Kinetic Theory of Gases. Chapter 33 1/6/2017. Kinetic Theory of Gases

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1 1/6/017 Kinetic Theory of Gases Kinetic Theory of Gases Chapter 33 Kinetic theory of gases envisions gases as a collection of atos or olecules in otion. Atos or olecules are considered as particles. This is based on the concept of the particulate nature of atter, regardless of the state of atter. A particle of a gas could be an ato or a group of atos (olecule). For gases following the relationship, PV = nrt (IGL); Observations KT Postulate Gas density is very low Particles are far apart Pressure is unifor in all directions IGL is independent of particle type Particle otion is rando Gas particles do not interact The Kinetic Theory relates the 'icro world' to the 'acro world. Dalton s Law of Partial Pressures nrt P V Gas particles do not interact KT (IGL): Applicable when particle density is such that the inter-particle distance >> particle size (point asses). Low pressures and high teperatures e.g 1at and roo tep. Postulates Gas particles are very far apart. Gas particles in constant rando otion. Gas particles do not eert forces on each other due to their large interolecular distances. Pressure in a gas is due to particle collisions (elastic) with the walls of the container fro translational otion - the icroscopic eplanation of pressure. Collisions with the wall are elastic, therefore, translational energy of the particle is conserved with these collisions. Each collision iparts a linear oentu to the wall, which results the gaseous pressure. In Newtonian echanics force defined as the change of oentu, here, due to the collision; pressure is force per unit area. In KT, the pressure arising fro the collision of a single olecule at the wall is derived and then scaled up to the collection of olecules in the container, to obtain the ideal gas law (IGL); PV = nrt 1

2 ()1/6/017 Gas kinetic theory derives the relationship between root-ean-squared speed and teperature. The particle otions are rando, therefore velocities along all directions are equivalent. Therefore the average velocity (vector) along any diension/direction will be zero. Now, the root-ean-squared velocity = root-ean-squared speed ; it is nonzero. A distribution of translational energies; therefore, any velocities would eist for a collection of gaseous particles. What is the distribution of the particle velocities? Velocity is a vector quantity (v). Speed is a scalar (). v = v v = v + v y + v z v = = + y + z <v > = < > ; sybol <> = average of. Most properties of gases depend on olecular speeds. The translational oveents of particles are aenable to treatent with classical Newtonian echanics (Justification, later). Root ean square velocity, translational energy: v v v v y z by definition v v v Therefore, y z rando otion v = v + v + v = v + v + v y z 3 v 3v 3 v y z v kt Using k T v 3 v 3 v kt Assue: (Chapter 31) v kt / 3kT = 3kT k T; v v 3 reeberlastslide1 For a gas saple of n oles occupying a volue V (cube), with an area of each side A. Consider a single particle of ass, velocity v. Particle collides with the wall. (elastic collisions) t Nuber density of particles = N nn V where N A = Avagadro Nuber Half of the olecules oving on ais with a (velocity coponent in the direction) within the volue v t collides with one surface in the direction. A v t Change of oentu p = v( v) v v ( v ) v N coll = nuber of collisions on the wall of area A in tie t.

3 1/6/017 Change of oentu on a surface during t = Pressure = Force per unit area = Force on the surface = Rate of change of oentu = Pressure arises because of the olecular otion of gases; icroscopic/olecular odel of pressure. PV = nrt sing k T ukt v Ideal gas law- IGL Pressure: icroscopic/epirical odel. Thus the kinetic theory describes the pressure of an ideal gas using a classical description of the otion of a single olecular collision with the walls and then scaling this result up to acroscopic proportions. The fact that coponent velocities of all olecules are not the sae, necessitates the definition of an average in each direction. Thus <v j > arises because of a probability distribution of v j values f(v j ) in each direction (j =, y, z) velocity distribution function eing a probability function; all ν j f( v ) dv 1 j j Derivation of distribution functions f(v i ) ln ( v, v, v ) ln f( v ) ln( v ) ln( v ) y z y z v v (v, v, v ) y z also fro another relationship v = v + v + v y z v [v + v + v ] 1 / y z for a olecule Chain rule Math suppleent 3

