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 conditions my not be negligible compred with the totl volume o the gs. (ii The orce eerted by the molecules on one nother my not be negligible. Let us now consider these ssumptions in some detil. (i olume o molecules: Under norml temperture nd pressure, the volume o the molecules is less thn 0.1 % o the totl volume o the gs. It hs been ound tht t pressure o bout 10 MP the molecules would occupy volume which is only bout 1 times their own volume. The eect o the inite size o the molecules will be to reduce the vilble ree spce or movement. As result, the number o impcts on the wlls o the continer, nd thereore the pressure on them, will be greter thn tht obtined by the use o the kinetic theory o gses. (ii Force eerted by the molecules on one nother: Liquids possess the cohesive property. The ct tht gses cn be trnsormed into liquids implies tht the molecules o gs possess ttrctive orce. The consequence o ttrctive orces mong the molecules is to mke the pressure less thn tht to be epected theoreticlly. Note: The Joule-Thomson eperiment provides direct eperimentl proo o moleculr ttrction. The generl observtion o this eperiment is tht gs cools itsel on pssing through porous plug. In pssing rom high to low pressure the gs does not perorm ny 4-
eternl work. Thereore, the cooling observed must be due to the work done in overcoming the mutul ttrction o the molecules. The n der Wls Eqution In 1873, J. D. n der Wls mde the irst successul ttempt to modiy the idel-gs eqution in order to correct or the volume nd moleculr ttrction. When gses t ordinry pressures nd tempertures re compressed, the volume is reduced by reducing the intermoleculr spce. Becuse the molecules themselves re not compressible, t very high pressures the entire volume is not inversely proportionl to the pressure. At comprtively low pressures, the molecules o rel gs do not eert ny orce on one nother, becuse they re widely seprted. Thereore, t reltively low pressures, rel gses behve like idel gses. At high pressures the molecules re brought closer together nd consequently the ttrctive orce between the molecules increses, which hs the sme eect s n increse in eternl pressure. It then ollows tht, i eternl pressure is pplied to volume o gs, prticulrly t low tempertures, the decrese in volume is slightly greter thn would be obtined by pressure only. This eect is more pronounced t low tempertures or the simple reson tht t such tempertures the molecules move more slowly nd hve less tendency to move prt on colliding with one nother. To derive the idel-gs eqution P nrt rom the kinetic theory o gses, number o ssumptions were mde. n der Wls modiied the idel-gs eqution to tke into ccount tht two o these ssumptions my not be vlid. The modiied ssumptions which re pplicble to 4-3
rel gses re s ollows. o (i The volume o the molecules my not be negligible in reltion to the volume occupied by the gs. o (ii The ttrctive orces between the molecules my not be negligible. Pressure correction Although there is no net orce cting on molecule inside continer rom the rest o the molecules, when net to the wll, ll the rest o the molecules reside on one side, thus producing net (ttrctive orce nd consequently the pressure on the wll will be diminished. The mesured pressure P is thus less thn the idel pressure given by the kinetic theory o gses. It is, thereore, necessry to dd correction term P to the mesured pressure P, so, the idel pressure is P P. P is proportionl to the product o the number o molecules striking unit re o the wll per second t ny instnt nd the number o molecules per unit volume behind them. For given volume o gs, both these numbers re proportionl to the density ρ o the gs. Thereore, the totl ttrctive orce which is relted to the correction term P' is proportionl to ρ. ρ 1/ Thus, the corrected pressure becomes: where is constnt depending on the gs. P P P / 1-67 4-4
The term / is clled the cohesion pressure. olume correction The volume o gs cnnot be compressed indeinitely becuse o the inl size o the molecules. A correction term b, known s the co-volume, must be subtrcted rom the mesured volume, so tht the corrected volume is -b. The term b is ctor depending on the ctul volume o the molecules. Note: b might be thought tht s being equl to the volume o the molecules but this is not the cse. According to vn der Wls, the co-volume is equl to our times the ctul volume o the molecules; other slightly dierent estimtes hve lso been mde. The eqution or the idel gses becomes (or one mole o gs: P ( b RT 1-68 When is lrge, both b nd / become negligible nd the vn der Wls eqution reduces to the idel-gs eqution P RT (or n 1. Note: At low pressures, the correction or intermoleculr ttrction is more importnt thn the correction b or moleculr volume. At high pressures nd smll volumes, the correction or the volume o the molecules becomes importnt, since the molecules re reltively incompressible nd they orm n pprecible prt o the totl volume. Under ordinry conditions, the devitions rom the gs lws re negligible. A n der Wls gs cn be liqueied nd it possesses criticl point which is given by the conditions 4-5
