Collisions of A with Stationary B. We want to find # collisions per unit time coll/ T = Z, so first lets get number of collisions of A

Transcription

1 Collisions of A with Stationary B We want to find # collisions per unit time coll/t = Z, so first lets get number of collisions of A

2 Collision Frequency The collision frequency Z A for one A molecule with N B B is Z A = d AB v A N B /V, s -1 When we have a density N A /V of A molecules, we express the total rate of A-B collisions as 1 d v N N AB A A B s Z AB 3 V cm Z AB is the Collision Density We have still ignored the fact that the B molecules are moving. Thus we need to replace v A with the average relative speed between A and B, v rel.

3 Going from one A molecule to N A A molecules/cm 3 When we have a density N A /V of A molecules, we express the total rate of A-B collisions as Z AB d v N N s V cm 1 AB A A B 3 Z AB is the Collision Density

4 Average Relative Velocity We have still ignored the fact that the B molecules are moving. Thus we need to replace v A with the average relative speed between A and B, v rel. <V B > <V A > 1 rel A B v v v

5 Replace v with relative velocity The A-B collision density for N A /V molecules with N B /V is Z AB 1 AB A B A B d v v N N V

6 We found The Simpler Case of A Pure Gas Z A d v N for a single A molecule with B molecules. If the B molecules become A s and are allowed to move, then and d AB becomes d A 1 A A A A v v v v So for a one component gas, AB A B V s 1 Collision frequency and the collision density Z A d v N A A A 1 s V d v N Z m s AA V A A A 3 1

7 Mean Free Path Mean Free Path () = Distance Traveled in unit time Number of Collisions in unit time u A d A uan A / V V d A N A

8 ConcepTest #1 In which of the following systems will the mean free path () be the shortest? A. Pure CCl 4, 1 Pa pressure B. Pure CCl 4, 10 Pa pressure C. Pure He, 1 Pa pressure D. Pure He, 10 Pa pressure

9 ConcepTest #1 In which of the following systems will the mean free path () be the shortest? A. Pure CCl 4, 1 Pa pressure B. Pure CCl 4, 10 Pa pressure C. Pure He, 1 Pa pressure D. Pure He, 10 Pa pressure

10 Collision Flux Collision Flux Number of collisions with an imaginary area in a given time interval, divided by the area and the time interval We already know that Collision Frequency is the number of hits per second, so if we just divide this by the area, we get the flux Consider a gas of A molecules with average speed v A and number density n A = N A /V. One can show (but we will not) that the flux, F, of molecules passing in one direction through a plane of unit area is given by n F A v 4 A molecules cm s 1

11 Molecular Speed Distributions There is a key, universal, fundamental concept: Mother Nature does not play favorites, but rather selects in an unbiased manner from the allowed outcomes. At any given temperature (total energy), collisions between gas molecules lead to exchange of energy between the molecules. However, when there are a large number of particles, the distribution function describing the molecular energies (or velocities) is essentially constant, at the equilibrium distribution.

12 General Principle!! Energy is distributed among accessible configurations in a random process. The ergodic hypothesis Consider fixed total energy with multiple particles and various possible energies for the particles Determine the distribution that occupies the largest portion of the available Phase Space. That is the observed distribution.

13 Possible arrangements of particles with fixed total energy While the real systems of interest typically have ~N A particles, we illustrate the concept on an embarrassingly small, but simple, system. Consider a system comprised of 4 particles. Any particle can have only discrete energies: E = 0, 1,, 3, 4... (quantized). Let the total energy of the system be 3. What is the average energy per particle, E? A. 3 B. 4 C D. 1 E. Not enough information given

14 Possible arrangements of particles with fixed total energy While the real systems of interest typically have ~N A particles, we illustrate the concept on an embarrassingly small, but simple, system. Consider a system comprised of 4 particles. Any particle can have only discrete energies: E = 0, 1,, 3, 4... (quantized). Let the total energy of the system be 3. What is the average energy per particle, E? A. 3 B. 4 C D. 1 E. Not enough information given

15 Possible arrangements of particles with fixed total energy Consider a system comprised of 4 particles. Any particle can have only discrete energies: E = 0, 1,, 3, 4... (quantized). Let the total energy of the system be three. What are the possible arrangements of energy? Energy I II. III. IV.

16 Possible arrangements of particles with fixed total energy Consider a system comprised of 4 particles. Any particle can have only discrete energies: E = 0, 1,, 3, 4... (quantized). Let the total energy of the system be three. With all possible arrangements enumerated, We make the ergodic assumption, namely that the system visits all of the available configurations with equal probability. What is the average occupancy of each energy level? Energy I II III Totals/ Plot n(e)vs. E Plot n(e) vs. E

17 Possible arrangements of particles with fixed total energy Energy I II III Totals/ Plot n(e) vs. E n(e) f( E) 3 E 0.75 e Energy Energy randomization leads to an exponential distribution of occupied energy levels at constant total energy

18 Maxwell distribution of molecular speeds The probability that the molecular velocity is between v x & v x +dv x, v y & v y +dv y, and v z & v z +dv z is given by dp x dp y dp z = B 3 exp[-mv x /k B T] x exp[-mv y /k B T] x exp[-mv z /k B T] dv x dv y dv z This expression simplifies using u = u x +u y +u z mv 3 kt B x y z x y z dp dp dp B e dv dv dv But we must convert to spherical polar coordinates to get the volume element in v!

19 Spherical Polar Co-ordinates

20 Maxwell distribution of molecular speeds Volume element (dv x dv y dv z ) transformation from cartesian to spherical polar coordinates: x = r sin cos, y = r sin sin z = r cos Volume element transform: dv x dv y dv z 4v dv mv 3 kt B 4 dp B v e dv For a normalized distribution, 0 P v dv 1 We use this constraint to set B, giving 3 3 mv Mv kt M B RT m Pvdv 4 ve dv 4 ve dv kt RT B

21 Characteristics of P(v) Mv M RT P v dv 4 v e dv RT 3

22 Characteristics of P(v) Mv M RT P v dv 4 v e dv RT 3

23 ConcepTest Maxwell speed distributions are shown for three situations. (I) If the curves represent 3 different gases at the same T, which curve shows the gas of greatest molar mass? (II) If the curves represent the same gas at 3 different T, which curve shows the highest T? xxxxxxxxxxxxxx blue green red xxxxxxxxxxxxxxxxxxxxxxxxxxxxx 400 m/s 800 m/s 100 m/s 1600 m/s A. blue; blue C. red; blue B. blue; red D. red; red

24 ConcepTest Maxwell speed distributions are shown for three situations. (I) If the curves represent 3 different gases at the same T, which curve shows the gas of greatest molar mass? (II) If the curves represent the same gas at 3 different T, which curve shows the highest T? xxxxxxxxxxxxxx blue green red xxxxxxxxxxxxxxxxxxxxxxxxxxxxx 400 m/s 800 m/s 100 m/s 1600 m/s A. blue; blue C. red; blue B. blue; red D. red; red

25 Using average speed of A, ua Mixture of A and B Using average relative speed of A and B, 1/ u AB u A u B Number of collisions of one molecule of A with B molecules (per unit time); SI unit: s 1 Z A d ABuAN B V Z A d AB ua ub 1 N B V Total number of A-B collisions (per unit volume per unit time); SI unit: m 3 s 1 Z AB d ABuAN A N B V Z AB Using average speed of A, ua Pure A d AB ua ub V Using average relative speed, u AA u A 1 N A N B Number of collisions of one molecule of A with A molecules (per unit time); SI unit: s 1 Z A d AuAN A V Z A d AuAN A V Total number of A-A collisions (per unit volume per unit time); SI unit: m 3 s 1 Z AA d AuAN A Z V AA d A V uan A

26 PhET Kinetic Theory Simulation properties Gas Properties PhET Simulations