Ideal Gas behaviour: summary

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Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or

In the contet of the kinetic oleculr theory n idel gs is one for which the en free pth λ (the distnce over which the gs olecules trvel efore they eperience collision) of the olecules is uch greter

1 Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or repulsive) eist ongst the olecules. These re not very resonle ssuptions. However the idel gs pproition is useful for ny gses t high tepertures or odertely low (< 0 t) pressures, i.e. dilute gses. In the contet of the kinetic oleculr theory n idel gs is one for which the en free pth λ (the distnce over which the gs olecules trvel efore they eperience collision) of the olecules is uch greter thn the collision dieter d. Also in n idel gs the only contriution to the internl energy coes fro the trnsltionl kinetic energy of the gs olecules. There is no contriution fro the potentil energy rising fro interctions of gs olecules with ech other. We now eine the ehviour of rel gses.

2 Brekdown of idel gs ehviour : Low teperture nd high pressure. Gses cnnot e sujected to n infinite copression. Gses liquefy t low tepertures nd t high pressures. Cheistry 3 Section 7.6, pp Kotz, Section.9, pp

3 Rel Gses Assue tht gs sple is copressed using piston. As the gs undergoes copression the individul olecules re rought closer together : the finite volue of the individul olecules will ecoe iportnt nd these olecules will interct with one nother. Hence finite oleculr size nd interoleculr interctions will e iportnt in the description of rel gses. Devitions fro idel gs ehviour will therefore e oserved s the gs ecoes ore dense. Moleculr interctions y e neglected. olue of individul gs olecules << overll gs volue. r Interoleculr forces y e either ttrctive or repulsive. Repulsive forces ssist epnsion of gs : significnt when olecules re close together (within single oleculr dieter) nd opertive t high pressure. Dilute gs Attrctive forces ssist copression of gses : cn hve influence over long distnce (close ut not touching). Opertive t odrerte pressures. Significnt oleculr interctions present. olue of individul gs olecules cnnot e neglected copred with overll volue of gs. Dense Gs Copressiility (Copression) Fctor We cn epress the etent of devition fro idel ehviour s function of pressure (which is relted to the density of the gs) y introducing quntity clled the Copressiility or Copression fctor Z. Z P n P For n idel gs Z, nd rel gses ehiit Z vlues different fro unity. Z vlues y e eplined in ters of the opertion of interoleculr forces. At low pressures the olecules re fr prt nd the predoinnt interoleculr interction is ttrction. The olr volue is less thn tht epected for n idel gs : interoleculr forces tend to drw the olecules together nd so reduce the spce which they occupy. Under such conditions we epect tht Z <. As the pressure is incresed the verge distnce of seprtion etween olecules decreses nd repulsive interctions etween olecules ecoe ore iportnt. Under such conditions we epect tht Z >. When Z >, the olr volue is greter thn tht ehiited y n idel gs: repulsive forces tend to drive the olecules prt. 3

4 iril eqution of stte The oservtion of Z fctor different fro unity cn e used to construct n epiricl or oservtion sed eqution of stte, y supposing tht the idel gs eqution of stte is only the first ter of ore cople epression which cn e epressed in ters of theticl power series. This is clled the iril eqution of stte. P P B C Z... B P C P P 0 Note tht the viril coefficients B, C, B nd C re otined y fitting the eperientl Z vs P dt to the viril eqution of stte. Their vlues depend on the identity of the gs nd Reflect the presence of interoleculr forces nd interctions. When the pressure P is sll the olr volue will e very lrge nd so the second nd third ters in the viril series will e very sll nd to good pproition the viril eqution of stte reduces to the idel gs eqution of stte. 4

5 Boyle Teperture In perfect gs dz/dp 0 (since Z ). In rel gs the result is different. dz B PC... B s P 0 dp dz B s d / ( ) At low T the initil dz/dp < 0, B is negtive. At high T the initil dz/dp > 0 nd B is positive. The teperture t which the initil slope is zero is Tered the Boyle Teperture T B where B 0. Here the rel gs ehves s n idel gs. n der Wls (dw) eqution of Stte Johnnes D n der Wls The viril eqution of stte (ES) cn ke sense of the eperientlly oserved Z versus P dt ut it is epiricl nd does not directly tell us how the idel gs eqution of stte is odified due to the opertion of interoleculr interctions. The n der Wls eqution of stte, lthough pproite, is sed on very siple physicl odel of interoleculr interctions. Repulsive Interctions. The opertion of repulsive interoleculr interctions iplies tht two gs olecules cnnot coe closer thn certin distnce of ech other. Insted of eing free to trvel nywhere in volue, the ctul volue in which the olecule cn trvel is reduced y n ount which is proportionl to the nuer of olecules present nd the volue which they eclude. The volue ecluding repulsive forces re odelled y chnging the volue ter in the idel gs eqution to -n, where represents the proportionlity constnt etween the reduction in volue nd the ount of olecules present in the continer: it is the ecluded volue per ole. Molecules hve finite size. Attrctive Interctions. Hence occupy spce. n P n Finite oleculr Size: ecluded olue correction Molecules in interior of gs volue eperience unifor force field The presence of ttrctive interctions etween olecules is to reduce the pressure tht the gs eerts. The ttrction eperienced y given olecule is proportionl to the concentrtion n/ of olecules in the continer. Attrctive forces slow the olecules down: olecules strike the wlls less frequently nd with less ipct. Pressure deterined y ipct of olecules on continer wlls nd is proportionl to rte of ipct ties verge strength of ipct. Both of these quntities re proportionl to the concentrtion. Molecules ner continer wll eperience net ttrctive force towrds ody of gs. 5

6 Reduction in pressure proportionl to (n/) (n/) where denotes constnt of proportionlity which tkes ccount of ttrctive interctions. Hence this correction fctor should e dded to the pressure P to ke up for this deficit, nd the Pressure ter in the idel gs eqution of stte is chnged fro P to P (n/). Mesured pressure n n P n Correction fctor to ccount for Interoleculr ttrctions Mesured volue Correction fctor to Account for finite oleculr size dw eqution of stte n P ( n) n P ( ) n Useful liits of the dw eqution of stte. n P ( n) n P ( ) When P is very low then is very lrge since P is proportionl to /. Then << nd / << P nd the dw epression reduces to P, the idel gs eqution s it should. At slightly higher pressures the following pertins. Hence P < nd decreses with incresing pressure P. The dip oserved in the curve of Z versus P is ttriuted to the / ter nd is due to ttrctive forces which will predointe when olecules re fr prt t low pressures. P P P Z ( ) ( ) P ( ) P ( ) P P P P Z P The dw epression y e recst in nother nner which proves useful t higher pressures. Now when P is lrge, the ter / / is sll nd cn e neglected. Now Z > nd increses with incresing pressure P. This rise in Z with P is due to the ter which rises fro the opertion of repulsive interctions since the ltter forces kick in when olecules re close together t high pressure. At the Boyle teperture T B dz/dp 0 nd we cn show tht T B /R. 6

7 7 Relting the iril nd dw equtions of stte. ( ) P P Z Copressiility fctor dw gs 3 << lid for oderte pressure where / << Z... C B Z iril eqution of stte dw eqution of stte C B Hence the second viril coefficient B incorportes ters rising fro repulsive nd ttrctive forces. At high teperture when >> then repulsive forces predointe, wheres t low teperture when <<, ttrctive forces predointe (gses condense to liquids t low teperture due to ttrctive forces). When, T T B the Boyle teperture of the gs.

Part I: Basic Concepts of Thermodynamics

Part I: Basic Concepts of Thermodynamics Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry