Equilibrium Composition and Thermodynamic Properties of Hydrogen Plasma

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1 Chatr- Equilibrium Comosition and Thrmodynami Prortis of ydrogn Plasma It is wll known that th thrmodynami and transort rortis dnd dirtly on th lasma omosition, whih furthr dnds uon th inlusion of ltronially xitd stats (EES) through th artition funtion. Thrmodynami rortis inlud mass dnsity, intrnal nrgy, nthaly, sifi hat and ntroy. Th artition funtion lays an imortant rol in th dtrmination of thrmodynami as wll as transort rortis. In this hatr, by valuating th dgr of ionization of hydrogn thrmal lasma, its quilibrium omosition and th thrmodynami rortis hav bn disussd both for GS and ES hydrogn lasmas ovr wid rang of tmratur and rssur. As th thrmodynami rortis dnd uon th dgr of ionization and th artition funtion f. Th artition funtion divrgs du to statistial wight g n n, th numbr of lvls to b insrtd in it ar obtaind by a siml utoff ritrion. Thus, th dgr of ionization varis with tmratur and rssur. Th quilibrium omosition of ground and xitd stat lasma has thus bn obtaind. Th thrmodynami rortis.g. sifi hat at onstant rssur v, th sifi hat at onstant volum v and th isntroi offiint ( / ) for both th ground and th xitd stat lasmas has bn alulatd. Th mthod of omutation of quilibrium omosition and th thrmodynami rortis has bn disussd in Stion.1. In Stion., Saha quation is usd for valuating th dgr of ionization and hn th quilibrium omosition for ground as wll as xitd stat hydrogn lasmas. Paramtrization of quilibrium omosition for hydrogn thrmal lasma and trunation of artition funtion using a utoff ritrion hav bn disussd in Stion.3. Th quilibrium omosition of ground and xitd stat lasmas has bn obtaind in Stion.4 and.5. Variation of dgr of ionization with tmratur has bn disussd in Stion.6. Th xrssions for th thrmodynami rortis for GS and ES hydrogn lasma has bn rsntd in Stion 0

2 .7. Th intrnal ontribution to sifi hat has bn saratly disussd in Stion.8. Th rol of EES on th variation of ths rortis for th two ass has bn dsribd grahially in Stion.9. Finally, th rsults and disussions hav bn rsntd in Stion Mthod of omutation Th mthod of omutation starts with th dtrmination of ltroni artition funtion whih through Saha quation givs th dgr of ionization ( n / nn ) with n and n N as th numbr dnsitis of ltrons and nuli rstivly. At a givn tmratur, th numbr of xitd stats to b insrtd in th artition funtion dnds uon rssur, thus th artition funtion varis with rssur. A strong dras of th ltroni artition funtion with inras of rssur has bn obsrvd. Th xrssions for th various thrmodynami rortis hav bn writtn in trms of th dgr of ionization and th artition funtion. In ordr to stimat th fft of inluding th EES in th atomi artition funtion and its drivativs, w hav sltd sifi hat at onstant rssur f ) whr f and r ar th frozn and rativ ontributions to th ( r sifi hat). Th frozn art f is furthr th sum of th translational and intrnal ontributions i.. f 5 RT int. Th ratios / f int and int / hav bn workd out for hydrogn thrmal lasma in th tmratur rang K and in th rssur rang 1-10 atm. At = 10 atm, th ontribution from ltroni xitation is highr than th orrsonding ontribution from th translational nrgy. Following th sam aroah, th ratios of sifi hats at onstant rssur to th sifi hat at onstant volum v i.. (alld th isntroi offiint) hav bn studid and th rol of ltroni xitation has bn disussd. Eltroni xitation aars in th frozn and rativ art of ths rortis with oosit signs rstivly. Finally, dtaild disussion of th rol 1

3 of ltronially xitd stats in affting th various thrmodynami rortis of hydrogn lasma has bn rsntd.. Saha s quation of ionization Saha s quation of ionization for hydrogn lasma is writtn as n n n 3 mkt h f Th total numbr dnsity n is givn by I x kt (.1) n n n n and th total rssur nkt. Equation (.1) an b rwrittn in trms of dgr of ionization = n n n as mkt (1 ) h 3 kt f I x kt (.) whr f is th ltroni artition funtion and I is th ionization nrgy of atomi hydrogn. n, n and n ar th numbr dnsitis of ltrons, - atoms and rotons rstivly. Total numbr dnsity n is givn by n n n n.3 Paramtrization of quilibrium omosition and utoff ritrion Th quilibrium omosition for hydrogn thrmal lasma as a funtion of tmratur has bn obtaind by using Saha s ionization quation, whih is writtn as 7 x / / T 1 x n atm T (.3) n whr x is th onntration of ions or ltrons and n atm is numbr of n atmoshrs. Firstly, th quilibrium omosition for th ground stat lasma (, +, ) is valuatd by using th abov quation and thn th atomi hydrogn has bn furthr dividd into xitd stats (n) by Boltzmann law (Equation (.5)).

