herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes of the ndvdual coponents. here are two ways to descrbe the coposton of a xture: ether by specfyng the nuber of oles of each coponent, called olar analyss, or by specfyng the ass of each coponent, called gravetrc analyss. Consder a gas xture coposed of coponents. he ass of the xture tot s the su of the asses of the ndvdual coponents, and the ole nuber of the xture N tot s the su of the ole nubers of the ndvdual coponents. + +... + and n n+ n2 +... + n n (.) tot 2 tot he rato of the ass of a coponent to the ass of the xture s called the ass fracton f, and the rato of the ole nuber of a coponent to the ole nuber of the xture s called the ole fracton y: f and tot y n (.2) ntot A lstng of the ass fractons of the coponents of a xture s soetes referred to as a gravetrc analyss. A lstng of the ole fractons of the coponents of a xture ay be called a olar analyss. An analyss of a xture n ters of ole fractons s also called a voluetrc analyss. he su of the ass fractons or ole fractons for a xture s equal to unty..e y and he ass of a substance can be expressed n ters of the ole nuber n and olar ass M of the substance as nm. hen the average olar ass a xture can be expressed as M nm avr tot ntot ntot n tot ym Copled by Ydneachew M. Page of 8 f (.3)
herodynacs II AA he gas constant R s dfferent for each gas and s deterned fro R u R (.4) M Where Ru s the unversal gas constant and M s the olar ass (also called olecular weght) of the gas. he constant R u s the sae for all substances, and ts value s 3.447 J / Kol. K. he average gas constant of a xture can be expressed as R avr R M u avr (.5) he olar ass of a xture can also be expressed as M n (.6) Mass and ole fractons of a xture are related by f nm M y nm M (.6).2 P-v- Behavor of Ideal Gas Mxtures Many therodynac applcatons nvolve xtures of deal gases. hat s, each of the gases n the xture ndvdually behaves as an deal gas. An deal gas s defned as a gas whose olecules are spaced far apart so that the behavor of a olecule s not nfluenced by the presence of other olecules a stuaton encountered at low denstes. he P-v- behavor of an deal gas s expressed by the sple relaton, whch s called the dealgas equaton of state. Pv R (.7) he P-v- behavor of real gases s expressed by ore coplex equatons of state or by Pv ZR, where Z s the copressblty factor. he predcton of the P-v- behavor of gas xtures s usually based on two odels: Dalton s law of addtve pressures and Aagat s law of addtve volues. Both odels are descrbed and dscussed below. Copled by Ydneachew M. Page 2 of 8
herodynacs II AA Dalton s law of addtve pressures: he pressure of a gas xture s equal to the su of the pressures each gas would exert f t exsted alone at the xture teperature and volue. Fgure. Dalton s law of addtve pressures for a xture of two deal gases. Aagat s law of addtve volues: he volue of a gas xture s equal to the su of the volues each gas would occupy f t exsted alone at the xture teperature and pressure. Fgure.2 Aagat s law of addtve volues for a xture of two deal gases. For deal gases, these two laws are dentcal and gve dentcal results. Dalton s and Aagat s laws can be expressed as follows: Dalton s law: P PV (, ) (.8) Aagat s law: V VV (, ) (.9) In these relatons, P s called the coponent pressure and V s called the coponent volue. he rato P /P s called the pressure fracton and the rato V /V s called the volue fracton of coponent. For deal gases, P and V can be related to y by usng the deal-gas relaton for both the coponents and the gas xture: PV (, ) nr u / V n y (.0) P n R / V n u VV (, ) nr u / P n y (.) V n R / P n u Copled by Ydneachew M. Page 3 of 8
