Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions. IGL proides a acroscopic explanation. Says nothing about the icroscopic behaior of the atos or olecules that ake up the gas.
Today.the Kinetic Molecular Theory (KMT) of gases. KMTG starts with a set of assuptions about the icroscopic behaior of atter at the atoic leel. KMTG Supposes s that t the constituent t particles (atos) of the gas obey the laws of classical physics. ccounts for the rando behaior of the particles with statistics, thereby establishing a new branch of physics - statistical echanics. Offers an explanation of the acroscopic behaior of gases. Predicts experiental phenoena that suggest new experiental work (Maxwell- oltzann Speed Distribution). Kotz, Section.6, pp.53-537 Cheistry 3, Section 7.4, pp.36-39 Section 7.5, pp.39-33. Kinetic Molecular Theory (KMT) of Ideal Gas Gas saple coposed of a large nuber of olecules (> 0 3 ) in continuous rando otion. Distance between olecules large copared with olecular size, i.e. gas is dilute. Gas olecules represented as point asses: hence are of ery sall olue so olue of an indiidual gas olecule can be neglected. Interolecular forces (both attractie and repulsie) are neglected. Molecules do not influence one another except during collisions. Hence the potential energy of the gas olecules is neglected and we only consider the kinetic energy (that arising fro olecular otion) of the olecules. Interolecular collisions and collisions ir at noral conditions: with the container walls are assued to ~.7x0 elastic. 9 olecules in c 3 of air be The dynaic behaiour of gas olecules Size of the olecules ~ (-3)x0-0, ay be described in ters of classical ewtonian echanics. Distance between the olecules ~ 3x0-9 The aerage kinetic energy of the The aerage speed - 500 /s olecules is proportional to the absolute teperature of the gas. This stateent in The ean free path - 0-7 (0.icron) fact seres as a definition of teperature. t any gien teperature The nuber of collisions in second - 5x0 9 the olecules of all gases hae the sae aerage kinetic energy.
Rando trajectory of indiidual gas olecule. ssebly of ca. 0 3 gas olecules Exhibit distribution of speeds. Gas pressure deried fro KMT analysis. The pressure of a gas can be explained by KMT as arising fro the force exerted by gas olecules ipacting on the walls of a container (assued to be a cube of side length L and hence of olue L 3 ). We consider a gas of olecules each of ass contained in cube of olue L 3. When gas olecule collides (with speed x ) with wall of the container perpendicular to x co-ordinate axis and bounces off in the opposite direction with the sae speed (an elastic collision) then the oentu lost by the particle and gained by the wall is Δp x. Pressure Δp ( ) x x x x z L The particle ipacts the wall once eery L/ x tie units. L Δt x y x x The force F due to the particle can then be coputed as the rate of change of oentu wrt tie (ewtons Second Law). Δp x x F Δt L L x - x 3
Force acting on the wall fro all olecules can be coputed by suing forces arising fro each indiidual olecule j. F L x, j j The agnitude of the elocity of any particle j can also be calculated fro the releant elocity coponents x, y, and z. + + x y z The total force F acting on all six walls can therefore be coputed by adding the contributions fro each direction. F L j x, j + y, j + z, j j j L j L { x, j + y, j + z, j} ssuing that a large nuber of particles are oing randoly then the force on each of the walls will be approxiately the sae. F 6 L j j 3 L j j The force can also be expressed in ters of the aerage elocity rs Where rs denotes the root ean square elocity of the collection of particles. rs F 3L j j rs j j The pressure can be readily deterined once the force is known using the definition P F/ where denotes the area of the wall oer which the force is exerted. F rs P 3L 3 rs L The fundaental KMT result for the gas pressure P can then be stated in a nuber of equialent ways inoling the gas density ρ, the aount n and the olar ass M. ogadro uber 6 x 0 3 ol - ρ ρ rs M P rs rs Mrs nm 3 3 3 3 3 M KMT result IGEOS rs Using the KMT result and the IGEOS we can derie a Fundaental expression for the root ean square elocity rs of a gas olecule. P nrt P 3 3 rs nm rs nmrs nrt 3 M 3RT rs rs 3RT M 4