4 As is wll known that th artition funtion of atomi hydrogn divrgs du to th dgnray fator. A siml utoff ritrion basd uon onfind- atom (CA) modl is adotd (Caitlli t al., 003), i.. by onsidring xitd stats with lassial Bohr radius not xding th intrartil distan. Th ritrion thus obtaind is a o n (.4) 1/ 3 max ( 1/ n ) whr a 0 is th Bohr radius and n / kt is th total numbr dnsity of hydrogn atoms. All th xitd stats with th rinial quantum numbr n nmaxshould b onsidrd in th artition funtion. Th maximum numbr nm axof xitd stats at diffrnt rssurs in tmratur rang K ar givn in Tabl.1. Thus, th artition funtion dislays its variation with tmratur at diffrnt rssurs. f dnds uon rssur and tmratur. Figur.1 f Figur.1. Partition funtion f vs. tmratur. Curvs (ur to lowr) rrsnt =1, 10, 100 & 1000 atm rstivly. 3

5 Tabl.1. Th maximum numbr n m ax of xitd stats to b insrtd in th artition funtion at diffrnt rssurs in tmratur rang K. Tmratur (K) =1 atm =10 atm =100 atm =1000 atm

6 .4 Equilibrium omosition of ground stat (GS) lasma Th onntration of ltron X, hydrogn ion X & hydrogn atom X in th ground stat lasma at =1, 10& 100 atm rstivly hav bn valuatd at diffrnt tmraturs. It has bn obsrvd that X at high rssur (i.. =100 atm) is X lss as omard to that at low rssur (=1or 10 atm) (Singh t al., 008) baus ionization of hydrogn atoms tak la at a high tmratur with inras in rssur..5 Equilibrium omosition of xitd stat (ES) lasma Atomi hydrogn in th xitd stat lasma is dividd into th diffrnt ossibl xitd stats dnding uon rssur and tmratur. W thus obtain th xitd stat lasma (, +, (n)) with n=1,, 3, alulation at =1, 10& 100atm rstivly. nm axwhr n m ax =1, 1 and 7 hav bn usd in th Th rlativ onntration of th ithatomi xitd stat is obtaind by ni n T gi E x( i ) Z( T ) kt (.5) with Z( T) Ei / kt gi whr n, g and i i E i ar th numbr dnsity, dgnray and nrgy of th th i atomi xitd stat rstivly. artition funtion of atomi hydrogn. n and Z(T ) ar th total numbr dnsity and ltroni T Tabl.,.3 &.4 rsnt th onntration of ltron X, hydrogn ion X, hydrogn atom X (1) in ground stat for xitd stat lasma at diffrnt rssurs. Conntration of various numbrs of xitd stats X (n) vs. tmratur at diffrnt rssurs ar ditd in Figur.. It has bn obsrvd that with inras in rssur, th onntration of th xitd stats inrass as wll as th maxima of X (n) shifts towards highr tmratur. 5

7 Tabl.. Conntrations of ltron X, hydrogn ion X, hydrogn atom X (1) in th ground stat for xitd stat lasma at =1 atm. =1 atm, n=1 X X X (1)

8 Tabl.3. Conntrations of ltron X, hydrogn ion ground stat X (1) for xitd stat lasma at =10 atm. X, hydrogn atom in =10 atm, n=1 X X X (1)

9 Tabl.4. Conntrations of ltron X, hydrogn ion in ground stat for xitd stat lasma at =100 atm. X, hydrogn atom X (1) =100 atm, n=7 X X X (1)

10 X (n) (a) X (n) (b) X (n) () Figur.. Conntration X (n) vrsus tmratur. Curvs (lowr to ur) rrsnt X (n) for (a) n=-1 at =1 atm (b) n=-1 at =10 atm and () n=-7 at =100 atm. 9

