复旦大学：《大学物理——热学》课程教学资源（PPT课件）第6次课

大学物理(热学) 金晓峰 复旦大学物理系 xijin@fudan.edu.cn 2006-928 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 2006-9-28

gas moving about in a cubical of a FIGURE 18-1 (a) Molecules container. (b) Arrows indicate the momentum of on molecule as it rebounds from the end wall (a) (b)

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(x)dx, where f(x) 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 M(x)0)(2kx劬d 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))f(z)=(x2+y2+z2) Solving this functional equation, we find f(e C d(r2) 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, Nvzx=M,∴C≈、l av兀

l54 SELECTED READINGS IN PHYSICS. KINETIC THEORY f(r)is therefore (ra) v Whence we may draw the following conclusions: Ist. The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is N v 2nd. The number whose actual velocity lies between v and u + du 4 N v2e-(o/adu 3rd. To find the mean value of D, add the velocities of all the particles together and divide by the number of particles; the result is mean velocity 4th. To find the mean value of u2, add all the values together and divide by N mean value of v This is greater than the square of the mean velocity, as it ougl

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