Gas 0 3 M/kg ol - rs /s - H.058 930 H O 8.058 640 8.0 55 O 3.00 480 CO 44.0 40 Internal energy of an ideal gas We now derie two iportant results. The first is that the gas pressure P is proportional to the aerage kinetic energy of the gas olecules. The second is that the internal energy U of the gas, i.e. the ean kinetic energy of translation (otion) of the olecules is directly proportional p to the teperature T of the gas. This seres as the olecular definition of teperature. P 3 P E erage kinetic Energy of gas olecule rs rs rs E 3 3 oltzann Constant n nrt RT kt R 8.34 J ol K 3 k.38x0 J K 3 6.0x0 ol kt E 3 3 E kt 3 3 U E kt nrt nr k 5
Maxwell-oltzann elocity distribution In a real gas saple at a gien teperature T, all olecules do not trael at the sae speed. Soe oe ore rapidly than others. We can ask : what is the distribution (spread) of olecular elocities in a gas saple? In a real gas the speeds of indiidual olecules span wide ranges with constant collisions continually changing the olecular speeds. Maxwell and independently oltzann analysed the olecular speed distribution (and hence energy distribution) in an ideal gas, and deried a atheatical expression for the speed (or energy) distribution f() and f(e). This forula enables one to calculate arious statistically releant quantities such as the aerage elocity (and hence energy) of a gas saple, the rs elocity, and the ost probable elocity of a olecule in a gas saple at a gien teperature T. Jaes Maxwell 83-879 3/ F () 4π exp π kt kt E E FE ( ) exp kt 3 π ( kt ) http://en.wikipedia.org/wiki/maxwell_speed_distribution http://en.wikipedia.org/wiki/maxwell-oltzann_distribution Ludwig oltzann 844-906 Maxwell-oltzann elocity Distribution function F( ) 4π particle ass (kg) k oltzann constant.38 x 0-3 J K - π k Gas olecules exhibit a spread or distribution of speeds. T 3/ exp F() kt The elocity distribution cure has a ery characteristic shape. sall fraction of olecules oe with ery low speeds, a sall fraction oe with ery high speeds, and the ast ajority of olecules oe at interediate speeds. The bell shaped cure is called a Gaussian cure and the olecular speeds in an ideal gas saple are Gaussian distributed. The shape of the Gaussian distribution cure changes as the teperature is raised. The axiu of the cure shifts to higher speeds with increasing teperature, and the cure becoes broader as the teperature increases. greater proportion of the gas olecules hae high speeds at high teperature than at low teperature. 6
Properties of the Maxwell-oltzann Speed Distribution. Maxwell oltzann (M) elocity Distribution 0.005 ax P r M 39.9595 kg ol - F() 0.000 0.005 0.000 T 300 K T 400 K T 500 K T 600 K T 700 K T 800 K T 900 K T 000K P (300K) 353.36 s - rs (300K) 43.78 s - (300K) 398.74 s - 0.0005 Features to note: The ost probable speed is at the peak of the cure. The ost probable speed increases as the teperature increases. The distribution broadens as the teperature increases. 0.0000 0 00 400 600 800 000 00 400 600 800 / s - Relatie ean speed (speed at which one olecule approaches another. rel 7
M elocity Distribution Cures : Effect of Molar Mass T 300 K 0.005 0.004 He M 4.0 kg/ol e M 0.8 kg/ol r M 39.95 kg/ol Xe M 3.9 kg/ol 0.003 F() 0.00 0.00 0.000 0 500 000 500 000 / s - Deterining useful statistical quantities fro M Distribution function. erage elocity of a gas olecule F 0 F ( ) 4 π () d π k T 3/ Maxwell-oltzann elocity Distribution function exp kt Most probable speed, ax or P deried fro differentiating the M distribution function and setting the result equal to zero, i.e. ax when df()/d 0. kt ax P Root ean square speed rs F( ) d 0 / RT M Yet ore aths! 3kT 3RT M Soe aths! M distribution of elocities enables us to statistically estiate the spread of olecular elocities in a gas 8kT 8RT π πm Mass of olecule Deriation of these forulae Requires knowledge of Gaussian Integrals. Molar ass < < P rs 8
< < P rs rs /s - /s - rel /s - P /s - Gas 0 3 M/ kg ol - H.058 930 775 50 570 H O 8.058 640 594 840 56 8.0 55 476 673 4 O 3.00 480 446 630 389 CO 44.0 40 380 537 33 00 000 800 600 H Typical olecular elocities Extracted fro M distribution t 300 K for coon gases. rs /s - 400 00 000 800 600 400 H O 00 0 0 0 30 40 50 0 3 M/kg ol - O CO rel 8kT 8RT π πm kt ax P rs RT M 3kT 3RT M Maxwell oltzann Energy Distribution 0.0005 F(E) 0.0000 0.0005 0.0000 r T 300 K T 400 K T 500 K T 600 K T700K T 800 K T 900 K T 000 K 0.00005 0.0000000000 0 000 4000 6000 E /J F( E) E ( kt ) π E E F( E) de 0 3/ / E exp kt E ( k T ) 3/ 3/ E exp de kt π k T 0 3 9
T θ ν L πνl θ pparatus used to easure gas olecular speed distribution. Rotating sector ethod. Cheistry 3 ox 7.3, pp.3-33. 0
