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-INVESTIGATIONS O N
THE THEORY .OF ,THE
BROWNIAN MOVEMENT
BY
ALBERT EINSTEIN, PH.D.
EDITED WITH NOTES BY
R. F ü R T H
TRANSLATED BY
A. D. COWPER
WITH 3 DIAGRAMS
DOVER PUBLICATIONS, INC. l.
MOLECULAR DIMENSIONS ' 37
III
A NEW DETERMINATION OF MOLECULAR
DIMENSIONS
(From the Annalen der Physik (4), 19, 1906,
pp. 289-306. Corrections, ibid., 34, 1911, pp.
591-5922.] (23)
T HE kinetic theory of gases made possible the earliest determinations of the actual
dimensions of the molecules, whilst physical
phenomena observable in liquids have not, up to
the present, served for the calculation of molecular
dimensions. The explanation of this doubtless
lies in the difficulties, hitherto unsurpassable,
which discourage the development of a molecular
kinetic theory of liquids that will extend tb details.
It will be shown now in this paper that the size
of the molecules of the solute in an undissociated
dilute solution can be found from the viscosity of
the solution and of the pure solvent, and from
the rate of diffusion of the solute into the solvent,
if the volume of a molecule of the solute is large
36
compared with the volume of a molecule of the
solvent. For such a sofute molecule will behave
approximately, with respect to its mobility in
the solvent, and in respect to its influence on the
viscosity of the latter, as a solid body suspended
in the solvent, and it will be allowable to apply
to the motion of the solvent in the immediate
neighbourhood of a molecule the hydrodynamic
equations, in which the liquid is considered homo-
geneous, and, accordingly, its molecular structure
is ignored. We will choose for the shape of the
solid bodies, which shall represent the solute mole-
cules, the sphericdi fom-*
I. ON THE EFFECT ON THE MoTroN OF A 1,IQWID
OF A VERY SMALL SPHERE SUSPENDED IN IT
As the subject of our discussion, Iet us take an
incompressible homogeneous liquid with viscosity
K, whose velocity-components W , V , W will be given
as functions of the Co-ordinates x, y, x, and sf the
time. Taking an arbitrary point go, yo, x,, we
wifl imagine that the functions u, V , W are de-
veloped according to Taylor's theorem as func-
tions of x - x,, y - yo, x - x,, and that a domain
G is marked out around this point so small that
within it only the linear terns in this expansion
38 THEORY OF BROWNIAN MOVEMENT
have to be considered. The motion of the liquid
contained in G can then be looked upon in the
familiar manner as the result of the superposition
of three motions, namely,
I. A parallel displacement of all the particles'
of the liquid without change of their
relative position.
2. A rotation of the liquid without change of
the relative position of the particles of
the liquid.
3. A movement of dilatation in three directions
at sight angles to one another (the prin-
cipal axes sf ation^^ ion^^
We wif€ imagine now a. spherical rigid body in
the domain G, whose centre lies at the point
yo, x,, and whose ~~~e~~~~~~ are very small com-
pared with those or" the domain G, We will
further assume that the motion under c~nsidera-
tion is so shw that the kinetic energy of the
sphere is negligible as we11 as that of the liquid.
It will be further assumed that the velocity com-
ponents of an element of sudace the sphere
show agreement with the corresponding velocity
components of the particles of the liquid in the
immediate neighbourhood, that is, that the contact-
layer (thought of as continuous) also exhibits
MOLECULAR DIMENSIONS 39
everywhere a viscosity-coefficient that is not
vanishingly small.
It is clear without further discussion that the
sphere simply shares in the partid motions I and 2,
without modifying the motion of the neigkibouring
liquid, since the liquid moves as a rigid body in
these partial motions ; and that we have ignored
the effects of inertia.
But the motion 3 will be modified by the pres-
ence of the sphere, and our next problem will be
to investigate the influence of the sphere on this
motion of the liquid. We will further refer the
motion 3 to a co-ordinate system whose axes are
parallel to the principal axes of dilatation, and we
x - x, = Q:
x - z , = 5,
then the motion can be expressed by the equations
will put
Y - YO = T',
uQ == ' f f
II) V0 = Bq,
q.J a ,
in the case when the sphere is not present.
A, B, C are constants which, on account of the
incompressibility of the liquid, must fulfil the
condition
(2) A + B + C = 0 * (24)
40 THEORY OF BROWNIAN MOVEMENT
Now, if the rigid sphere with radius P is intro-
duced at the point x,, yo, q,, the motions of the
liquid in its neighbourhood are modified. In the
foliowing discussion we will, for the sake of con-
venience, speak of €’ as “ finite ” ; whilst the
values of 6, 9, 5, for which the motions of the
liquid are no longer appreciably influenced by the
sphere, we will speak of as “ infinitely great.”
Firstly, it is clear from the symmetry of the
motions of the liquid under consideration that
there can be neither a translation nor a rotation
of the sphere accompanying the motion in ques-
tion, and we obtain the limiting conditions
u = v = = w = o w h e n p = P
where we have put
p = JP 3- ?I2 4- P > 0.
Here u, V , w are the velocity-components of the
motion now under consideration (modified b~7 the
sphere). If we put
a4 = 4 3- u,,
W = cg + wi, (3) = Br] 4- v1,
’ since the motion defined by equation (3) must be
transformed into that defined by equations (I)
in the “ infinite” region, the velocities ul, V,, wl
will vanish in the latter region.
