Kinetic Molecular Theory of Ideal Gases

Transcription

1 ecture /. Kinetic olecular Theory of Ideal Gases ast ecture. IG is a purely epirical law - solely the consequence of eperiental obserations Eplains the behaior of gases oer a liited range of conditions. IG proides a acroscopic eplanation. Says nothing about the icroscopic behaior of the atos or olecules that ake up the gas.

2 Today.the Kinetic olecular Theory (KT) of gases. KTG starts with a set of assuptions about the icroscopic behaior of atter at the atoic leel. KTG Supposes that the constituent 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 eplanation of the acroscopic behaior of gases. Predicts eperiental phenoena that suggest new eperiental work (awell- oltzann Speed Distribution). Kotz, Section.6, pp.5-57 Cheistry, st edition: Section 7.4, pp.6-9; Section 7.5, pp.9-. nd edition: Section 8.4, pp.54-57, section 8.5, pp Kinetic olecular Theory (KT) of Ideal Gas Gas saple coposed of a large nuber of olecules (> 0 ) 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. olecules do not influence one another ecept 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 with the container walls are assued to be elastic. The dynaic behaiour of gas olecules ay be described in te of classical ewtonian echanics. The aerage kinetic energy of the olecules is proportional to the absolute teperature of the gas. This stateent in fact seres as a definition of teperature. t any gien teperature the olecules of all gases hae the sae aerage kinetic energy. ir at noral conditions: ~.70 9 olecules in c of air Size of the olecules ~ (-)0-0, Distance between the olecules ~ 0-9 The aerage speed /s Theeanfreepath (0.icron) The nuber of collisions in second

3 Rando traectory of indiidual gas olecule. ssebly of ca. 0 gas olecules Ehibit distribution of speeds. Gas pressure deried fro KT analysis. Deriation: o 8. pp The pressure of a gas can be eplained by KT as arising fro the force eerted by gas olecules ipacting on the walls of a container (assued to be a cube of side length and hence of Volue ). We consider a gas of olecules each of ass contained in cube of olue V =. When gas olecule collides (with speed ) with wall of the container perpendicular to co-ordinate ais 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 p. Pressure p ( ) z The particle ipacts the wall once eery / tie units. t y The force F due to the particle can then be coputed as the rate of change of oentu wrt tie (ewtons Second aw). p F t -

4 4 Force acting on the wall fro all olecules can be coputed by suing forces arising fro each indiidual olecule. F, The agnitude of the elocity of any particle can also be calculated fro the releant elocity coponents, y, and z. z y The total force F acting on all si walls can therefore be coputed by adding the contributions fro each direction. z y z y F,,,,,, ssuing that a large nuber of particles are oing randoly then the force on each of the walls will be approiately the sae. F 6 The force can also be epressed in te of the aerage elocity where denotes the root ean square elocity of the collection of particles. F 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 eerted. V F P V The fundaental KT result for the gas pressure P can then be stated in a nuber of equialent ways inoling the gas density, the aount n and the olar ass. n PV V ogadro uber = 6 0 ol - n PV KT result PV nrt IGEOS RT RT nrt n Using the KT result and the IGEOS we can derie a Fundaental epression for the root ean square Velocity of a gas olecule.

5 Gas 0 /kg ol - V /s - H H O O CO 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 to the teperature T of the gas. This seres as the olecular definition of teperature. P V P E erage kinetic Energy of gas olecule E V V oltzann Constant n nrt RT kt R 8.4 J ol K V V V k.80 J K 6.00 ol nr k kt E V V E kt U E kt nrt 5

6 awell-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. awell and independently oltzann analysed the olecular speed distribution (and hence energy distribution) in an ideal gas, and deried a atheatical epression 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 elocity, and the ost probable elocity of a olecule in a gas saple at a gien teperature T. Jaes awell / F () 4 ep kt kt E E FE ( ) ep kt kt udwig oltzann awell-oltzann elocity Distribution function F( ) 4 = particle ass (kg) k = oltzann constant = J K - k Gas olecules ehibit a spread or distribution of speeds. T / ep 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 aority 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 aiu 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