T P T 0 0 The criticl pressure, volume nd temperture re given by: P c / 7 b c 3b T c 8 / 7bR The vn der Wls constnts At very low pressures, both nd b my be neglected, nd Eq. 1-68 becomes P RT. nd under these conditions the gs obeys the idel-gs lw, which my be considered s representing the limiting behvior o gses t etremely low pressures. At slightly higher pressures, it is generlly possible to ignore b reltive to. Then Eq. 1-68 becomes or P ( RT 1-69 P RT / 1-70 Eq. 1-70 shows tht P < RT nd decreses with incresing pressure. 4-6
At modertely high pressures, becomes smller, so b cnnot be ignored. However, / is smll compred with the high vlue o P. Thus Eq. 1-68 becomes or P ( b RT P RT P b 1-71 Eq.6 shows tht P > RT nd increses with incresing pressure. It cn be shown tht when the product P is plotted ginst P, it hs minimum (or P 0 clled Boyle Point T B. Rewriting vn der Wls eqution or P: or P P RT b RT b Dierentiting the eqution with respect to P t constnt temperture Note: We get P ( P T RT b where RT b RT ( b 1-7 T 4-7
For minimum, ( P T 0 which gives RT b RT ( b 0 or, When P0, >>0, nd RT T B b RT B Rb b b 1-73 1-74 At ll tempertures greter thn the Boyle point the vlue o P will lwys increse s pressure increses. Since the vlues o or hydrogen nd helium re very smll, or these two gses, T B will be reltively low. I we substitute the vlues o nd b or hydrogen (.5 10 - Jm 3 /mol; b.67 10-5 m 3 /mol nd R 8.31 J/mol.K, we obtin T B 11 K. The eperimentl vlue is ound to be 106 K. Conclusion: The Boyle point T B is deined s the lowest temperture bove which P increses continuously with incresing P. It cn be shown tht i the pressure is not too lrge, vn der Wls eqution reduces to: P RT B 1-75 4-8
iril Eqution o Stte I the pressure P nd the volume o n moles o gs held t constnt temperture re mesured over wide rnge o vlues o the pressure, nd the product Pυ, where υ / n, is plotted s unction o 1/υ. The reltion between Pυ nd 1/ υ my be epressed by mens o power series or viril epnsion o the orm Pυ And the other series in powers o P: B C D υ υ υ RT(1... 3 1-75 Pυ RTB P C P. 1-76 where B, C,..., re clled second, third, etc. viril coeicients; their vlues depend on the substnce nd re unction o the temperture. The irst viril coeicient is 1 becuse Pυ must pproch RT s 1/υ pproches zero. This type o eqution is useul or purpose o etrpoltion. All the known orms o eqution o stte, i.e. the Clusius, the Berthelot, etc. cn be epressed in the viril orm i desired, the viril coeicients being determined by the constnts o the vrious equtions. O ll the viril coe. The second coe. Is the most importnt; it depends on the moleculr interction. In the pressure rnge 0-40 tms, the reltion between Pυ nd 1/υ is lmost liner, so tht only the irst two terms in the epnsion re signiicnt. In generl, the greter the pressure rnge, the greter re the number o terms in the viril epnsion. The viril eqution my be derived rom some ssumptions. For emple, i we ssume T nd P to be independent vribles, then P is unction o T nd P. Mthemticlly, 4-9
P (T, P where denotes some unction. As P 0, P RT with no pprecible devition even t severl tmospheres. Thereore, it is possible to epnd (T, P in Mclurin series bout P 0 t constnt temperture. Thus,, 1 ( T, P ( T, 0 ( T, P P 0 T T P ( T, P 0, P! P T T where (T, 0 RT. I we tke... ( T, P P 0, T T P B 1! ( T, P P 0, T T P C nd so on, then the Mclurin series becomes P (T, P (T, 0 BP CP... P RT BP CP... This eqution is identicl with the viril Eqution 1-76. I there re no ttrctive nd repulsive intermoleculr orces within mss o gs, then it cn be shown rom sttisticl mechnics tht the viril coeicients B, C,... identiclly vnish, nd the viril eqution reduces to P RT (or 1 mol o gs, which is the idel-gs lw. Thereore, we cn sy tht gs which obeys P RT t ll pressures hs no intermoleculr orces cting between its molecules. However, it should be remembered tht, in relity, null intermoleculr orces never occur. Consequently some properties o rel gses do not pproch their idel-gs vlues s pressure tends to zero. 4-10
Note: The Mclurin series A unction ( epnded in the power series ( 0 1 3 3 By dierentiting the series term by term,... (... ( d d d d 3 3 1 6 3 By substituting 0 in (, (, (, we get (0 0 ; (0 1 ; (0! ; Thus, the bove series becomes (in the rnge where it is convergent... (!... (! ( ( ( 0 0 0 0 n n n which is Mclurin series. 4-11
Enthlpy preliminry considertions The quntity enthlpy ws introduced into thermodynmics by the Americn physicist nd physicl chemist J. Willrd Gibbs, who ws one o the ounders o modem chemicl thermodynmics. Chemicl rections re usully perormed t constnt pressure. In these situtions usully the volume chnges with work being done. 1 Pd P 1-77 The net work being done cn be seprted into two terms, one t constnt pressure nd the rest: W net W totl P For isobric processes, the unction Enthlpy is: nd is stte unction. H U P 1-78 I we replce or idel gses P nrt, H U nrt 1-79 4-1