Brownian Motion Is Caused by the Irregular Thermal Movements of It Is to Be Hoped That Some Enquirer May Succeed the Molecules of the Liquid

-INVESTIGATIONS ON THE THEORY .OF ,THE BROWNIAN MOVEMENT BY ALBERT EINSTEIN, PH.D.

EDITED WITH NOTES BY e R. FüRTH This new Dover edition, first published in 1956, is an unabridged .and unaltered republication of the TRANSLATED BY translation first: published in 1926. It is published through special A. D. COWPER arrangément with Methuen and Co., Ltd., and the estate of Albert Einstein. Manufactured in the United'States WITH 3 DIAGRAMS of America.

DOVER PUBLICATIONS, INC.l. INVESTIGATIONS ON THE THEORY OF THE BROWNIAN MOVEMENT

I ON THE MOVEMENT OF SMALL PARTICLES SUSPENDED IN A STATIONARY LIQUID DEMANDED BY THE MOLECULAR- KINETIC THEORY OF HEAT N this paper it will be shown that according I to the molecular-kinetic theory of heat, bodies of microscopically-visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat. It is possible that the movements to be discussed here are identical with the so-called ‘‘ Brownian

molecular motion ” ; however, the information available to me regarding the latter is so lacking in precision, that I can form no judgment in the matter (I). If the movement discussed here can actually be observed (together with the laws relating to 2 THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES 3 it that one would expect to find), then classical thermodynamics-at least when the foreë of gravity thermodynamics can no longer be looked upon (which does not interest us here) is ignored-we as applicable with precision to bodies even of would not expect to find any force acting on the dimensions distinguishable in a microscope : an partition ; for according to ordinary conceptions exact determination of actual atomic dimensions the “ free energy ” of the system appears to be is #en possible. On the other hand, had the independent of the position of the partition and prediction of this movement proved to be in- of the suspended particles, but dependent only correct, a weighty argument would be provided on the total mass and qualities of the suspended against the molecular-kinetic conception of heat. material, the liquid and the partition, and on the pressure and temperature. Actually, for the cal- 3 I. ON THE OSMOTIC PRESSURE TO BE ASCRIBED culation of the free energy the energy and entropy TO THE SUSPENDED PARTICLES of the boundary-surface (surface-tension forces) Let z gràm-molecules of a non-electrolyte be should also be considered ; these can be excluded dissolved in a volume V* forming part of a if the size and condition of the surfaces of contact quantity of liquid of total volume V. If the do not alter with the changes in position of the volume V* is separated from the pure solvent partition and of the suspended particles under by a partition permeable for the solvent but consideration. impermeable for the solute, a so-called “ osmotic But a different conception is reached from pressure,’’ is exerted on this partition, which 9, the standpoint of the molecular-kinetic theory of satisfies equation the heat. According to this theory a dissolved mole- $V*= RTz . (4 cule is differentiated from a suspended body when V*/zis sufficiently great. soZeZy by its dirhensions, and it is not apparent On the.other hand, if small suspended particles why a number of suspended particles should not are present in the fractional volume V* in place produce the same osmotic pressure as the same of the dissolved substance, which particles are also number of molecules. We must assume that the unable to pass through the partition permeable to suspended particles perform an irregular move- the solvent : according to the classical theory of ment-even if a very slow one-in the liquid, on 4 THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES 5 account of the molecular movement of the liquid ; a. physical system which completely define the if they are prevented from leaving the volume V* instantaneous condition of the system (for ex- by the partition, they will exert a pressure on the ample, the Co-ordinates and velocity components partition just like molecules in solution. Then, of all atoms of the system), and if the complete if there are fi suspended particles present in the system of the equations of change of these variables volume V*, and therefore %/'V* = V in a unit .of of state is given in the form volurne, and if neighbouring particles are sufficiently far separated, there be a corresponding will ?&3t = +.