7 Properties of the awell-oltzann Speed Distribution. awell oltzann () Velocity Distribution a = P r = 9.95 kg ol - F() T = 00 K T = 400 K T = 500 K T = 600 K T = 700 K T = 800 K T = 900 K T = 000K P (00K) = 5.6 s - (00K) = 4.78 s - (00K) = s 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 / s - Relatie ean speed (speed at which one olecule approaches another. rel 7

8 Velocity Distribution Cures : Effect of olar ass T = 00 K He = 4.0 kg/ol e = 0.8 kg/ol r = 9.95 kg/ol Xe =.9 kg/ol 0.00 F() / s - Deterining useful statistical quantities fro Distribution function. erage elocity of a gas olecule F 0 F( ) 4 d k T / awell-oltzann elocity Distribution function ep kt ost probable speed, a or P deried fro differentiating the distribution function and setting the result equal to zero, i.e. = a when df()/d = 0. a Root ean square speed P kt F( ) d 0 / RT Yet ore aths! kt RT Soe aths! distribution of elocities enables us to statistically estiate the spread of olecular elocities in a gas 8kT 8RT ass of olecule Deriation of these forulae Requires knowledge of Gaussian Integrals. olar ass P 8

9 P /s - /s - V rel /s - P /s - Gas 0 / kg ol - H H O O CO H Typical olecular elocities Etracted fro distribution t 00 K for coon gases. /s H O /kg ol - O CO rel a 8kT 8RT kt P RT kt RT awell oltzann Energy Distribution F(E) r T = 00 K T = 400 K T = 500 K T = 600 K T = 700 K T = 800 K T = 900 K T = 000 K E /J F( E) E k T E E F( E) de 0 / / E ep kt E k T / / E ep de kt k T 0 9

10 T pparatus used to easure gas olecular speed distribution. Rotating sector ethod. Cheistry st edition: o 7., pp.-. nd edition: o 8., pp

11 Effusion and Diffusion of Gases. Effusion is the process by which gas olecules pass through a sall hole such as a pore in a ebrane. Diffusion occurs when two or ore gases coe together and i. Diffusion arises due to the presence of a concentration gradient. Grahae s aw of effusion: t a gien teperature and Pressure the rate of effusion (nuber of olecules Passing through hole per second) is inersely proportional to square root of olar ass of gas. R effusion K K = constant For a iture of two gases and With olar asses and the Relatie effusion rate is gien by: R R We conclude that gases with different olar asses Will effuse at different rates. gas with a low olar ass (e.g. He) will effuse Faster than a gas with a higher olar ass (e.g. ). itrogen passes through ebrane.6 ties ore Slowly than heliu. R He 8gol 7.6 R 4gol He Gas iing takes tie. Rate of diffusion gien by Diffusion coefficient D (units: c s - ). ean distance traelled by a diffusing olecule is: 6Dt olecular collisions in gases. Cheical reactions occur when olecules collide With each other. We can use KTG to estiate collision frequency in gases. olecular size will affect probability of collision. olecules, will collide if the center of one (olecule ) Coes within a distance of two radii one diaeter- of olecule. This area is the target presented by. The area of the target for olecule to hit olecule is a circle of area s called the Collision Cross Section. d The collision cross section is related to olecular size.

12 Collision frequency & ean free path. Collision frequency Z = ean nuber of collisions that a olecule undergoes per second. uber of collisions per second depends on distance traelled, and nuber of olecules per unit olue (concentration). This is the pressure. In a gien tie period, a olecule will collide with any other olecule whose centre lies within a cylinder with cross sectional area. Using Kinetic theory we can show that the collision frequency is: Z P RT Knowing aerage speed that olecules oe and nuber of collisions they undergo, the distance traelled between collisions can be deterined. This is called the ean free path. RT Z P RT P Eaple: gas at STP. Collision Cross Section = 0.4 n. ean speed <> = 475 s -. Collision Frequency Z: Z = / (6.00 ol - )(475 s - )( ){(0 5 Pa)/(8.4JK - ol - )(98K)} Hence collision frequency Z = s -. ean free path: = RT/ / P, Hence ={ (8.4 Jol - K - ) (98K)}/{ / (6.00 ol - )( (0 5 J - )} So ean free path = = 67 n. ote significant orders of agnitude for : olecular size typically 0. n. ean free path typically 67 n. t atospheric pressure gas olecules collide eery 0. ns.