8 h Block Invesigaions Concerning he Viscosiy of Gases and he Mean Free Pah of he Molecules English-speaking physical chemisry laboraory classes Auumn 009/010 wrien by Soma Veszergom 1.) Inroducion The aim of our presen laboraory pracice is he deerminaion of he viscosiy of four approximaely ideal gases. A vial condiion of he successful compleion of his class is ha our sudens are aware of he basics of he kineic gas heory and he ranspor phenomena. During our measuremens some gas samples having he same volume are flowing hrough a capillary. By measuring he average ime of ouflow for all he samples one can calculae he viscosiies given ha he viscosiy of one of he samples (in our case ) is known. The viscosiy of a gas in quesion can be calculaed using he direc proporionaliy given in Eqn. 1.1: unknown unknown (1.1).) Theoreical Grounds.a) The Viscous Flow Le us consider wo plaes having he same area laying parallel o each oher: le he space beween hem be filled wih a fluid (eiher gas or liquid). If he upper plae is dragged by a force F in he x direcion hen i shall move wih a consan u velociy. So he plae will move wih a consan velociy having zero acceleraion: he sum of he forces ha ac on he plae is zero. Thus agains he force F ha is dragging he plae a fricional force having he same magniude bu he opposie direcion as F is acing. I s an empirical fac ha in his case he fluid layers near he plaes are no moving as compared o he plaes; so his fricional force does no arise on he solid-fluid inerface bu beween he fluid layers: he adjacen layers are moving wih differen velociies. 1
y x Moving plae u The velociy of he layers drops from u o 0 as we advance in he y direcion from he moving plae o he sagnan plae. Here he gradien of he velociy is consan which is no always rue. d u Gradien of he velociy Sagnan plae Fig..1. For he explanaion of viscous flow. So when dealing wih viscous flow every fluid layers chafe on heir neighbours: he molecules of he moving layer penerae ino he adjacen layer rying o pull i wih hem while he layer wih smaller velociy is seing back he quicker one. For wo fluid layers ineracing wih each oher hrough a surface A laying a an infiniesimal disance dy o each oher and having d u x difference in heir velociies in he direcion of x (ha is orhogonal o he direcion of y) one can make he following saemen: F A du (.1) ha is commonly referred o as Newon s law. The coefficien arising in Eqn..1. is called he viscosiy of he fluid. The ranspor of momenum beween wo gaseous layers a a moderae pressure was explained by Maxwell on he basics of he kineic heory of gases..b) Some Imporan Resuls of he Kineic Theory of Gases The kineic heory of gases uses he Maxwell Bolzmann densiy funcion o deermine he disribuion of he velociies of he gas molecules 1 : f 3 Mv M (.) πrt RT v 4π v e so he mean velociy of he molecules can be given as: 8RT v v f v dv. (.3) π 0 M Using his formula one can express he so-called collision frequency as p z σ vn σ v (.4) kt
where σ is he effecive cross-secion of he molecules and N denoes he densiy of he molecules (number of molecules/volume). The average disance ha is covered by a paricle beween wo successive impacs is called he mean free pah and may be expressed as follows: v kt λ. (.5) z σp.c) The Viscosiy of Ideal Gases Le us consider he flowing gas layers as recangular cuboids having a heigh of λ slipping on each oher on heir basis wih an area A in he direcion of x. As an approximaion we migh assume ha every gas molecule is moving wih a velociy of same magniude v in any of he direcions of he hree orhogonal coordinae axes (see Fig...): his approximaion is equivalen o he neglecion of noncenral impacs. Anoher serious approximaion and also a cause of many errors of his model is ha we assume ha he saisical velociy disribuion is no affeced by convecion. Fig... For he kineical basics of viscous gas flow. One migh see ha in his case he number of molecules passing hrough he boundary of he 1 layers from one side is vna over ime uni. Accordingly he mass flow passing hrough from one 1 side is v ρa where ρ is he densiy of he gas. Since every molecule moving in he layers have covered a disance of λ since he las collision one can sae ha a he layer boundary in he quicker 3