(pl . pl) (V = I, 2, * . . Z) osmotic pressure fi of magnitude given by pz--RTn RT whence V*N"Ñ'v' where N signifies the actual number of molecules contained in a gram-molecule. It will be shown in the next paragraph that the molecular-kinetic then the entropy of the system is given by the theory of heat actually leads to this wider con- expression ception of osmotic pressure. fj-2, OSMOTIC PRESSURE FROM THE STANDPOINT OF THE MOLECULAR-KINETIC THEORY OF where T is the absolute temperature, E the energy HEAT (*) of the system, E the energy as a function of fiv. If pl, P,, . . . @J are the variables of state of The integral is extended over all possible values I (*) In this paragraph the papers of the author on the of 9. consistent with the conditions of the prob- '' Foundations of Thermodynamics " are assumed to be lem. x is connected with the constant N referred familiar to the reader (Ann. d. Phys., 9, p. 4r7, ; 1902 to before by the relation zxN = R. We obtain 11, p. 170, 1903). An understanding of the conclusions reached in the present paper is not dependent on a hence for the free energy F, knowledge of the former papers or of this paragraph of the present paper. 6 THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES 7 Now let us consider a quantity of liquid enclosed dy2, dz,, . . . dx,, dy,, dzn, lying wholly within in a volume V ; let there be n solute molecules V*, The value of the integral appearing in the (or suspended particles respectively) in the por- expression for F will be sought, with the limita- tion 'V* of this volume 'V# which are retained in tion tilat the centres of gravity of the particles the volurne V* by a semi-permeable partition ; lie within a domain defined in this manner. The the integration limits of the integral B obtained integral can then be brought into the form in the expressions for S and F will be affected dB dX1 dyl dZn . J, accordingly. The combined. volume of the solute where J is independent of axl, dy,, etc., as well as molecules (or suspended particles) is taken as of V*, i.e. of the position of the semi-permeable small compared with V*. This system will be partition. But J is also independent of any completely defined according to the theory under special choice of the position of the domains of discussion by the variables of condition pl . . . pl. the centres of gravity and of the magnitude of If the molecular picture were extended to deal V*, as will be shown immediately. For if a with every single unit, the calculation of the second system were given, of indefinitely small integral B would offer such difficulties that an domains of the centres of gravity of the particles, exact- calculation of F could be SCarceIy contem- and the latter designated dx:dyl'dzl' ; dx;dy,'dz[ plated. Accordingly, we need here only to know . . . dx,'dyn'dzn', which domains differ from those how F depends on the magnitude of the volume originally given in their position but not in their V*, in which all the solute molecules, or suspended magnitude, and are similarly all contained in V*, bodies (hereinafter termed briefly " particles an analogous expression holds :- are contained. dB' = dxl'dy; . . . dz,,' , J'. We will call x,, yI, x, the rectangular Co-ordinates of the centre of gravity of the first particle, Whence x,, y,, x, those of -the second, etc., x,, y,, x, those dXIdy1 dzn = dxl'dyl' . . . dza'. of the last particle, and allocate for the centres Therefore of gravity of the particles the indefinitely small dB-- - J domains of parallelopiped form dg,, dy,, dzl ; dxzt dB' - TfT 8 THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES g But from the molecular theory of Heat given in the and 3F RT RT paper quoted, (*) it is easily deduced that dB/B (4) p=-alr*=Y*lCi=Nv* (or dB'/B respectively) is equal to the probability that at any arbitrary moment of time the centres It has been shown by this analysis that the exist- of gravity of the particles are included in the ence of an osmotic pressure can be deduced from domains (dx, . . . dz,) or (2%: . . . dzn') respec- the molecular-kinetic theory of Heat ; and that tively. Now, if the movements of single particles as far as osmotic pressure is concerned, solute are independent of one another to a sufficient molecules and suspended particles are, according degree of approximation, if the liquid is homo- to this theory, identical in their behaviour at geneous and exerts no force on the particles, then great dilution. for equal size of domains the probability of each 6 3. THEORY OF THE DIFFUSION OF SMALL of the two systems will be equal, so that the follow- SPHERES IN SUSPENSION ing holds : Suppose there be suspended particles irregularly dB dB' -x- dispersed in a liquid. We will consider their B B' state of dynamic equilibrium, on the assumption But from this and the last equation obtained it that a force K acts on the single particles, which follows that force depends on the position, but not on the time. J = J'. It will be assumed for the sake of simplicity that We have thus proved that J is independent both the force is exerted everywhere in the direction of of V* and of x,, yr, . . . x,. By integration we the x axis. obtain Let Y be the number of suspended particles per B = /]dxl. . . dzn = J.V*n, unit volume ; then in the condition of dynamic and thence equilibrium V is such a function of x that the varia- tion of the free energy vanishes for an arbitrary virtual displacement Sx of the suspended substance. We have, therefore, (*) A. Einstein, Ann. d. Phys., 11, p. 170, 1903. 8F = 8.E - TSS O. IO THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES II