MOLECULAR DIMENSIONS 41
The functions u, V , W must satisfy the hydro-
dynamic equations with due reference to the
viscosity, and ignoring inertia. Accordingly, the
following equations will hold :- (*)
where A stands for the operator
3 2 3% -+-+- 3e2 392 352
and 9 for the hydrostatic pressure.
Since the equations (I) are solutions of the
equations (4) and the latter are linear, according
to (3) the quantities u,, V,, w1 must also satisfy the
equations (4). I have determined u,, V,, q , and P,
according to a method given in the lecture of
Kirchhoff quoted in $ 4 (t), and find
(*) G. Kirchhoff, “ Lectures on Mechanics,” Lect. 26.
(t j ‘‘ From the equations (4) it follows that ap = o,
If p is chosen in accordance with this condition, and a
function V is determined which satisfies the equation
A V = ;P,
then the equations (4) are satisfied if we put
and chose u’, V’, W‘, so that Au’ = o, Av’ = o, and
AW’ = o, and
42 THEORY OF BROWNIAN MOVEMENT
Now if we put
and in agreement with this
and
the constants a, b, e can be chosen SO that when p P,
= W = W = O. By superposition of three similar
solutions we obtain the solution given in the equations
( 5 ) and W *
where
(5a)
MOLECULAR DIMENSIONS 43
It is easy to see that the equations (5) are solu-
tions of the equations (4). Then, since
I 2 A t = o, A - = o,
P A p = P
and
A($) = - ${A( ; ) } = o,
we get
But the last expression obtained is, according to
the first of the equations (S) , identical with dpldE.
In similar manner, we can show that the second
44 THEORY OF BROWNIAN MOVEMENT
and third of the equations (4) are satisfied. We
obtain further-
But since, according to equation (sa),
it' follows that the last of the equations (4) is
satisfied. As for the boundary conditions, our
equations for zd, V , W are transformed into the
equations (I) only when p is indefinitely large.
By inserting the value of D from the equajion
(sa) in the second of the equations (5) we get
We know that u vanishes when p = P. On the
grounds of syrnmetry the same holds for V and W .
We have now demonstrated that in the equations
(5) a solution has been obtained to satisfy both
MOLECULAR DIMENSIONS 45
the equations (4) and the boundary conditions of
the problem.
It can also be shown that the equations (5) are
the only solutions of the equations (4) consistent
with the boundary conditions of the problem.
The proof will only be indicated here. Suppose
that, in a finite space, the velocity-components of
a liquid u, V, W satisfy the equations (4). Now, if
another solution U, V , W of the equations (4) can
exist, in which on the boundaries of the sphere in
question U = zc, V = V , W = W, then (U - u,
V - V , W - W ) will be a solution of the equa-
tions (4), in which the velocity-components vanish
at the boundaries of the space. Accordingly, no
mechanical work, can be done on the liquid con-
tained in the space in question. Since we have
ignored the kinetic energy of the liquid, it follows
that the work transformed into heat in the space
in question is likewise equal to zero. Hence we
infer that in the whole space we must have zc = u',
ZI = V', W = W', if the space is bounded, at least
in part, by stationary walls. By crossing the
boundaries, this result can also be extended to
the case when the space in question is infinite, as
in the case considered above. We can show thus
that the solution obtained above is the sole
solution of the problem.
46 THEORY OF BROWNIAN MOVEMENT
We will now place around the point x,, yo, x, a
sphere of radius R, where R is indefinitely large
compared with P, and will calculate the energy
which is transformed into heat (per unit of time)
in the liquid lying within the sphere. This energy
W is equal to the mechanical work done on the
liquid. If we call the components of the pressure
exerted on the surface of the sphere of radius R,
Xn, Yn, 2%) then
where ,the integration is extended over the surface
of the sphere of radius R.
Here
x, = - . (Xr 6 + x? + X<-) , 6
2% = - (zt 8 - + 23 + Zr),
86’. Y,=Z,=-k($+?)
Y , = $ - z k - , 3v Z t = X g = - k ( z + g ) am 3%
P P P
Yn = - (y{ + y? + Y$),
. P P
6
P P
where
W X t = p - ~ k -
311
MOLECULAR DIMENSIONS 47
The expressions for u, V , W are simplified when we
note that for p -.- R the terms with the factor
P6/p5 vanish.
We have to put
For P we obtain from the first of the equations (5)
by corresponding onaissi~ms
p = - 5kP 4- Bq2 + CC2
P5
+ const.
We obtain first
A‘2 25kP3 t2(At2+Rrlt+CC2) x,= - 2kA+10kP3--
Pb P7
and from this
With the aid of the expressions for Yn and Zn,
obtained by cyclic exchange, we get, ignoring all
48 THEORY OF BROWNIAN MOVEMENT
terms which involve the ratio .P /p raised to any
power higher than the third,
- 5kP-8(A2f2 + B2q2 + Cece) + 1 5 k # A f ~ + B q ~ + C S ~ ) 2 . ( ~ 3 ) P3
If W é integrate over the sphere and bear in mind
that
P*
5 ds = 4R%,
5 eds = 5 q2ds '= 5 = SmR4,
5 [4ds 5 $ds = 5 {*as = h R 6 ,
5 q2{2ds = S t2E2dS = 5 t2q2dS = fTnRg,
5 (A[2+3q2+Cc2)2ds =