Further aspects of KMT Ideal Gases. The KMT of ideal gases can be deeloped further to derie a nuber of further ery useful results. Cheistry 3 pp.33-36. It is used to deelop expressions for the ean free path λ (the distance traelled by a gas olecule before it collides with other gas olecules).. The nuber of olecules hitting a wall per unit area per unit tie can be deried. 3. The rate of effusion of gas olecules through a hole in a wall can be deterined. 4. The nuber of collisions per unit tie (collision frequency) between two olecules (like or unlike olecules) can also be readily deried. This type of expression is useful in describing the icroscopic theory of cheical reaction rates inoling ggas phase olecules (tered the Siple Collision Theory (SCT)). 5. The transport properties of gases (diffusion, theral conductiity, iscosity) can also be described using this odel. The KMT proposes expressions for the diffusion coefficient D, theral conductiity κ and iscosity coefficient η which can be copared directly with experient and so it is possible to subject the KMT to experiental test. Wall collision flux and effusion KMT proides expressions for rate at which gas olecules strike an area (collision flux) and the rate of effusion through a sall hole. # collisions uber density uber of olecules per unit olue (unit: -3 ) Gas olecules n τ 8 kt ω n τ 4 4 π wall rea of surface ( ) uber of particles striking surface per unit area per unit tie (particle flux) dp 8 Ch. pp.755-756 Tie taken (s) ω Z W τ P n n k T erage olecular speed Z W 8 Z n n kt n kt W 4 4 π π P πk T P 00 kpa ( bar) T 300 K Z W ~ 3 x 0 3 c - s -
f E This expression for Z W also describes the rate of effusion f E of olecules through a sall hole of area 0. Confirs Grahae s experiental Law of Effusion that states that the olecular flux is inersely proportional to M /. Z W p0 P0 πk T π MRT 0 Diffusion - One gas ixing into another gas, or gases, of which the olecules are colliding with each other, and exchanging energy between olecules. Effusion - gas escaping fro a container into a acuu. There are no other (or few ) for collisions. Effusion of an Ideal Gas - the process of a gas escaping through a sall hole (a << l ) into a acuu the collisionless regie. The nuber of olecules that escape through aholeofarea in sec, h,intersofp(t ), T. h Δ p x P Δt P h h h Δt Δt x x x kt, x x h - Δ, where is the total # of olecules in a syste kt P Δt kt Δt Δt kt Δ k Δ T k T k T Δ t τ t ( t) (0) exp, τ τ k T Depressurizing of a space ship, - 50 3, of a hole in a wall 0-4 3 50 τ 4 0 7 30.7 0 kg.38 0 J/K 30 K 0 6 0 0.3s 3000s 3
The Mean Free Path of Molecules Energy, oentu, ass can be transported due to rando theral otion of olecules in gases and liquids. The ean free path λ - the aerage distance traeled by a olecule btween two successie collisions. Reference : dp 8 Ch., pp.75-755 Cheistry 3 Ch.7, pp.36-330. To ealuate the ean free path we exaine how we describe collisions between olecules of like size in a gas. We consider two olecules and approaching each other and assue initially that is stationary and is oing. The olecules will collide if the centre of one () coes within a distance of two olecular radii (a diaeter) of olecule. The area of the target for olecule to hit olecule is a circle with an area σ πd. This area is called the collision cross section. r r σ πd πr r r r 4 σ ( + ) 4π r r r r Particle trajectory C uber density # olecules per unit olue n rea σ Δt Collision tube # collisions in tie Δt n x( tube ) n.. Δt. σ # collisions per unit tie Z n σ The aerage tie interal between successie collisions - the collision tie: Z σ P k T P n n k T τ λ λ τ μ r 8kT πμ + Mean free path λ (total distance traelled in tie Δt)/(# collisions in tie Δt) Δt/n σδt erage relatie speed λ More generally all olecules oe with Z n σ n aerage speed. kt r kt M speed λ distribution ( r ) nσ σ n Pσ Pσ 3
λ Soe ubers: MFP inersely proportional to gas density, inersely proportional to gas pressure and directly proportional for an ideal gas: to gas teperature. P k T P n k T σ n λ n T P air at nor. conditions: the interol. distance.38 0 J/K 300K 40 n P 3 k T.38 0 J/K 300K 5 0 Pa 3 9 ~ 3 0 d 6 3 P 0 5 Pa: λ ~ 0-7 - 30 ties greater than d P 0 - Pa (0-4 bar): λ ~ (size of a typical acuu chaber) - at this P, P there are still ~.5 0 olecule/c 3 (!) λ /3 /3 n P The collision tie at nor. conditions: τ ~ 0-7 / 500/s 0-0 s d For H gas in interstellar space, where the density is ~ olecule/ c 3, λ ~ 0 3 - ~ 00 ties greater than the Sun-Earth distance (.5 0 ) 4