It will be assumed that the liquid has unit area 2. A process of diffusion, which is to be looked of cross-section perpendicular .to the x axis and upon as a result of the irregular movement of the is bounded by the planes x = o and x = 1. We particles produced by the thermal molecular have, then, movement. 6E = - {:Kv6xdx If the suspended particles have spherical form (radius of the sphere = P), and if the liquid has and a coefficient of viscosity k, then the force K im- parts to the single particles a velocity (*)

The required condition of equilibrium is therefore and there will pass a unit area per unit of time -KV+ RT--- av N ax-O vK or 67rkP particles. If, further, D signifìes the coefficient of diffusion The last equation states that equilibrium with the of the suspended substance, and p the mass of a force K is brought about by osmotic pressure particle, as the result of diffusion there will pass forces. across unit area in a unit of time, Equation (I) can be used to find the coefficient of diffusion of the suspended substance. We can - D'M grams bX look upon the dynamic equilibrium condition considered here as a superposition of two processes or proceeding in opposite directions, namely i- - D-3V particles. I. A movelment of the suspended substance 3X under the influence of the force K acting on each (*) Cf. e.g. G. Kirchhoff, " Lectures on Mechanics," single suspended particle. Lect. 26z 8 4. 12 TE3:EOR.Y OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES ~3 Since there must be dynamic equilibrium, we different intervals of time must be considered as must have mutually independent processes, so long as we VK 31, think of these intervals of time as being chosen D- = O. =P- 3x not too small. We can calculate the coefficient of diffusion We will introduce a time-interval T in our discussion, which is to be very small compared with from the two conditions (I) and (2) found for the dynamic equilibrium. We get the observed interval of time, but, nevertheless, of such a magnitude that the movements executed D=------RT I by a particle in two consecutive intervals of time N 61rkP * r are to be considered as mutually independent The coefficient of diffusion of the suspended sub- phenomena (8). stance therefore depends (except for universal Suppose there are altogether n suspended par- constants and thk absolute temperature) only on ticles in a liquid. In an interval of time r the the coefficient of viscosity of the liquid and on the x-Co-ordinates of the single particles will increase size of the suspended particles. by d, where d has a different value (positive or negative) for each particle. For the value of d fj 4. ON THE IRREGULAR MOVEMENT OF PARTICLES a certain probability-law will hold ; the' number SUSPENDED IN A LIQUID AND THE RELATION d% of the particles which experience in the time- OF THIS TO DIFFUSION interval r a displacement which lies between d We will turn now to a closer consideration of and d + dA, will be expressed by an equation of the irregular movements which arise from thermal the form molecular movement, and give rise to the diffusion dn = n+(A)d& investigated in the last paragraph. where Evidently it must be assumed that each single [+OO+(A)dd-00 = I particle executes a movement which is indepen- and + only differs from zero for very small values dent of the movement of all other particles ; the of d and fulfils the condition movements of one and the same particle after 14 THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES 15 We will investigate now how the coefficient of On the right-hand side the second, fourth, etc., diffusion depends on 4, confining ourselves again terms vanish since +(x) = #(- x) ; whilst of the to the case when the number V of the particles per first, third, fifth, etc., terms, every succeeding unit volume is dependent only on x and t. telm is very small compared with the preceding. Putting for the number of particles per unit Bearing in mind that volume V = f(x, t), we will calculate the distribution of the particles at a time t + T from the distribution at the time t. From the definition and putting of the function +(A), there is easily obtained the +m number of the particles which are located at the. time t + T between two planes perpendicular to the x-axis, with abscissz! x and x + ax. We get and taking into consideration only the first and A==+ m f(x, t + 7)dx = dx. J f(x + A)#(A)dA. third terns on the right-hand side, we get from A= - m this equation Now, since T is very small, we can put

