxn'dyn'dzn', the function can be written as: dB' = dx1'dy1'...dzn'*J If dx1dy1...dzn = dx1'dy1'...dzn', then equation becomes: dB/dB' = J/J'

Based on this theory, dB/B or dB/B' is equal to probability that centers of gravity of particles at specific moment in domains. So, if movement of single particle is independent of one another and is homogenous with no force exerted to particle, probability will be equal, thus dB/B = dB'/B' which ultimately establishes the equation: J = J' This proves that J is independent of both V* and centres of gravity.

Integrating dB gives: B = ______ Thus, F = and Osmotic pressure, p =

This proves that existence of osmotic pressure by utilizing from molecular-kinetic theory of Heat.

Example

References

https://www.mit.edu/~kardar/research/seminars/motility/eins_brownian.pdf https://www.britannica.com/science/Brownian-motion https://en.wikipedia.org/wiki/Brownian_motion