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e This new Dover edition, first published in 1956, is an unabridged .and unaltered republication of the translation first: published in 1926. It is published through special arrangément with Methuen and Co., Ltd., and the estate of Albert Einstein. Manufactured in the United' States .OF ,THE BROWNIAN MOVEMENT BY ALBERT EINSTEIN, PH.D. EDITED WITH NOTES BY R. FüRTH TRANSLATED BY A. D. COWPER WITH 3 DIAGRAMS DOVER PUBLICATIONS, INC. l. INVESTIGATIONS ON THE THEORY OF THE BROWNIAN MOVEMENT I ON MOLECULAR- I suspended perform movements easily observed Brownian molecular motion ” ; however, the information available to me (I). here can be observed 2 THEORY no longer be looked as applicable with bodies even : exact determination actual atomic #en possible. other hand, had movement proved be incorrect, a kinetic conception 3 I. ON THE OSMOTIC PRESSURE TO BE ASCRIBED TO THE SUSPENDED PARTICLES Let z gràm-molecules of a nonelectrolyte be V* forming part of a quantity of liquid of total volume V. If the volume V* is separated from the pure solvent by a partition permeable solute, a “ osmotic pressure,’’ 9, exerted on this partition, the equation $V*= RTz . (4 when V*/z is sufficiently great. On the.other hand, if small suspended particles are present in the fractional volume V* in place of pass through : according to the classical theory of MOVEMENT OF SMALL PARTICLES 3 thermodynamics-at least when the foreë not interest us for according “ free energy ” partition and material, and the partition, and and temperature. Actually, for of and entropy of tension forces) should also ; can be do not alter of partition and particles under a different conception of differentiated from a soZeZy dirhensions, not apparent a number same osmotic pressure molecules. We assume that the suspended particles perform an irregular move- ment-even if a very slow one-in liquid, on 4 THEORY OF BROWNIAN MOVEMENT account of molecular movement ; they are V* a pressure solution. Then, fi suspended particles present in the volume V*, and therefore %/'V* = V in a unit .of volurne, and if neighbouring a corresponding fi of magnitude given by RTn RT where N next paragraph of osmotic pressure. pz-- V*N"Ñ'v' fj-2, OSMOTIC PRESSURE FROM THE STANDPOINT OF THE MOLECULAR-KINETIC THEORY OF HEAT (*) If pl, P,, . . . @J are the variables of state of (*) In this paragraph the papers of the author on the '' Foundations of Thermodynamics " are assumed to be familiar to the reader (Ann. d. Phys., 9, p. 4r7, 1902 ; 11, p. 170, 1903). An understanding of present paper is not dependent former papers of present paper. 5 a. completely define of the equations of change of these variables of state is given in the form ?& 3t = +.(pl . pl) (V = I, 2, * . . Z) is given T absolute temperature, E the energy of the system, E a function fiv. The integral extended over I of 9. consistent with of the prob- lem. x connected with N referred to before by the relation zxN = R. free energy F, 6 BROWNIAN MOVEMENT Now let us consider a quantity of liquid enclosed in a volume V ; let there n suspended particles respectively) 'V* of this volume 'V# which are retained in the volurne V* by a semi-permeable partition ; integration limits of the integral B obtained in the expressions for S and F molecules (or V*. This system completely theory under pl . . . pl. If with every B would offer that an F could be contem- plated. Accordingly, we need here only to know how F depends on the magnitude of the volume V*, in which all the solute molecules, or suspended bodies (hereinafter termed briefly " particles are contained. We will call x,, yI, x, the rectangular Co-ordinates of the centre of first particle, x, those of -the second, etc., x,, y,, x, allocate for indefinitely small parallelopiped form dg,, dy,, dzl ; dxzt MOVEMENT OF SMALL PARTICLES 7 dy2, dz,, . . . dx,, dy,, dzn, lying wholly within V*, integral appearing F sought, with tilat lie within a domain integral can dX1 dyl dZn . J, where J is independent of axl, dy,, etc., as well as of V*, i.e. of the position J gravity and V*, indefinitely small dx:dyl'dzl' ; dx;dy,'dz[ . . . dx,'dyn'dzn', differ from in their not in their V*, analogous expression holds :- dB' = dxl'dy; . . . dz,,' , J'. Whence dXIdy1 dzn = dxl'dyl' . . . dza'. Therefore dB J dB' - TfT -- - 8 BROWNIAN MOVEMENT given in paper quoted, (*) it is easily deduced that dB /B (4) (or dB'/B included in (dx, . . . dz,) or (2%: . . . dzn') and exerts will be equal, so ing holds : dB dB' B B' But from this and the last equation obtained it follows that J = J'. We have thus proved that J is independent both of V* and of x,, yr, . . . x,. By integration we obtain -x- B = /]dxl. . . dzn = J. V*n, and thence (*) A. Einstein, Ann. d. Phys., 11, p. 170, 1903. MOVEMENT OF SMALL PARTICLES g and 3F RT RT p=-alr*=Y*lCi=Nv* been shown pressure can be ; as far theory, identical behaviour at great dilution. 