11 .6 Variation of dgr of ionization with tmratur and rssur Using omutr rogram basd uon Equation (.), dgr of ionization has bn valuatd at diffrnt rssurs ovr a wid rang of tmratur (i.. from K). Its variation with tmratur has dislayd in Figur.3 for both GS and ES lasmas at =1& 100 atm. It is lar that with inras of rssur, th dgr of ionization for ES lasma bom lss than that for GS lasma GS ES (a) GS ES (b) Figur.3. Dgr of ionization vs. tmratur for GS and ES (ur and lowr) hydrogn lasma at (a) =1 atm and (b) =100 atm. 30

12 .7 Thrmodynami rortis: Exrssions Th xrssions for thrmodynami rortis suh as nthaly, sifi hat at onstant rssur and isntroi offiint for GS and ES hydrogn lasma has bn rsntd. Th nthaly of thr omonnt lasma (, +, ) has bn dfind and its dndn uon th ltroni artition funtion has bn disussd in Stion.7.1. Exrssions for th sifi hat at onstant rssur and th isntroi offiint ( / ) hav bn givn in Stions.7. &.7.3. v.7.1 Enthaly Among th thrmodynami rortis, th most imortant for lasma modlling ar th nthaly, its drivativ with rst to tmratur T, and th sifi hat at onstant rssur. Th aks on ths urvs orrsond to dissoiation mhanisms at low or intrmdiat tmraturs (around 4000 K for, and 7000 K for N at atmoshri rssur) and to ionization at high tmraturs (mainly around 15000K for many sis having an ionization nrgy around V). Starting with on mol of atomi hydrogn, thn at a givn tmratur, + and hav 1, and mols rstivly, whr is th dgr of ionization dfind by whr N n / n N n n n, (.6) n, n and n ar th numbr dnsitis of nuli, atoms and rotons rstivly. Total nthaly of this thr omonnt lasma (, +, ) is whr ( 1) (.7), and ar th molar nthalis of atoms, rotons and ltrons givn by 5 D RT E 5 RT 5 D RT I 31

13 whr 5 RT, D, I and E ar th translational molar nthaly of ah sis, dissoiation nrgy of, th ionization nrgy of atoms and th ltroni nrgy of atomi hydrogn rstivly. Thus, nthaly of atomi hydrogn givn by 5 (1 D and D I ar th hmial and rotons. Thus, total nthaly of hydrogn lasma is ) RT (1 ) E D I Th sond trm in th abov quation is th intrnal nthaly with E givn by E ln f RT T whr f is th intrnal artition funtion dfind by (.8) with 1 1 n n I and g n = n f g x( n n ) kt (.9) whr n dnots th rinial quantum numbr of atomi hydrogn and g n is th statistial wight..7. Sifi hat at onstant rssur Whn th drivativ of total nthaly is takn with rst to tmratur T at onstant rssur, th dndn of th dgr of ionization with tmratur must b onsidrd. Thus, w dfin two sifi hats, th first on, alld frozn sifi hat is obtaind by taking drivativ of th total nthaly with rst to tmratur at onstant whras in th sond on, alld rativ sifi hat of th lasma, this onstraint is liminatd. Thus, th total sifi hat of th hydrogn lasma is givn by (.10) f r whr th frozn sifi hat f is 3

14 whr 5 f (1 ) RT (1 ) T V is th intrnal sifi hat of atomi hydrogn and is givn by V (.11) V E T V ln f R ln T ln f ln T with f as th ltroni artition funtion of atomi hydrogn. Th rativ sifi hat is givn by r 5 RT I T, E T (.1) r 1 RT (1 ) 5 I E (.13) whr th dgr of ionization α and its drivativ ( α/ T) hav bn obtaind from th Saha s quation..7.3 Isntroi offiint For th isntroi offiint / v, w hav 1 RT 1 RT (1 )( I (1 ) ( I () 5 RT E 3 RT E 5 ) (1 ) R (1 ) 3 ) (1 ) R (1 ) V V (.14).8 Intrnal sifi hat Th intrnal sifi hat of atomi hydrogn drivativ of th artition funtion and an b writtn as V R 11, 600 E E t V dnds uon first and sond whih is th rodut of two fators. Th first fator is shar akd urv at low rssur du to larg numbr of xitd stats whras it is a flattnd urv at high rssur du to smallr numbr of xitd stats. Th sond fator is 33

