[ Pobierz całość w formacie PDF ]vicinity, so that an amount of water whose volume is about
three times larger than the volume of the sugar molecule is
attached to the sugar molecule. 33
Hence we may say that a dissolved molecule of sugar
(i.e , the molecule together with the water attached to it)
behaves hydrodynamically like a sphere with a volume of
2 45 · 342/N cm3, where 342 is the molecular weight of
sugar and N is the number of actual molecules in one gram-
molecule. 34
4. On the Diffusion of an Undissociated
Substance in a Liquid Solution
Let us consider a solution of the kind discussed in section 3.
If a force K acts upon a molecule, which we assume to be a
sphere with radius P , the molecule will move with a velocity
É, which is determined by P and the coefficient of viscosity
61
PAPER 1
k of the solvent. Indeed, the following equation holds:3
K
É = (1)
6ÀkP
We use this relation to calculate the coefficient of diffu-
sion of an undissociated solution. If p is the osmotic pres-
sure of the dissolved substance, the only motion-producing
force in such a dilute solution, then the force acting on the
dissolved substance per unit volume of solution in the di-
rection of the X-axis equals -´p/´x. If there are Á grams
per unit volume, and m is the molecular weight of the dis-
solved substance and N the number of actual molecules
in one gram-molecule, then Á/m · N is the number of
(actual) molecules per unit volume, and the force exerted
on a molecule by virtue of the concentration gradient is
´p
m
K =- (2)
ÁN ´x
If the solution is sufficiently dilute, the osmotic pressure is
given by the equation:
R
p = ÁT (3)
m
where T is the absolute temperature and R = 8 31 · 107.
From equations (1), (2), and (3) we obtain for the migration
velocity of the dissolved substance
´Á
É =-RT 1 1 (4)
6Àk NP Á ´x
3
G. Kirchhoff, Vorlesungen über Mechanik, 26. Vorl. (Lectures on Me-
chanics, Lecture 26), equation (22).
62
DETERMINATION OF MOLECULAR DIMENSIONS
Finally, the amount of the substance passing per unit time
through a unit cross section in the direction of the X-axis is
´Á
RT 1
ÉÁ =- ·
6Àk NP ´x
Hence, we obtain for the coefficient of diffusion D
RT 1
D = · 35
6nk NP
Thus, from the coefficients of diffusion and viscosity of the
solvent we can calculate the product of the number N of ac-
tual molecules in one gram-molecule and the hydrodynam-
ically effective molecular radius P .
In this derivation the osmotic pressure has been treated as
a force acting on the individual molecules, which obviously
does not agree with the viewpoint of the kinetic molecular
theory; since in our case according to the latter the os-
motic pressure must be conceived as only an apparent force.
However, this difficulty disappears when one considers that
the (apparent) osmotic forces that correspond to the concen-
tration gradients in the solution may be kept in (dynamic)
equilibrium by means of numerically equal forces acting on
the individual molecules in the opposite direction, which can
easily be seen by thermodynamic methods.
´p
1
The osmotic force acting on a unit mass - can be
Á ´x
counterbalanced by the force -Px (exerted on the individual
dissolved molecules) if
´p
1
- - Px = 0
Á ´x
Thus, if one imagines that (per unit mass) the dissolved
substance is acted upon by two sets of forces Px and -Px
that mutually cancel out each other, then -Px counterbal-
ances the osmotic pressure, leaving only the force Px, which
63
PAPER 1
is numerically equal to the osmotic pressure, as the cause
of motion. The difficulty mentioned above has thus been
eliminated.4
5. Determination of Molecular
Dimensions with the Help of the
Obtained Relations
We found in section 3 that
k" 4 3
= 1 + Õ = 1 + n · ÀP 36
3
k
where n is the number of dissolved molecules per unit vol-
ume and P is the hydrodynamically effective radius of the
molecule. If we take into account that
Á
n
=
N m
where Á denotes the mass of the dissolved substance per
unit volume and m its molecular weight, we get
3 3 m k"
NP = - 1 37
4À Á k
On the other hand, we found in section 4 that
RT 1
NP =
6Àk D
These two equations enable us to calculate separately the
quantities P and N , of which N must be independent of
the nature of the solvent, the dissolved substance, and the
temperature, if our theory agrees with the facts.
4
A detailed presentation of this line of reasoning can be found in Ann.
d. Phys. 17 (1905): 549. [See also this volume, paper 2, p. 86.]
64
DETERMINATION OF MOLECULAR DIMENSIONS
We will carry out the calculation for an aqueous solution
of sugar. From the data on the viscosity of the sugar solution
cited earlier, it follows that at 20æ%C,
3
NP = 200 38
According to the experiments of Graham (as calculated by
Stefan), the diffusion coefficient of sugar in water is 0.384 at
9 5æ%C, if the day is chosen as the unit of time. The viscosity
of water at 9 5æ% is 0.0135. We will insert these data in our
formula for the diffusion coefficient, even though they have
been obtained using 10% solutions, and strict validity of our
formula cannot be expected at such high concentrations. We
obtain
NP = 2 08 · 1016
Neglecting the differences between the values of P at 9 5æ%
3
and 20æ%, the values found for NP and NP yield
P = 9 9 · 10-8 cm
N = 2 1 · 1023
The value found for N shows satisfactory agreement, in
order of magnitude, with values found for this quantity by
other methods. 39
(Bern, 30 April 1905)
editorial notes
1
A factor k is missing on the right-hand side of the last equation in this
line; this error is corrected in Albert Einstein, Eine neue Bestimmung
der Moleküldimensionen, Ann. d. Phys. 19 (1906), pp. 289 305, cited
´
hereafter as Einstein 1906. Note that denotes partial differentiation
´
"
modern .
"
65
PAPER 1
2
The denominator on the right-hand side should be ´¾2; this error is
corrected in ibid.
3
The denominator of the first term on the right-hand side should be
´¾2; this error is corrected in ibid. A reprint of this article in the Einstein
Archive shows marginalia and interlineations in Einstein s hand, the first
1
of which refer to this and the following equation. The term +g was
Á
added to the right-hand side of the equations for V and then canceled.
These marginalia and interlineations are presumably part of Einstein s
unsuccessful attempt to find a calculational error; see note 13 below.
1
´
Á
4
The equation for u should be, as corrected in ibid., u =-2c .
´¾
In the reprint mentioned in note 3, the first derivative with respect to
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