Analytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases

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1 Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN (Prt) & ISSN (Ole) Alytcl Approch for the Soluto of Thermodymc Idettes wth Reltvstc Geerl Equto of Stte Mxture of Gses Skr Chttopdhyy Cetre for Theoretcl Plsm Reserch, 346, R.K.Plly (Blk), P.O. Sorpur, Kolkt 50, West Begl, INDIA. & Deprtmet of Mthemtcs, Dr J.L.N. Vdyly, P.O. Thkurber, Dst. South 24 Prgs, West Begl, INDIA. Dpkr Ry (Retd.) Deprtmet of Mthemtcs, Jdvpur Uversty, Kol.- 32, West Begl, INDIA. Abstrct: D.P. Mso d A.M.Kgth tegrted some thermodymc dettes for del gs equto of stte p = kt where p s the pressure, s the prtcle umber desty, k s the Boltzm costt d T s the bsolute temperture.the preset uthors exted tht work for mxture of del gses wth the geerl equto of stte p = p(,t) s specl cse d foud the totl eergy desty fucto ( ) d the etropy per ut volume (S) s fucto of other thermodymc vrbles d T. PACS Number: 05.70, Ce Keywords: Thermodymc vrbles. INTRODUCTION I collso domted equlbrum for gs, two thermodymc vrbles descrbe the thermodymc stte of the system completely. D.P.Mso d A.M.Kgth[] cosdered the thermodymc vrbles of reltvstc gs collso domted equlbrum s fuctos of the prtcle umber desty () d the etropy per ut volume (S). For oe compoet reltvstc gs, the geerl form for collso-domted equlbrum dstrbuto fucto s, exp f x p x x p Where K, kt, k = Boltzm s costt, K = Chemcl potetl, kt T = bsolute temperture, p s the four mometum of prtcle t pot x of spce-tme, u, u u u, u d u re the kemtc d dymc me four-velctes of the k D k D gs, hs dfferet vlues for dfferet dstrbuto.for reltvstc Bose Este dstrbuto, Ferm Drc dstrbuto, Mxwell Boltzm dstrbuto 0.Actully Olver d Dvs[2] hve show tht the bsolute temperture T s homogeeous fucto of degree oe d S reltvstc gs collso domted equlbrum wth equto of stte p = where p s the sotropc pressure d s the totl eergy desty. But Mso d Kgth took the terestg results of Olver d Dvs dfferet wy d exteded ther () ARC Pge 6

2 Skr Chttopdhyy & Dpkr Ry results to the equto of stte other th p = for oe compoet gs. I recet pper Mso d Kgth [] hve show tht for ll vlues of the equlbrum dstrbuto fucto () oe compoet gs the followg two well-kow thermodymc dettes d TdS Kd (2) TS K p (3) re stsfed d for reltvstc Mxwell-Boltzm dstrbuto the bove two dettes c be tegrted for reltvstc del gs equto of stte p = kt (4) where s the totl eergy desty, T s the temperture Kelv, s the prtcle umber desty, K s the chemcl potetl per prtcle, p s the pressure, S s the etropy per ut volume d k s the Boltzm costt. They obted the geerl soluto from equtos (2) to (4) for the bsolute temperture T(,S), the chemcl potetl per prtcle K(,S) d the totl eergy desty fucto (,S). I ths pper the preset uthors cosder mxture of gses wth more geerl equto of stte p = p(t, ) (5) Whle the thermodymc dettes tke the form d TdS K d (6) (7) TS K p Where d K re respectvely the prtcle umber desty d chemcl potetl of the -th gs whle, T, S d p re s before. Flly the cse of mxture of del gses s cosdered s specl cse. The prcpl m of ths pper s to solve ler dfferetl equto for obtg the expresso of the totl eergy desty fucto ( ) d etropy per ut volume (S) by lytcl pproch. Ths techque my be used dfferet brches of Physcs d egeerg sceces. I secto 2, we wrte dow the bsc equtos from the thermodymc dettes d foud the expresso for totl eergy desty fucto ( ) d etropy per ut volume (S) fter soluto of equto (5). I secto 3, mxture of del gses topc s dscussed d expressos for d S for those mxture of del gses re gve. The etre dscusso s gve secto 4. Secto 5 cots cocluso d the future plg of the problem. 2. FORMATION OF THE DIFFERENTIAL EQUATION AND ITS SOLUTION From the equto (6) oe gets 2 S,,,... (8) T S K Hece vew of the equto (5), oe c reduce the equto (7) to (9) (0) Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Pge 7

