-INVESTIGATIONS ON THE THEORY ,THE BROWNIAN MOVEMENT

ALBERT EINSTEIN,

EDITED WITH BY

new edition, published an unabridged unaltered republication of the TRANSLATED BY translationfirst: published in It is published A. D. COWPER Methuen and and the Albert Einstein. Manufactured the United'States WITH DIAGRAMS of America.

DOVER PUBLICATIONS, INVESTIGATIONS ON THE THEORY OF THE BROWNIAN MOVEMENT

ON THE MOVEMENT OFSMALL PARTICLES SUSPENDEDIN A STATIONARY LIQUID DEMANDED BY T H E KINETIC THEORY OF HEAT N this paper it will be shown that according to the molecular-kinetic theory of heat, bodies of microscopically -visible size 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, thatI can form no judgment in the matter If the movement discussed here can actually be observed (together with the laws relating to OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES it that one would expect to find), then classical thermodynamics - atleastwhen the of gravity thermodynamics can no longer be looked upon (which does not interest us here) is ignored- we applicable 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. the liquid and the partition, and on the pressure and temperature. Actually, for the cal- I . ON THE OSMOTIC PRESSURE TO ASCRIBED culation the free energy the energy and entropy TO THE SUSPENDED PARTICLES the boundary -surface (surface -tension forces) Let a non-electrolyte be should also be considered these can be excluded dissolved in a volume forming part of a if the size and condition of the surfaces of contact quantity liquid of total volume If the do not alter with the changes in position the volume is separated from the pure solvent partition and of the suspended particles under a partition for the solvent but consideration. impermeable for the solute, a so-called osmotic But a different conception is reached from is exerted on this partition, which pressure,’’ the standpoint of the molecular-kinetic theory equation satisfies heat. According to this theory a dissolved mole - cule is differentiated from a suspended body when is sufficiently great. by its and it is not apparent On hand, if small suspended particles why a number of suspended particles should not are present in the volume in place produce the same osmotic pressure as the same the dissolved substance, which particles are also number of molecules. We must that the unable to pass through the partition permeable to suspended particles perform an irregular move- the solvent according to the classical theory ment- even a very the liquid, on THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES account the molecular movement of the liquid physical system which completely define the if they are prevented from leaving the volume 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 suspended particles present in the system the equationsof change of these variables volume and therefore in a unit of state is given in the form and if particles are suffi - ciently far separated, there willbe a corresponding osmotic pressure of magnitude given by whence where 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 osmotic pressure.

OSMOTIC PRESSURE FROM THE STANDPOINT OF THE MOLECULAR -KINETIC HEORY OF T is the absolute temperature, the energy HEAT where of the system, the energy as a function of If are the variables state of The integral is extended over all possible values In this paragraph the papers of the author on the of consistent with the conditions the prob - Foundations of Thermodynamics are assumed to be lem. is connected with the constant referred familiar to the reader p. We obtain p. An understanding the conclusions to before by the relation reached in the present paper is not dependent on a hence for the free energy knowledge of the former papers or of this paragraph the present paper. THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES let us consider a quantity of liquid enclosed lying wholly within a volume let there be solute molecules The value of the integral appearing in the (or suspended particles respectively) in the por- expression for will be sought, with the limita - tion this volume which are retained in tion the centres of gravity of the particles the volurne by a partition lie within a domain defined in this manner. The the integration limits the integral obtained integral can then be brought into the form in the expressions forS and will be affected dB accordingly. The combined. volume of the solute where is independent of etc., as well as molecules (or suspended particles) is taken as of the position of the semi -permeable This system will be small compared with partition. But is also independent of any defined according to the theory under special choice of the position of the domains of discussion by the variables of condition the centres of gravity and of the magnitude of the molecular picture were extended to deal as will be shown immediately. For if a with every single unit, the calculation of the second system were given,of indefinitely small integral would offer such difficulties that an domains of the centresof gravity of the particles, exact-calculation of could be contem- and the latter designated plated. Accordingly, we need here only to know which domains differ from those depends on the magnitude of the volume originally given in their position but not in their in which all the molecules, or suspended magnitude, and are similarly all contained in bodies (hereinafter termed briefly particles an analogous expression holds are contained. dB' We call the rectangularCo -ordinates of the centre gravity of the first particle, Whence x,,y,, those -the second, etc., those of thelast particle, and allocate for the centres Therefore of gravity of the particles the indefinitely small domains of form THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES But from the moleculartheory of Heat given in the and paper quoted, it is easilydeduced that dB (or 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 or 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 at geneous and exerts no force on the particles, then great dilution. for equal size of domains the probability of each THEORY THE DIFFUSION of the two systemswill be equal, so that the follow - SPHERES IN SUSPENSION holds Suppose there be suspended particles irregularly 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 acts on the single particles, which follows that force depends on the position, but not on the time. It will be assumed the sake of simplicity that We have thus proved that is independent both the force is exerted everywhere in the direction of and of By integration we the axis. obtain Let be the number of suspended particles per J. unit volume then in the condition of dynamic and thence equilibrium is such a function of that the varia - tion of the free energy vanishes for an arbitrary virtual displacement of the suspended sub - stance. We have, therefore, Ann. d. O. THEORY BROWNIANMOVEMENT MOVEMENTOF SMALL PARTICLES It will be assumed that the liquid has unit area process of diffusion, which is to be looked cross-section perpendicular .to the axis and upon as a result of the irregular movement of the is bounded by the planes and We particles produced by the molecular have, then, movement. If the suspended particles have spherical form (radius of the sphere P), and if the liquid has and a coefficient of viscosity then the force im- parts to the single particles a velocity