du layer he molecules have a velociy in he x direcion of u λ while in he slower layer he x- du direcional velociy of he molecules a he boundary is u λ. Summing he momena of x-direcion over ime uni coming from he wo sides a he layer boundary he viscous force can be wrien as 1 du 1 du 1 du F vρa u λ vρa u λ vρaλ. (.) 3 Comparing his resul o Eqn..1. he viscosiy of ideal gases can be expressed as: 1 vρλ (.7) 3 3.) Experimenal deails 3 A scheme of he experimenal seup can be seen in Fig. 3.1. Before he measuremens are aken he measured gas is conneced o he inle. Afer opening he upload ap and closing he baffle we cauiously le he gas in he device: he level of he paraffin oil ha fills he reservoir should slowly drop beneah he lower sign. By sopping he gas flow and closing he upload ap one may choose using he baffle wheher o le he gas ou hrough he oule or hrough he measuring capillary. Fig. 3.1. Scheme of a gas viscomeer. Repeaing he procedure described above hree or four imes he device should be rinsed properly wih he gas ha is currenly measured. I s imporan ha he measuring capillary should also be flushed a he las sep. As he equipmen is now ready for he measuremens o ake place i would be filled again wih he gas: by urning he baffle in he direcion of he measuring capillary we 4
measure he ime ha is needed for he oil level o rise from he lower sign o he upper sign. We make four-five measuremens wih each of he four gases ( nirogen oxygen and helium). Afer he experimen is over we also measure he pressure and he emperaure in he laboraory using a hermomeer placed near he gas viscomeer and a digial pressure monior. 4.) Evaluaion of he Experimenal Resuls 4.a) The Process of Calculaions We calculae he mean of ime values for each gas. The half-widh of he confidence inerval should be esimaed using Eqn. 4.1.: SD 005 f (4.1) n Δ 0 05 f is he value of he random variable wih a Suden-disribuion a a significance level of where 95% wih degrees of freedom f n 1 and SD is he sandard deviaion of our n measured values. Exreme values should be negleced in hese calculaions (one may reason for he neglecion using he mehods of g-saisics) 3. We obain he relaive viscosiies of oxygen nirogen and helium using Eqn. 4.: rel (4.) and we can calculae he half-widh of he confidence inervals of relaive viscosiies based on Gauß s concep of he spread of errors so Δ Δ Δ rel (4.3) To calculae he absolue viscosiy of he gases we shall need o know he viscosiy of a he emperaure a which he measuremens were aken. If unknown his value can be esimaed using he merely empirical Suherland formula: 3 0 0 T C T 0 (4.4) T C T where he superscrip 0 denoes a known reference value 0 0 T 9115 K 187 10 Pa s and C is a consan wih a value of C 10 K. Knowing he viscosiy of he cenre of he confidence inervals of absolue viscosiies of he oher hree gases may be calculaed as (4.5) rel 5
while Δ Δ (4.) rel Once he absolue viscosiies are known he confidence inervals of he mean free pahs for oxygen nirogen and helium can be obained as 3 RTπ λ (4.7) p 8M 3Δ RTπ Δλ (4.8) p 8M where if we use he molar mass M of he paricles he pressure p and he emperaure T in heir SI 1 base unis kg mol Pa and K respecively we should obain he mean free pah in meers. 4.b) Expecaions Concerning he Lab Repors The lab repor should conain he following measured and calculaed resuls: Four or five measured ime values for each of he four gases. The emperaure (K) and pressure (hpa) in he laboraory. 95% confidence inervals of ouflow ime values for each of he gases. The absolue dynamic viscosiy of as calculaed using Eqn. 4.3. 95% confidence inervals of he relaive and absolue viscosiy (Pa s) of nirogen oxygen and helium. 95% confidence inervals of he mean free pahs (nm) of hese hree gases. 5.) Noes 1. Furher informaion abou he kineic heory of gases may be found in P. W. Akins: Physical Chemisry. 8 h ed. Oxford Universiy Press Par 1. (Equilibrium) pp. 8 30. and in Hungarian language Erdey-Grúz Tibor Schay Géza: Elmélei fizikai kémia. Első kiadás Tankönyvkiadó Budapes 1955. I. köe pp. 4-43.. This approximaion is naurally a bi rough bu he model i leads o describes he sysem well in ime-average. 3. Based on he work: Farkas József Kaposi Olivér Mihályi László Mika József Riedel Miklós: Bevezeés a fizikai-kémiai mérésekbe. Tankönyvkiadó Budapes 1988. I. köe pp. 33 35. available only in Hungarian language. 4. Concerning he spread of errors and g-saisics see: Szalma József Láng Győző Péer László: Alapveő fizikai kémiai mérések és a kísérlei adaok feldolgozása. Eövös Kiadó Budapes 007. pp. 5. The supervisor of his class wishes all of You good work a he lab Soma Veszergom