Further, we can expand j(x + d, t) in powers This is the well-known differential equation for of A :- diffusion, and we recomise that D is the coeecient of diffusion. Another important consideration can be related We can bring this expansion under the integral to this method of development. We have assumed sign, since only very small values of A contribute that the single particles are all referred to the anything to the latter. We obtain same Co-ordinate system. But this is unneces- +m ax +W sary, since the movements of the single particles f-/--~or=fj Q(d)dA+jfS d+(A)dA -m -00 are mutually independent. We wilI now refer the motion of each particle to a Co-ordinate 16 THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES 17 system whose origin coincides at the time t = o of this equation the displacement Xz in the direc- with the position of the centre of gravity of the tion of the X-axis which a particle experiences on particles in question ; with this difference, that an average, or-more accurately expressed-the f(x, t)dx now gives the number of the particles square root of the arithmetic mean of the squares whose x Co-ordinate has increased between the of displacements in the direction of the X-axis ; time t = o and the time t = t, by a quantity it is which lies between x and x + dx. In this case Aa = 45 = JZt . ' (11) also the function f must satisfy, in its changes, the equation (I). Further, we must evidently . The mean displacement is therefore propor- tional to the square root of the time. It can have €or x > o and t = o, < easily be shown that the square root of the mean f(x, t) = o and [+wj(x, t)dx = n. of the squares of the total displacements of the --m particles has the value &J3 . . (12) The problem, which accords with the problem of the diffusion outwards from a point (ignoring pos- 5 5. FORMULA FOR THE MEAN DISPLACEMENT OF sibilities of exchange between the diffusing par- SUSPENDED PARTICLES. A NEW METHOD OF ticles) is now mathematically completely defined DETERMINING THE REAL SIZE OF THE ATOM (9) ; the solution is In 5 3 we found for the coefficient of diffusion D xcr -- of a material suspended in a liquid in the form of small spheres of radius P- The probable distribution of the resulting displacements in a given time t is therefore the same as that of fortuitous error, which was to be ex- Further, we found in 5 4 for the mean value of the pected. But it is significant how the constants in displacement of the particles in the direction of the exponential term are related to the coefficient the X-axis in time t- of diffusion. We wil€ now calculate with the help Aa = Jak 18 THEORY OF BROWNIAN' MOVEMENT By eliminating D we obtain

This equation shows how )C, depends on T,k, and P. II We will calculate how great & is for one second, ON THE THEORY OF THE BROWNIAN if N is taken equal to 6.10~3in accordance with the MOVEMENT kinetic theory of gases, water at 17" C. is chosen as the liquid (K = 1-35 IO-^), and the diameter of the particles -001mm. We get

&, = 8*10-~cm. = 0.8,~. OON after the appearance of my paper (*) The mean displacement in one minute would be, Son the movements of particles suspended therefore, about 6p. in liquids demanded by the molecular theory of On the other hand, the relation found can be heat, Siedentopf (of Jena) informed me that he used for the determination of N. We obtain and other physicists-in the first instance, Prof. Gouy (of Lyons)-had been convinced by direct N=-*-I RT ha2 3wkP' observation that the so-called Brownian motion is caused by the irregular thermal movements of It is to be hoped that some enquirer may succeed the molecules of the liquid. shortly in solving the problem suggested here, (t) Not only the qualitative properties of the which is so important in cqnnection with the Brownian motion, but also the order of magnitude theory of Heat. (13) of the paths described by the particles correspond Berne, May, 1905. completely with the results of the theory. I will (Received, II May, 1905.) not attempt here a comparison of the slender experimental material at my disposal with the (*) A. Einstein, Ann. d. Phys., 17, p. 549, 1905. (t)M. Gouy, Jouyn. de Phys. (z), I,561, 1888. 19