Topics covered so far:

Transcription

1 Topics covered so far: Chap 1: The kinetic theory of gases P, T, and the Ideal Gas Law Chap 2: The principles of statistical mechanics 2.1, The Boltzmann law (spatial distribution) 2.2, The distribution of speeds (Maxwellian distribution) 2.3, The Equipartition of energy 1

2 Class 4 Previous class: * Maxwellian distribution f(v x,v y,v z )dv x dv y dv z / e mv2 x /2kT e mv2 y /2kT e mv2 z /2kT * Equipartition of energy Each degree of freedom has an average kinetic energy of 1/2 kt. * For a molecule with r, s, l degrees of freedom for translational motion, rotational motion, and vibrational motion, the average kinetic energy of all molecules is <K.E.> = 1/2 kt * (r+s+l). Treating vibrational motion as that of a harmonic oscillator, then the average total energy is <E> = 1/2 kt * (r + s + 2*l) * U = N * <E> = > NkT = 2 U / (r+s+2l) = PV = (ɣ-1)u => ɣ = 2/(r+s+2l) +1 For diatomic molecules, we showed that experiments do not support this classical theory. The resolution of the difficulty could be found from Quantum Mechanics. 2

3 2.3 (cont d) Equipartition of energy Fig (a) A sensitive light-beam galvanometer. Light from a source L is reflected from a small mirror onto a scale. (b) A schematic record of the reading of the scale as a function of the time. 3

4 Kinetic energy: T = 1/2 mv 2 Assuming it s a harmonic oscillator: V = 1/2 kx 2 In thermal equilibrium: 4

5 Chap 3: Diffusion and Drift Two important means of transport of molecules 3.1 The Brownian motion Background The Brownian movement was discovered in 1827 by Robert Brown, a botanist. While he was studying microscopic life, he noticed little particles of plant pollens jiggling around in the liquid he was looking at in the microscope, and he was wise enough to realize that these were not living, but were just little pieces of dirt moving around in the water. Exact mechanism was explained by Albert Einstein in The explanation served as definitive confirmation that atoms and molecules actually exist. Papers published by Einstein in 1905 (The Miracle Year ): On a Heuristic Viewpoint Concerning the Production and Transformation of Light (June 9) On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat (July 18) On the Electrodynamics of Moving Bodies (Sept 26) Does the Inertia of a Body Depend Upon Its Energy Content? (Nov 21) 5

6 Reproduced from the book of Jean Baptiste Perrin, Les Atomes, three tracings of the motion of colloidal particles of radius 0.53 µm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2 µm). 6

7 3.2 The random walk (a stochastic process) Fig. The progress made in a random walk. The horizontal coordinate N is the total number of steps taken; the vertical coordinate DN is the net distance moved from the starting position. 7

8 We know that the average distance is <DN> = 0. So we normally look for < DN > or <DN 2 >, the latter is the conventionally used one. We know: And xn = xn-1+1 or xn-1-1 with 1/2 probability. Note: Replace D by x, save D for the diffusion coefficient So: So: The root-mean-squared distance 8

9 Or if each step takes t time, then we have <xn 2 > = N t = t. Traditionally we define a diffusion coefficient D as, in this case, D = <xn 2 >/2t = 1/2. (the meaning of diffusion will be explained later). Now let s get back to the Brownian motion in 1 dimension: Like the case analysed above, we are not interested in <x> which is about 0, but we look for <x 2 >, averaged over a large number of particles. Now Newton s ma = F. m d2 x 2 = dx + F ext the viscous drag the fluctuating force representing the incessant impacts of the molecules of the liquid on the Brownian particle. 9

10 Since we are interested in <x 2 >, we multiple both sides of the equation by x and obtain mx d2 x 2 = xdx + xf ext m " d x dx dx 2 # = 2 dx 2 + xf ext And now take an average over all Brownian particles, we have m 2 d 2 hx 2 i 2 hmv 2 xi = 2 dhx 2 i + hxf ext i = 0, because of F is fluctuating and is independent of x 10

11 One can also solve the following equation directly: m 2 d 2 hx 2 i 2 kt = 2 dhx 2 i which can be written as m 2 dy + 2 y = kt where dhx 2 i y This equation has a general solution in the form is about 0 dhx 2 i y = 2kT + C exp ( t/m) m/ 10 8 s by Langevin (very small) 11

12 So we have dhx 2 i = 2kT the diffusion coefficient: D = 1 2 dhx 2 i = kt 12

13 3.3 The evolution of the distribution function (Brownian motion, Einstein s explanation) Consider a one dimensional system in x. The Brownian particles are constantly moving around randomly. Now choose a time interval τ that is much smaller than the time scale of interest, but larger enough so that in two successive time intervals, the motions executed by the particle can be thought of as events which are independent of each other. (a Markov process) For a particle, the x-coordinate of Brownian particles will increase by an amount. And has a probability distribution ɸ Z 1 1 ( )d =1 and the number of dn of particles which experience a shift which is between and +d dn = n ( )d ( )= ( ) 13

14 Let f(x,t) be the number of particles per unit volume. Z 1 f(x, t + )dx =dx 1 f(x +,t) ( )d f(x, t + ) @f(x, t) f(x +,t)=f(x, t) = f Z 1 1 ( Z 1 1 ( )d 2 Z ( )d 1 Z ( )d D 14

15 = 2 A diffusion equation If all particles start from the origin (x=0) at t=0, then f(x, t) = p n e x2 /4Dt p 4 D t 15

16 f(x,t) Fig. The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a Dirac delta function, indicating that all the particles are located at the origin at time t=0, and for increasing times they become flatter and flatter until the distribution becomes uniform in the asymptotic time limit. (Wikipedia) 16