Einstein: 1905 Paper #2 on Molecular Dimensions

Doctoral Thesis completed in April 1905 and revised in 1906.
Published in Annalen der Physik 19 (1906) 289-305
A New Determination of Molecular Dimensions
Highlights in bold and comments in color by Vinaire

In this paper Einstein explores a new theoretical method to calculate the molecular dimensions through the observation of physical phenomenon in liquids. The main part of the thesis involves hydrodynamics and the relation between coefficients of viscosity, followed by a determination of diffusion-coefficient from fundamentals.

Some of the mathematical symbols used by Einstein in his thesis are as follows:

k: coefficient of viscosity of the solvent
k*: coefficient of viscosity of the solution
x, y, z, t: the coordinates and time
x₀, y₀, z₀: an arbitrary point (center of the molecule; molecule has a contact layer)
u, v, w: velocity components of the solvent
u₀, v₀, w₀: dilation motion in the absence of the sphere (at infinity)
u₁, v₁, w₁: dilation motion in the presence of the sphere—would have to vanish at infinity
p: hydrostatic pressure (Section 1)
G: small region around this arbitrary point x0, y0, z0
P: radius of the rigid sphere
R: a sphere of radius R, where R is indefinitely large compared with P
X_n, Y_n, Z_n: the components of the pressure exerted on the surface of the sphere of radius R
n: the number of solute molecules per unit volume
W: mechanical work done on the liquid (per unit of time) in the liquid lying within the sphere R = the energy which is transformed into heat
ds: surface element
ϕ: the fraction of the volume occupied by the spheres.
s: specific volume of the sugar present in solution
p: the osmotic pressure of the dissolved substance (Section 4)
ρ: grams of solute in a unit volume (section 3)
m: molecular weight of the solute
N: the number of actual molecules’ in a gram-molecule (Avogadro’s Number)
P: the hydrodynamically-effective radius of the solute molecule (Section 3)

Here is a summary of the steps taken by Einstein in his thesis:

1. When a very small sphere is suspended in a liquid, how does it effect the motion of that liquid?

Set up hydrodynamical equations for viscous fluids for random motions in the liquid.
Solve it using the method given by Kirchhoff getting expressions for p (hydrostatic pressure) and for u, v, w (the velocity components of solvent).
These solutions are unique for the boundary conditions at the surface of the sphere (a particle of solute).
Use these solutions to calculate the mechanical work done in the liquid (per unit of time).
The energy expression is obtained in terms of the viscosity of the solvent when the sphere is present.
Obtain the energy expression when the sphere is absent. There is a lot of mathematics and simplifying assumptions.

2. Calculation of the viscosity-coefficient of the liquid in which a large number of small spheres are suspended in irregular distribution.

Use the results from above to calculate the energy for a large number of spheres suspended in the liquid.
Express the energy in terms of the parameters for the solution in a manner similar to above.
Compare the energy expressions for the solvent and the solution, to determine the ratio of their viscosities.
This ratio of viscosities is obtained in terms of the total volume of the solute present in the solution per unit volume.
Conclusion: If very small rigid spheres are suspended in a liquid, the coefficient of internal friction is thereby increased by a fraction which is equal to 2.5 times the total volume of the spheres suspended in a unit volume, provided that this total volume is very small.

3. On the volume of a dissolved substance of molecular volume large in comparison with that of the solvent.

Einstein applies the above conclusion to the viscosity data for dilute solution of sugar in water.
A gram of sugar dissolved in water has the same effect on the viscosity as small suspended rigid spheres of total volume 0.98 cc.
But the a gram of solid sugar has a volume of 0.61 cc.
Einsteins reasons that the sugar molecules present in solution limit the mobility of the water immediately adjacent, so that a quantity of water, whose volume is approximately one-half the volume of the sugar-molecule, is bound on to the sugar-molecule
Using the molecular weight of sugar, Einstein calculates the volume of a sugar molecule in terms of the Avogadro’s number.

4. On the diffusion of an undissociated substance in solution in a liquid

Use Kirchhoff’s equation to calculate the diffusion-coefficient of an undissociated solution.
In this calculation osmotic pressure is treated as a force acting on the individual molecules.
This expression of diffusion-coefficient includes “the product of the number N of actual molecules in a gram-molecule and of the hydrodynamically-effective radius P of the molecule.

5. Determination of molecular dimensions with the help of relations already obtained

The expressions of the ratio of viscosities and the expression of the diffusion-coefficient give us two equations in “N” and “P”.
Einstein solves these equations for the aqueous sugar solution.
The values of “N” and “P” are found to have the same order of magnitude as determined by other methods.

Einstein’s thesis serves to validate his theoretical approach and the fundamental expressions derived for the viscosities and for the diffusion coefficient.

Vinaire's Blog