On The Movement of Small Particles Suspended in Stationary Liquids Required By The Molecular-Kinetic Theory of Heat

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First entry by Kelly Miller, AP225 Fall 2011

Introduction

Einstein's paper on diffusion can be found here: [1]

In this paper, Einstein uses the kinetic-molecular theory of heat to describe how "bodies of microscopically visible size suspended in liquids" move randomly through the medium

By entertaining the possibility that the motion he described in his paper is the same as "Brownian molecular motion", Einstein makes a significant contribution to the debate between classical thermodynamics and the kinetic-molecular theory of heat (which later developed into the field of statistical mechanics).

"If it is really possible to observe the motion to be discussed here, along with the laws it is expected to obey, then classical thermodynamics can no longer be viewed as strictly valid even for microscopically distinguishable spaces, and an exact determination of the real size of atoms becomes possible. Conversely, if the prediction of this motion were to be provided wrong, this fact would provide a weighty argument against the molecular-kinetic conception of heat"(Einstein, A.)


Keywords

Diffusion, Brownian motion

Context for This Article

In 1905 (when this paper was written), there was a controversy between two schools of thought:

Classical Thermodynamics:

-a model that provided an exact description of the behavior of macroscopic materials by treating them as a continuum (as opposed to a set of discrete particles) -a deterministic model - could be used to determine how a material would behave with a change in pressure or temperature (for example)

Molecular Kinetic Theory of Heat:

- Boltzmann and Maxwell had developed a model that eventually developed into statistical mechanics -model that described matter as the movement and interrelations of a large number of discrete particles (on the order of Avogadro's number) -a statistical model - views everything (pressure, temperature, etc) as a statistical quantity (Andelman, Diamant)

Findings

Einstein begins his paper by pointing out the discrepancy between classical thermodynamics and the molecular-kinetic theory of heat in predicting the osmotic pressure produced by small suspended bodies -according to the classical theory of thermodynamics - a small suspended particle would not exert osmotic pressure because "the 'free energy' of the system does not seem to depend on the position of the wall and of the suspended bodies"(Einstein, A.)

-but - according to the molecular-kinetic theory of heat - the only difference between a suspended particle and a dissolved particle is SIZE and therefore, a suspended body should produce the same osmotic pressure as an equal number of dissolved molecules

Einstein goes on to show that a suspended particle in a liquid behaves like a big atom in equilibrium with the liquid - the liquid itself is a collection of smaller particles, moving at random and colliding with one another and that the particles in suspension diffuse through the medium - colliding with the liquid particles

-Einstein showed that the "existence of osmotic pressure is a consequence of the molecular-kinetic theory of heat, and that, according to this theory, at great dilutions numerically equal quantities of dissolved molecules and suspended particles behave completely identical with regard to osmotic pressure"(Einstein, A.)

-Einstein showed that the movement of these suspended particles in the liquid fulfill a statistical law so that the distance the particle moves will increase as the square root of the time <math>xo=\sqrt{6Dt}</math> (where D is the diffusion coefficient)

Screen shot 2011-09-12 at 1.49.03 PM.png


The average velocity of a particle is the ratio between the distance X0 and the time t required to travel this distance. In the figure the red line shows constant velocity as a function of time. For a Brownian particle, however, the distance grows as <math>\sqrt{t}</math> and so we find an average velocity proportional to <math>\sqrt{1 / t}</math>, as shown by the blue curve. This means that as we measure average velocity over smaller and smaller time periods, the velocity will grow enormously. This experimental fact had confounded investigators of the Brownian motion before Einstein’s theory was published.

(Andelman, D., Diamant, H.)

-Einstein showed that the movement of suspended particles is the function of a dynamic equilibrium between two opposing processes: a) a force acting on each individual suspended particle (which is a function of the particles' position) b) diffusion - which is the result of the random motion of the particles due to thermal molecular motion

-from the establishment of a and b in dynamic equilibrium - Einstein concluded that the coefficient of diffusion depends only on the coefficient of friction of the liquid and the size of the suspended particles (aside from universal constants and absolute temperature)


Significance

-by showing that suspended particles behave, osmotically, the same as dissolved particles, Einstein was able to illustrate the connection between diffusion (which is a macroscopic process) and the random movement of individual particles (which is a microscopic process)

-Einstein was the first to show the importance of Brownian motion as experimentally verifying the statistical mechanics model of physics

-this led to our current understanding that thermodynamics and statistical physics are not contradictory models - they predict the same outcome; statistical physics on a microscopic scale and thermodynamics on a macroscopic scale)

References

1. Einstein, A. "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat", Annalen der Physik, 17, 1905, 549-560 2. Andelman, D., Diamant, H. "100 Years Since Einstein's Less Known Revolution: From the pollen dance to atoms and back"[2]