15 arabolially drasing with tmratur. Th variation of V at low and high R rssurs is rsntd in Figur.4. Th intrnal ontribution int to th frozn art f of th sifi hat is again th rodut of two fators (i) (1 - fration of atoms and (ii) in Figur.5. V R. Th variation of int with tmratur is ditd R V R Figur.4. V vs. tmratur. Curvs (lowr to ur) rrsnt =1, 10, 100 R & 1000 atm rstivly. int R Figur.5. int R vs. tmratur. Curvs a, b, & d rrsnt =1, 10 &100atm rstivly. 34

16 .9 Efft of ltronially xitd stats on thrmodynami rortis Th rol of ltronially xitd stats (EES) in affting thrmodynami rortis of hydrogn thrmal lasma has bn xamind ovr a wid rang of tmratur and rssur by taking GS and ES lasmas. Th nthaly for GS and ES lasma, total sifi hat at onstant rssur along with its frozn and rativ ontributions hav bn workd out in dtail for both th ass. Thir rssur and tmratur dndn has bn ditd grahially. Th ratios and / int rrsnt th ontribution of ltroni xitation to th frozn and total sifi hats rstivly. Th isntroi offiints xhibit strong dndn on ltroni xitation. Stion.9.1 dsribs ratio of nthaly for GS and ES lasmas. In Stion.9. th frozn, rativ and total sifi hat at onstant rssur for both th ass has bn dsribd. In ordr to disuss th rol of EES on th sifi hat at onstant rssur, intrnal ontribution to th frozn and total sifi hat has bn obtaind in Stion.9.3.Th total isntroi offiint has bn obtaind at diffrnt rssurs in Stion.9.4, whih shows strong minima (sially at high rssurs)..9.1 Enthaly Th nthaly for th hydrogn lasma has bn valuatd. Th fft of ltronially xitd stats on th nthaly has bn grahially ditd by th lot of ratio of / int f =1000atm =100atm ES GS =10atm =1atm Figur.6. Ratio of nthaly rssurs. ES GS for GS and ES hydrogn lasma at diffrnt 35

17 nthaly for GS and ES lasmas at diffrnt tmraturs and rssurs and is dislayd in Figur Sifi hat at onstant rssur In ordr to stimat th fft of ltronially xitd stats on th frozn f, rativ ontributions r and th total sifi hat of hydrogn thrmal lasma, th numbr of xitd stats to b inludd in th artition funtion is first dtrmind using a siml utoff ritrion (basd on onfind atom (CA) modl). Thn a omutr rogram has bn dvlod to omut th dgr of ionization and th various ontributions to th sifi hat at diffrnt rssurs in th tmratur rang K. Th rsults thus obtaind for f and r ar rsntd in Figurs.7 &.8 for =1, 10& 100 atm. Th omarison of rsults of for th xitd stat (ES) lasma with thos of th ground stat (GS) lasma hav bn mad in Figur.9 whih dislays th variation of th ratio tmratur for =1, 10, 100 and 1000 atm rstivly. ( ES) ( GS) with.9.3 Intrnal ontribution to sifi hat Th rol of ltroni xitation on th frozn ontribution of sifi hat int has bn disussd and is givn by: int (1 ) V Whr th dgr of ionization and th intrnal sifi hat of atomi hydrogn V dnd uon ltroni artition funtion. Comarison of int with f and has bn attmtd by lotting th ratios / / int f and int with tmratur at diffrnt rssurs in Figurs.10&.11 rstivly. It may b mntiond hr that f is th sum of intrnal and translational ontributions i.. f 5 (1 ) RT int. 36

18 ES f (J/g/K) GS (a) (b) f (J/g/K) f (J/g/K) () Figur.7. Frozn sifi hat f vs. Tmratur for ES and GS hydrogn lasmas at (a) =1 atm (b) =10 atm and () =100 atm. 37

19 r(j/g/k ) ES (a) r(j/g/k) (b) r (J/g/K) ES () Figur.8. Rativ sifi hat vs. tmratur for ES and GS hydrogn lasmas at (a) =1 atm (b) =10 atm and () =100 atm. r 38

20 d ( ES) ( GS) a b Figur.9. ( ES) ( GS) vs. tmratur. Curvs a, b, & d rrsnt = 1, 10, 100& 100 atm rstivly. int f Figur.10. / int f vs. tmratur. Curvs a, b, & d rrsnt =1,10, 100&1000 atm rstivly. 39