required condition of equilibrium is there - fore and there will pass unit area per unit of time

or particles. further, the coefficient of diffusion The last equation states that equilibrium with the the suspended substance, and the mass of a force 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 grams look upon the dynamic equilibrium condition con - sidered here as a superposition two processes proceeding in opposite directions, namely particles. of the suspended substance under the influence of the force acting on each Cf. Kirchhoff, Lectures on Mechanics, " single suspended particle. BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES there must be dynamic equilibrium, we different intervals time must be considered as must have mutually independent processes, so long as we think of these intervals of time as being chosen not too small. We can calculate the coefficient diffusion We will introduce a time-interval in our dis - from the two conditions(I ) and (2) found for the cussion, which is to be very small compared with dynamic equilibrium. We get the observed interval of time, but, nevertheless, of such a magnitudethat the movements executed N by a particle in two consecutive intervals of time 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 suspended par - constants and absolute temperature) only on ticles in a liquid. In an interval of time the the coefficient of viscosity of the liquid and on the -xCo-ordinates of the single particles will increase sizeof the suspended particles. by where has a different value (positive or negative) for each particle. For the of ON THE IRREGULAR MOVEMENT OF PARTICLES a certain probability -law will hold the'number SUSPENDED IN LIQUID AND THE RELATION the particles which experience in the time - OF THIS TO DIFFUSION interval a displacement which lies between will turn now to a closer consideration and will be expressed by an equation the irregular movements which arise from thermal the form molecular movement, and give rise to the diffusion investigated in the last paragraph. where Evidently it must be assumed that each single indepen particle executes a movement which is - and only differs from zero for very small values dent the movement all other particles the of and the condition movements one and the same particle after THEORY OF BROWNIAN MOVEMENT MOVEMENT OF SMALL PARTICLES

We will investigate now how the coefficient of the right -hand side the second, fourth, etc., diffusion depends on confining ourselves again vanish since whilst the to the case when the number of the particles per first, third, fifth, etc., terms, every succeeding unit volume is dependent only on and is very small compared with the preceding. Putting for the number particles per unit Bearing in mind that volume we will calculate the distri - bution the particles at a T from the distribution at the time From the definition the function +(A),there is easily obtained the and putting number of the particles which are located at the. time between two planes perpendicular to the x-axis, with and We get and taking into consideration only the first and on the right -hand side, we get from m third this equation Now, since T is very small, we can put

Further, we can expand in powers Thisis the well -known equation for A diffusion, and that is the diffusion. Another important consideration can be related We can bring this expansion under the integral to thismethod of development. We have sign, since only very small values contribute that the single particles are all referred to the anything to the latter. We obtain Co-ordinate system. But this is unneces - sary, since the movements the single particles are mutually independent. We now refer the motion of each particle to a THEORY OF BROWNIAN MOVEMENT OF SMALL PARTICLES system whose origin coincides at the time equation the displacement in the direc - with the position of the centre of gravity of the tion of the X -axis which a particle experiences particles in question with this difference, that average, or - more accurately expressed - the now gives the number of the particles root of the arithmetic mean the squares whose Co-ordinate has increased between the displacements in the direction the X-axis time o and the time by a quantity it is which lies between and In this case also the function satisfy, in its changes, the equation(I ). Further, we must evidently The mean displacement is therefore propor - tional to the square root the time. can have€or o and It easily be shown that the square root of the mean o and of squares of the total displacements of the particles has the value The problem, which accords with the problem of the diffusion outwards from a point (ignoring pos- FORMULA THE MEAN DISPLACEMENT OF sibilities of exchange between the diffusing par - SUSPENDED PARTICLES. NEW METHOD OF ticles) is now completely defined DETERMINING THE REAL SIZE OF THE the solution is In we found for the coefficient diffusion a suspended in a liquid in the form of small spheresof radius P- The probable distribution of the resulting dis - placements in a given time is therefore the same as thatof fortuitous error, which was to be ex - Further, we found in for the mean value the pected. But it is significant how the constants in displacementof the particles in the direction the exponential are related to the coefficient the X-axis in of diffusion. We now calculate with the help THEORY OF BROWNIAN' MOVEMENT By eliminating we obtain

This equation showshow dependson and P. II We will calculate how great is for one second, THE THEORY OF THE BROWNIAN is taken equal to in accordance with the if MOVEMENT kinetic theory of gases, water at C. is chosen as the liquid and the diameter of the particles mm. We get cm. OON after the appearance of my paper The mean displacement in one minute would be, the movements of particles suspended therefore, about liquids demanded by the molecular theory of On the other hand, the relation found can be (of Jena) informed me that he used for the determination of We obtain other physicists - in the first instance, Prof. I (of Lyons)- had been convinced by direct observation that the so-called Brownian motion is caused by the irregular thermal movements It is to be hoped that some enquirer may succeed the molecules of the liquid. shortly in solving the problem suggested here, Not only the qualitative properties of the which is so important with the Brownian motion, but also the order of magnitude theory Heat. (13) of the paths described by the particles correspond Berne, completely with the results the theory. I will (Received, II not attempt here a comparison the slender experimental material at my disposal with the d. de