herodynacs II AA herefore P V n y (.2) P V n he quantty y P s called the partal pressure (dentcal to the coponent pressure for deal gases), and the quantty y V s called the partal volue (dentcal to the coponent volue for deal gases). Note that for an deal-gas xture, the ole fracton, the pressure fracton, and the volue fracton of a coponent are dentcal..3 Propertes of Gas Mxture Dalton s law was re-forulated by Gbbs to nclude a second stateent on the propertes of xtures. he cobned stateent s nown as the Gbbs-Dalton law, and s as follows he nternal energy, enthalpy, and entropy of a gaseous xture are respectvely equal to the sus of the nternal energes, enthalpes, and entropes, of the consttuents. Each consttuent has that nternal energy, enthalpy and entropy, whch t could have f t occuped alone that volue occuped by the xture at the teperature of the xture. hs stateent leads to the followng equatons : ( u) u + u +... + u Or 2 2 ( u) u (.3) u f u (.4) ( h) h + h +... + h Or ( h) 2 2 h (.5) h f h (.6) ( s) s + s +... + s Or 2 2 ( s) s (.7) s f s (.8).4 Adabatc Mxng of Perfect Gases Fgure.3 shows two gases A and B separated fro each other n a closed vessel by a thn daphrag. If the daphrag s reoved or punctured then the gases x and each then occupes Copled by Ydneachew M. Page 4 of 8
herodynacs II AA the total volue, behave as f the other gas were not present. hs process s equvalent to a free expanson of each gas, and s rreversble. he process can be splfed by the assupton that t s adabatc; ths eans that the vessel s perfectly therally nsulated and there wll therefore be an ncrease n entropy of the syste. Fgure.3 Gases before and after xture In a free expanson process, the nternal energy ntally s equal to the nternal energy fnally. U nc A vaa + nc B vbb and U2 ( nc + nc ) A va B vb If ths result s extended to any nuber of gases, we have (.9) U nc v and U2 nc U hen U U 2 nc v 2 nc v nc v U nc v v (.20) When two streas of flud eet to for a coon strea n steady flow, they gve another for of xng Copled by Ydneachew M. Page 5 of 8
herodynacs II AA Fgure.4 Flud xture Applyng steady-flow energy equaton to the xng secton (neglectng changes n netc and potental energy), we get h + h + Q h + h + W A A B B A A2 B B2 In case of adabatc flow: Q 0, and also W 0 n ths case Also h c p, hence, h + h h + h A A B B A A2 B B2 c + c c + c A pa A B pb B A pa B pb (.2) (.22) (.23) For any nuber of gases ths becoes c c p p c p c Also, C p Mc p and M /n p nc p c p (.24) nc p Hence, nc p (.25) Copled by Ydneachew M. Page 6 of 8
herodynacs II AA Eqns. (9.28) and (9.29) represent one condton whch ust be satsfed n an adabatc xng process of perfect gas n steady flow. In a partcular proble soe other nforaton ust be nown (e.g., specfc volue or the fnal pressure) before a coplete soluton s possble..5 Mxng of Ideal Gases ntally at dfferent Pressure and eperature Consder three deal gases A, B and C ntally at dfferent pressure and teperature and separated by parttons. Let the gases be xed by reovng the parttons. he total volue and ass occuped by the xture are gven by V VA + VB + VC and A + B + C (.26) In ters of the xng asses, the nternal energy of the gases after xng can be related to the nternal energy of the gases before xng by expresson: u aua + bub + cuc (.27) Snce u for a perfect gas s equal to c v, ths relaton ay be rewrte as: cv acvaa + bcvbb + ccvcc (.28) c + c + c c a va a b vb b c vc c (.29) v c + c + c c + c + c a va a b vb b c vc c a va b vb c vc (.30) Applng the perfect gas law.e, PV R PV c PV c PV c Ra Rb Rc PV c PV c PV c R R R a a va b b vb c c vc a a va b b vb c c vc a a b b c c (.3) Multplyng each ters n the nuerator and denonator of equaton by the respectve olecular weghts: Copled by Ydneachew M. Page 7 of 8
herodynacs II AA M PV c M PV c M PV c MR a a MR b b MR c c M PV c M PV c M PV c MR MR MR a a a va b b b vb c c c vc a a a va b b b vb c c c vc a a a b b b c c c (.32) he product Mc v s the sae for deal gases and the product MRR u s a constant for all deal gases. Cancellaton of these quanttes fro equaton results n: Where : RPV/ PV a a + PV b b + PV c c PV PV PV a a b b c c a b c PV + PV + PV R (.33) a a b b c c (.34) If the pressure are all equal before xng then, then fro equaton, the resultng xng teperature wll be obtaned fro: Va + Vb + Vc V V V a b c a b c (.35) If the volue are all equal before xng then, then fro equaton, the resultng xng teperature wll be obtaned fro: Pa + Pb + Pc Pa Pb Pc a b c (.38) Copled by Ydneachew M. Page 8 of 8