21 int Figur.11. / int vs.tmratur.curvs (lowr to ur) rrsnt =1, 10, 100& 1000 atm rstivly..9.4 Isntroi offiint Following th similar rodur, omarison of rsults for total isntroi offiint hav bn rortd for both GS and ES lasmas in Figurs.1 at =1 and 100atm. 40

22 (a) Tmratur ES (b) Figur.1. Isntroi offiint vs. Tmratur for ES and GS hydrogn lasmas at (a) =1 atm and (b) =100 atm. 41

23 .10 Rsults and Disussion (i) From Figur., it has bn obsrvd that as rssur inrass, th oulation of EES inrass du to th fat that ionization of atoms ours at high tmratur and hn onntration of EES inrass. (ii) It has bn obsrvd from Figur.7, that for th ground stat (GS) lasma 5 f ( 1) R and it bhavs in a similar way as that of th dgr of ionization with tmratur for all rssurs whras th ak obsrvd for th xitd stat (ES) lasma is du to th addition of int to th abov xrssion, thrby indiating th rol of EES. With inras of rssur, ionization shifts towards high tmratur thrby, shifting tmratur (Figur.5). (iii)th valu of th ak of int for ES lasma towards high r for ES lasma is lowr than that of th GS lasma in th rgion whr ltroni xitation is dominant (Figur.8). Ground stat rsults ovrstimat th rativ ontribution by about 15% at =100 atm. This is du to th fat that th ltroni nrgy E of th atomi hydrogn aars in th xrssion for r with ngativ sign [Equation (.13)]. (iv) At low rssur =1 atm, th intrnal ontributions in f and r anl ah othr lading to a sort of omnsation whih is not obsrvd at high rssurs = atm.th dviation of rsults for GS and ES lasmas ar mor mhatially dislayd in Figur.9 whr th ratio ( ES) dos not bhav ( GS) monotonially, rathr maxima and minima ar obsrvd at a givn rssur. Th diffrns strongly inras with inras of rssur. But for 1atm ths diffrns ar ngligibl i.. th ratio is ratially indndnt of tmratur. This fat mislad rsarhrs in ast for not onsidring ltronially xitd stats in alulating thrmodynami rortis of LTE lasmas. (v) Rgarding th stimation of intrnal ontribution to sifi hat, Figur.10 dislays maxima of int at all rssurs. At =10 3 atm, this ratio is gratr than 0.5 f 4

24 i.. ontribution du to ltroni xitation is highr than th translational on. On th othr hand, th ratio int / (Figur.11) bhavs in a similar way to that of int / f but th orrsonding maxima dras du to th ngativ rol layd by th ltroni xitation in th rativ ontribution r of th sifi hat. At =10 3 atm, / int not ngligibl at high rssurs. =0.5 indiating thrby that th intrnal ontribution is (vi) In th absn of ltroni xitation i.. for th ground stat lasma f =5/3. Th obsrvd minima ar du to th ltroni xitation and strongly dnd uon rssur ( 1. 5 at =10 3 atm). Th ontribution of ltroni xitation tnds f to disaar in th total isntroi offiint whih inlud translation, ltroni and rativ omonnts (Figur.1 (a)). In fat, th omutd valus for GS and ES lasmas diffr by not mor than 7.5% du to som omnsation in diffrnt trms of [Eq. (.14)]. Thus, th ontribution of ltroni xitation maks its rsn flt to total sifi hat at high rssurs but not as mhatially as in th as of frozn sifi hat. Thus, thrmodynami rortis of a LTE hydrogn lasma dnd uon th numbr of ltronially xitd stats (EES) to b insrtd in th artition funtion. This inlusion of EES inrass th artition funtion thrby affting th dgr of ionization and th intrnal sifi hat of high tmratur atomi hydrogn lasmas sially at high rssurs. Its strong dndn on th frozn sifi hat of th lasma mixtur has bn obsrvd but th fft is ngligibl on th total sifi hat du to omnsation btwn th frozn and rativ ontributions to th total sifi hat at low rssur. This omnsation fails at rssurs highr than on atm. As th quilibrium rortis suh as nthaly and sifi hat dnd uon EES, thrfor it is of intrst to s how ths EES afft various transort rortis of thrmal lasmas and th sam hav bn disussd in th subsqunt hatrs. 43

Electron Transport Properties for Argon and Argon-Hydrogen Plasmas

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