6 3. THEORY OF THE DIFFUSION OF SMALL there be particles irregularly dispersed in dynamic equilibrium, on K will be assumed for force is of the x axis. Let Y be the number of suspended particles per unit volume ; then in V is such x that the energy vanishes Sx have, therefore, 8F = 8.E - TSS O. IO THEORY OF BROWNIAN MOVEMENT It will be assumed that the liquid has unit area of cross-section perpendicular .to the x axis and x = o have, then, 6E = - {:Kv6xdx and and x = 1. We The required condition av -KV+ --- N ax-O The last equation equilibrium with K brought about can be dynamic equilibrium condition sidered here a superposition of opposite directions, namely i- I. A movelment of the suspended substance under the influence of the force K single suspended II 2. A by the thermal particles have spherical form = P), and if the liquid has a coefficient of viscosity k, then the force K a velocity (*) and there will pass a unit area vK 67rkP particles. If, further, D signifìes the coefficient of diffusion substance, and p area in - D'M grams bX or 3V 3X - D- particles. (*) Cf. e.g. G. Kirchhoff, " Lectures on Mechanics," Lect. 26z 8 4. 12 TE3:EOR.Y OF BROWNIAN MOVEMENT Since there must be dynamic equilibrium, VK 31, =P- 3x D- = O. can calculate of dynamic equilibrium. RT I D=------ N 61rkP * The coefficient of diffusion of thk absolute temperature) fj 4. ON THE IRREGULAR MOVEMENT OF PARTICLES SUSPENDED IN A LIQUID AND THE RELATION OF THIS TO DIFFUSION We will turn now to a closer consideration of give rise investigated in last paragraph. each single of the movement of all other particles ; the movements of one and the same particle after MOVEMENT OF SMALL PARTICLES ~3 different intervals of time must these intervals being chosen T in our disbe very small compared with such a movements executed particle in two consecutive r mutually independent there are altogether n r the x-Co-ordinates of the single particles will increase by d, where d a different value (positive negative) for value of d will hold ; d% of experience in r a displacement d and d + dA, by an of the form where dn = n+(A)d& [+OO+(A)dd -00 = I and + from zero for very small values d and fulfils the condition 14 BROWNIAN MOVEMENT We will 4, confining ourselves again to the case when the number V particles per is dependent x and t. Putting for the number of particles per unit volume V = f(x, t), we will calculate the distri- bution of the particles at a time t + T from the distribution at the time t. From the definition of the function +(A), there is easily obtained the number of the particles which are located at the. time t + T two planes perpendicular abscissz! x and x + ax. We get f(x, t + 7)dx = dx. J f(x + A)#(A)dA. A==+ m A= - can expand j(x + d, t) in powers of A :- since only very small values of A contribute anything to the latter. We obtain f-/--~o r=fj Q(d)dA+jfS d+(A)dA +m ax +W -m -00 MOVEMENT OF SMALL PARTICLES 15 On fourth, etc., terms vanish since +(x) = #(- x) ; whilst of third, fifth, etc., terms, telm very small Bearing in +m and taking into consideration only the first and third terns on the right-hand side, we get from this equation This is the well-known differential equation for we recomise that D is the coeecient of diffusion. Another important consideration can assumed that the single particles are all referred to the same Co-ordinate system. But this is unneces- sary, since the movements of are mutually independent. wilI now refer the motion of each particle to a Co-ordinate 16 origin coincides t = o with the position of the centre of gravity of the particles in question ; with this difference, that f(x, t)dx now gives the number of the particles whose x increased between t = o and the time t = t, by a quantity which lies between x and x + dx. In this case also the function f must satisfy, in x � o and t = o, f(x, t) = o and [+wj(x, --m must evidently t)dx = n. accords with a point mathematically completely defined (9) ; the solution is xcr -- resulting displacements in a t to be constants in term are related wil€ calculate with MOVEMENT OF SMALL PARTICLES 17 of this equation the displacement Xz in the direc- tion of the X-axis which a particle experiences on an average, or-more accurately expressed-the square root of the arithmetic mean of the squares of displacements in the direction of the X-axis ; it is Aa = 45 = JZt . ' (11) . therefore propor of the time. It can easily be shown that the square root of the mean of the squares of the total displacements of the particles has the value &J3 . . (12) 5 5. FORMULA FOR THE MEAN DISPLACEMENT OF SUSPENDED PARTICLES. A NEW METHOD OF DETERMINING THE REAL SIZE OF THE ATOM In 5 3 found for of diffusion of a material suspended in liquid in small spheres 5 4 mean value of the displacement of the particles in the direction of the X-axis in time t- Aa = Jak 18 THEORY OF BROWNIAN' MOVEMENT By eliminating D This equation )C, depends on T, k, and P. We will calculate how great & is for one second, N is taken equal 6.10~3 17" C. is chosen as the liquid (K = 1-35 IO-^), and the diameter of the particles -001 mm. We get &, = 8*10-~ cm. = 0.8,~. The mean displacement in one minute would be, therefore, about 6p. found can N. We obtain I RT ha2 3wkP' N=-*- problem suggested in cqnnection with the theory of Heat. (13) Berne, May, 1905. (Received, II May, 1905.) II ON THE THEORY OF THE BROWNIAN MOVEMENT S OON after the appearance of (*) on the movements of particles suspended in liquids demanded by the heat, Siedentopf (of Jena) informed me that he and instance, Prof. Gouy been convinced by the irregular thermal of the molecules of the liquid. (t) Not only qualitative properties of the theory. I will not attempt here a comparison of disposal with (*) A. Einstein, Ann. d. Phys., 17, p. 549, 1905. (t) M. Gouy, Jouyn. de Phys. (z), I, 561, 1888. 19