3 Alytcl Approch for the Soluto of Thermodymc Idettes wth Reltvstc Geerl Equto of Stte Mxture of Gses () S p,, 2,... S S p beg gve fucto of the vrbles dcted the equto () s the equto to be solved for. Now we defe m, m for 2 S, m, m,... S,,, (2) (3),,,... pt,,,... q T m m So tht S S 2 2 m m m m 2,3,... m m (4) The equtos (2) to (4) log wth the lst three reltos reduce the equto () to S m q, m, m,... S m S 2 Where the equto (5) cots m 2, m 3 etc. s mere prmeters. Ths equto c be solved the followg wy: Dfferettg the equto (5) wth respect to S whle tretg m, m 2,... s costts d usg the equtos (9) d (3) oe gets q T T S m 0 T S m Now sted of tretg S, m, m 2,... s depedet f oe trets T,m,m 2,... s depedet vrbles the bove equto tkes the form q S T S T, m, m2,... m m Ths s ler equto S d s solved by tegrtg ths equto. Thus the soluto s P S m (6) T q T, m, m,... P T m m dm Where,,, m (7) Also tegrtg by prts the equto (9) d usg the equtos (6) d (7) oe gets (5) Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Pge 8

4 Skr Chttopdhyy & Dpkr Ry P m T P m, m2,... T Where s rbtrry fucto of the vrbles dcted. Now puttg the lst equto bck to the equto () d usg the equtos (9), (6) d (7) oe gets m, m,... m.costt 2 Hece the soluto for s gve the followg form: P m T P (8) T But t s foud from the equto (8) tht c be esly bsorbed the rbtrry fucto P where m 2, m 3,... re treted s costts, s rbtrry fucto of the vrbles dcted, the fucto q ( T, m, m 2,...) s relted to the gve fucto p ( T,, 2,...) through the equto (4) whle m, m 2,... re relted to, 2,... through the equtos (2). As the prevous equtos (6) to (8) log wth the equtos (2) d (4), we express d S s fuctos of T,, 2,... d thereby we express s fucto of S,, 2,... wth T s the prmeter. 3. MIXTURE OF IDEAL GASES [SPECIAL CASE] Restrctg to mxture of del gses d usg the lw of prtl pressure the equto (5) tkes the form p kt (9),2 Where k s the Boltzm costt. Puttg ths equto (4) d usg the equtos (2), (3), (6), (7) d (8) oe gets 2 3 T,,,... S k l (20) T 2 3 T,,, T T,,,... T Where s rbtrry fucto of the vrbles dcted s before? 4. DISCUSSION I reltvstc gs wth collso domted equlbrum for equto of stte p =, Olver d Dvs took the pror work for the determto of the bsolute temperture T d showed tht T s homogeeous fucto d S.Further t c be observed tht the reltvstc perfect gs lw p = kt holds for the reltvstc Mxwell-Boltzm dstrbuto fucto. But for qutum gs, the equto of stte (4) s ot stsfed wheres for o-qutum gs collso-domted equlbrum descrbed by the reltvstc Mxwell-Boltzm dstrbuto, the equto of stte (4) s stsfed. Moreover Mso d Kgth foud the two depedet thermodymc vrbles ( d S) for the two thermodymc dettes (2) d (3) wth the del gs equto of stte (4). But ths result s vld for lmted rge of eerges. After tht the preset uthors tegrted the sme dettes for mxture of gses wth more geerl equto of stte p = p (, T) for the vrbles, S d K. It s more geerl result th tht of Mso d Kgth from whch the result for sgle gs s obted s specl cse. (2) Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Pge 9

5 Alytcl Approch for the Soluto of Thermodymc Idettes wth Reltvstc Geerl Equto of Stte Mxture of Gses 5. CONCLUSION I summry, the thermodymc dettes gve by equtos (6) d (7) for mxture of gses whose equto of stte s gve by the equto (5), hve bee tegrted to gve the equtos (6) to (8) where q ( T, m, m 2,...) s coected to p ( T,, 2,...) through the equto (4) whle m, m 2,... etc. re coected to, 2,...etc. through the equtos (2). I the specl cse of mxture of del gses where the equto of stte s gve by the equto (9), S d re gve by the equtos (20) d (2). Ths work s the geerlzed exteso of the prevous work of Mso d Kgth[]. Our future pl s to solve the sme thermodymc dettes (2) d (3) for my compoet qutum gs wth modfed equto of stte. ACKNOWLEDGEMENT Oe of the uthors, S.Chttopodhyy, would lke to thk Dr. S.N.Pul for hs vluble suggestos d dscussos the preprto of ths pper to ts preset form. REFERENCES [] D.P.Mso d A.M.Kgth, J.Mth. Phys. 32 (99) 493 [2] D.R.Olver d W.R.Dvs, A. Ist. H. Pocre 30 (979) 339 Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Pge 0