大学物理(热学) 金晓峰 复旦大学物理系 xijin@fudan.edu.cn 2006-9-21 2021/9/5
2021/9/5 1 大学物理（热学） 金晓峰 复旦大学物理系 xfjin@fudan.edu.cn 2006-9-21
y=f(r 0
y 2(, f(x)) P(a, f(a) f(x)-fla x-a 0 a x
y Q P 0 x
mI m2 vI 2 n 2 2 Fig. 39-2. Atoms of two different monatomic gases are separated by a movable piston
m1 v1 n1 m2 v2 n2
Fig. 39-3. A collision between un equal molecules, viewed in the CM system
MAXWELL: THE DYNAMICAL THEORY OF GASES so that the probability is independent of that is, all directions of Let N be the whole number of particles. Let x, y, z be the com ponents of the velocity of each particle in three rectangular direc- Prop. III. Given the direction and magnitude of the velocities of tions, and let the number of particles for which x lies between x and wo spheres before impact, and the line of centres at impact; to find x+ dx, be N/(x)dx, where f(x)is a function of x to be determined the velocities after impact. The number of particles for which y will be Nf)dy tween y and y+ dy Let 04, OB represent the velocities before impact, so that if there e number for which z lies between z and had been no action between the bodies they would have been at A and B at the end of a second, Join AB and let g be their centre of t dz will be Nf(z)dz, where f always stands for the same function. Now the existence of the velocity x does not in any way affect that of the velocities y or z, since these are all at right angles to each other and independent, so that the number of particles whose velocity lies between x and x t dx, and also between y and y dy, and also between z and z+dz, is Nf(ruf(z)dx dydz If we suppose the N particles to start from the at the same instant, then this will be th (dx dy dz)after unit of time gravity,the position of which is not affected by their mutual action d to unit of volume will be Draw GN parallel to the line of impact(not necessarily in the plane 4OB). Draw aGb in the plane AGN, making NGa = NGA N(x)Uy)八(z) and Ga GA and Gb= GB; then by Prop I Ga and Gb will be But the directions of the coordinates are perfectly arbitrary, and the velocities relative to G; and compounding these with OG, we herefore this number must depend on the distance from the origin have Oa and Ob for the true velocities after impact. lone, that is By Prop. II. all directions the line aGb are equally fA()f(e)=p(x2+y2 +22) It appears therefore that the velocity after impact is comp the velocity of the centre of gravity, and of a velocity equ m Solving this functional equation, we find velocity of the sphere relative to the centre of gravity, which may with equal probability be in any direction whatever If a great many equal spherical particles were in motion in a If we make A positive, the number of particles will increase with perfectly elastic vessel, collisions would take place among the the velocity, and we should find the whole number of particles particles, and their velocities would be altered at every collision; so infinite. We therefore make A negative and equal to -1 a2, so that that after a certain time the vis viDa will be divided among the the number between x andx dy particles according to some regular law, the average number of NCe-( /an) dx particles whose velocity lies between certain limits being ascertain able, though the velocity of each particle changes at every collision Integrating from x =-ao to x =+ oo, we find the whole number Prop. iV. To f nd the average number of particles whose velocities of particles tween given limits, after a great number of collisions among a number of equal particles N,∴C
SELECTED READINGS IN PHYSICS: KINETIC THPORY E DYNAMICAL THEOR f()is therefore e-(/a there be n particles of the first system, and Nof the second Whence we may draw the following conclusions: then NN is the whole number of such pairs. Let us consider the Ist. The number of particles whose velocity, resolved in a certain velocities in the direction of x only; then by Prop. IV. the number of the first kind, whose velocities are between x and x + dx, is direction, lies between x and x +dx is la)dx 2nd. The number whose actual velocity lies between u and D The number of the second kind, whose velocity is between x +y and x+y+dy,is (2) 1。(x+》P rd. To find the mean value of v, add the velocities of all the where B is the value of a for the second system articles together and divide by the number of particles; the result is The number of pairs which fulfil both conditions is mean velocity =2a NN Br e-t/e*at?"8)dx dy 4th. To find the mean value of u2, add all the values together and ivide by N Now x may have any value from -oo to ao consistently with the mean value of u2=影2 difierence of velocities being between y and y +dy; therefore itegrating between these limits, we find This is greater than the square of the mean velocity, as it ought It appears from this proposition that the velocities are distributed NNa+I among the particles according to the same law as the errors are distributed among the observations in the theory of the"method of for the whole number of pairs whose difference of velocity lies be- least squares. "The velocities range from 0 to oo, but the number of tween y and y+小 those having great velocities is comparatively small. In addition to This expression, which is of the same form with(1)if we put NN these velocities, which are in all directions equally, there may be a for N, 2+B2 for a2, and y for x, shews that the distribution of general motion of translation of the entire system of particles which relative velocities is regulated by the same law as that of the velocities must be compounded with the motion of the particles relatively to themselves, and that that the mean relative velocity is the square root of one another. We may call the one the motion of translation, and the sum of the squares of the mean velocities of the two systems. the other the motion of agitation Since the direction of motion of every particle in one of the systems Prop. V. Two systems of particle may be reversed without changing the distribution of velocities, it iw stated in Prop. IV. to find the number of pairs of particles, one that the velocities compounded of the velocities of two of each system, whose relative velocity lies between given limits es, one in each system, are distributed according to the same ula(5)as the rel
MAXWELL: THE DYNAMICAL THEORY OF GASES Phil Mag. 19(1860)19153 Let n be the whole number of particles. Let x, y, z be the com ponents of the velocity of each particle in three rectangular direc- tions, and let the number of particles for which x lies between x and x + dx, be Nf(r)dx, where f(r) is a function of x to be determined The number of particles for which y lies between y and y dy will be Nfdy; and the number for which z lies between z and z dz will be Nf(z)dz, where f always stands for the same function Now the existence of the velocity x does not in any way affect that of the velocities y or z, since these are all at right angles to each other and independent, so that the number of particles whose velocity lies between x and x dx, and also between y and y t dy, and also between z and z dz, is Nf()fo)f(z)dx dy dz If we suppose the N particles to start from the origin at the same instant, then this will be the number in the element of volume (dx dy dz) after unit of time, and the number referred to unit of volume will be Nf(fo)f(z)
Phil. Mag. 19 (1860) 19
But the directions of the coordinates are perfectly arbitrary, and therefore this number must depend on the distance from the origin alone, that is f(x)jUy)∫(z)=φ(x2+y2+z) Solving this functional equation, we find fx)=Ce,φ2)=C3e If we make A positive, the number of particles will increase with the velocity, and we should find the whole number of particles infinite. We therefore make A negative and equal to- 1/, so that the number between x and x+ dx is NCe-la)dx Integrating from x=-oo to x =+oo, we find the whole number of particles, NCvr=M,∴C≈、l a√兀