4 1/6/017 In general? ecause derivatives of three independent variables are equal, the derivatives ust be constant, say = - ;(>0). Upon rearrangeent and integration, Siilarly where A = integration constant Note the distribution (probability) function! Evaluating A: Distribution function: probability of a gas particle having a velocity within a given range, e.g. v and v +dv. Distribution function even function (Assuption slide 9) Math suppleent Mean/average averaging Now, Substituting for in f (v j ); where kt use tables Math suppleent 4

5 1/6/017 F( v) velocity distribution function Deriving the distribution function for v Changing the volue eleent (in Cartesian) to variable v, spherical coordinates. replace dv dv dv by 4 d y z and v vy vz by v rise eponential decay Notice the shape, blue. ave s of Xe, H, He? Earth and Jupiter (300) 5

6 and 1/6/017 and fied varied t v t rise eponential decay Notice the shape, blue. ave s of Xe, H, He? Earth and Jupiter (300) At lower angular velocities slower oving olecules go through the second slit. Detector signal proportional to the nuber of particles reaching the detector. Most probable velocity v p Most probable velocity v p : differentiate F(v), set to zero. Notice the shape. Mean (average) velocity: Root ean square velocity: Using: (Assuption slide 4) v v v v 3 v y z v 3 v v rs 6

7 1/6/017 Nuber of collisions per unit tie on the wall of area A: average coponent of velocity <v > = # olecules in light blue volue of the cube colliding per unit tie = N c = nuber of particles colliding. A Rate of collisions on surface = <v > Collisional Flu Z c : Nuber of collisions on the wall per unit tie per unit area. & Substituting in Z c ; Particle collision rates: (Hard sphere odel) Particles interact when spheres attept to occupy the sae region of the phase. (consider one oving particle orange, label 1; all other particles stationary are red label ecause the collisional partners are oving too in reality, an effective speed, <v 1 >, of orange particle will be considered in the odel to eulate the collisions the orange particle encounters; = collisional cross-section. V v dt cyl ave 7

8 1/6/017 ecause the collisional partners are oving, an effective speed <v 1 > used to odel the syste; N Collisional partner (red) density = V Volue covered by orange in dt = V cyl Vcyl vavedt N Collisions by it in tie dt = V cyl V Particle collisional frequency of it = z ; reduced ass For a saple of one type of gas; Total collisional frequency, two types of gases Z 1 : Total nuber of collisions in the gaseous saple. For a saple of one type of gas we have; Accounts for double counting Mean Free Path: Effusion: Average distance a particle would travel between two successive collisions two types of olecules, say 1 and. For one type of olecules, Effusion is the process in which a gas escapes through a sall aperture. This occurs if the diaeter of the aperture is considerably saller than the ean free path of the olecules (effusion rate = nuber of olecules that pass through the opening (aperture) per second). Once the particle passes through it generally wont coe back because of the low partial pressure on the other side Pressure of the gas and size of the aperture is such the olecules do not undergo collisions near or when passing through the opening. 8

9 1/6/017 Collisional Flu Z C : High P Z C = nuber of collisions per unit tie per unit area. (by one type of olecule); definition. Low P d < Left effusion; right - diffusion. Effusion occurs through an aperture (size A~d) saller than the ean path of the particles in otion whereas diffusion occurs through an aperture through which any particles can flow through siultaneously. d > Note: Upon substitution for v, ave N ;using IGL and siplification; Effusion rate decreases with tie because of the reduction in gas pressure inside the container due to effusion/diffusion. And rate of loss of olecules Integration yields; Upon substitution -v 0 v 0 9

10 1/6/017 o f ( v v v ) f( v ) dv f( v ) dv o -v 0 v 0 v kt v kt f ( v0 v v0) e d 0 0 v kt 0 0 f ( v0 v v0) e d 0 v / v v kt / 0 v / kt z erf ( z) e d 0 erf ( z) f ( zv z) z erf ( z) e d 1 erf () 1 e d covers v kt / 0 0 erf ( z) probability v kt /? 10