Thermodynamical analysis for a variable generalized Chaplygin gas

Transcription

1 Available olie at WS 66 (7) 49-6 EISS 9-9 hermodyamical aalysis for a variable geeralized haplygi gas Mauel Malaver Departmet of asic Scieces, Maritime iversity of the aribbea, atia la Mar, eezuela address: mmf.umc@gmail.com ASRA osiderig the work of aigrahi (5) ad aigrahi ad hatterjee (6), i this paper we study some properties of variable geeralized haplygi gas (GG) model, exotic matter used i / some cosmological theories whose equatio of state is / where. We obtai a aalytical expressio for Joule-homso coefficiet ad we have deduced the iversio temperature for variable ad variable geeralized haplygi gas. We ivestigate the thermodyamical behavior for GG model i differet process. Keywords: ariable geeralized haplygi gas, ariable haplygi gas, Exotic matter, Joule- homso coefficiet, Iversio temperature. IRODIO he evideces of the cosmic acceleratio of the uiverse [-] has caused a primordial challege i the fudametal theories of physics ad cosmology. his acceleratio is frequetly attributed to a hypothetical eergy called dark eergy ad the haplygi type of gas cosmology is oe of the most reasoable explaatios of this recet pheomea. he form

2 World Scietific ews 66 (7) 49-6 of the haplygi gas equatio of state is the followig where is the pressure of the fluid, is the eergy desity of the fluid ad is a costat. he thermodyamical behaviour of the haplygi gas model was studied by Satos et al. [4], Guo ad Zhag [5], Myug [6], aigrahi [7] ad Malaver [8,9]. Satos et. al. [4] have studied the stability i haplygi gas model ad determied that the variable geeralized haplygi gas (GG) is thermodyamically stable durig ay expasio process. Guo ad Zhag [5] proposed a ew geeralized haplygi gas model that icludes the origial haplygi gas (G) model as a special case ad foud that the backgroud evolutio for the model is equivalet to that for a coupled dark eergy model with dark matter. Myug [6] obtais a ew geeral equatio of state that describes the haplygi gas completely ad cofirms that the G model could shows a uified picture of dark eergy ad eergy which cools dow through the uiverse expasio without ay critical poit. aigrahi [7] cocludes that the volume icreases whe temperature falls durig adiabatic expasios i the variable haplygi gas (G) model, which also is observed i a gas ideal []. Malaver [8,9] respectively, foud that the thermodyamic efficiecy of arot cycle for G model oly deped o the limits of maximum ad miimal temperature as i case of the ideal gas ad the photos gas ad that the adiabatic compressibility for this model oly will deped o the pressure ad the thermal capacity at costat pressure is always positive. More recetly, aigrahi ad hatterjee [] have showed that GG model satisfies the third law of thermodyamics. Ökcü ad Aydier [] studied the Joule-homso effects for charged AdS black holes ad obtaied iversio temperatures ad curves. I this paper a expressio is deduced for the Joule-homso coefficiet J from the thermal equatio of state of the GG model. With the equatio for J we obtai the coditio for the iversio temperature i this model. We also studied ad derived expressios as fuctios of temperature, pressure ad volume for differet thermodyamical process. he article is orgaized as follows: i Sectio, it presets the defiitio of the Joule- homso coefficiet; i Sectio, we show the deductio for the Joule- homso coefficiet ad the coditio of the iversio temperature for a haplygi gas; i Sectio 4, we have derived thermal equatios for the GG model for some thermodyamical process; i Sectio 5, we coclude.. E JOLE-OMSO OEFFIIE I this sectio, we preset the coefficiet of Joule-homso expasio [,]. he Joule-homso expasio is a throttled process where a gas at a high pressure passes through a porous valve to a regio with low pressure i a thermally isulated tube uder coditios of costat ethalpy. he chage of the temperature with the pressure is give by J () -5-

3 World Scietific ews 66 (7) he quatity () is called Joule-homso coefficiet. We ca express ( / ) i fuctio of heat capacity at costat pressure p ad. As the ethalpy is a state fuctio that depeds of pressure ad temperature, we ca write: ), ( ad d d d () Followig Dickerso [], the Joule-homso coefficiet µ J ca be obtaied derivig () with respect to pressure uder coditios of costat ethalpy: () Substitutig ( / ) =, p = ( / ) ad µ J =( / ) i () J (4) Rearragig the equatio (4), J (5) ad µ J ca be writte as J (6) With the defiitio of ethalpy it is possible calculate ( / ). is give by (7) With (7) we obtai (8)

4 World Scietific ews 66 (7) ca express as (9) ad () Replacig (9) ad () i (8), µ J ca be writte as follow J () At the iversio temperature ( / ) = ad µ J =. Sice ( / ) = for a ideal gas, accordig to the eq. (6), the Joule-homso coefficiet for a ideal gas is always zero [].. E IERSIO EMERARE I A ALYGI GAS I this sectio, we have determied the iversio temperature of Joule-homso expasio for the G ad GG models. For the G model, the thermal equatio of state for the pressure [7] is give by 6 () where is a positive uiversal costat, is a costat, 6 ad is a uiversal costat with dimesio of temperature. For the iteral eergy as fuctio of ad ()

5 World Scietific ews 66 (7) With the eq. (), we obtai 6 6 (4) hat implies that / 6 / 6 (5) From the eq. () we have deduced (6) ad agai for () (7) Substitutig (5), (6) ad (7) i () ad rearragig terms we have J (8) he expressio for Joule-homso coefficiet (8) is a explicit fuctio of the temperature ad the volume. I the iversio temperature i, µ J = that is i (9) With the objective of determiig i i GG model, the thermal equatio of state [] is

6 World Scietific ews 66 (7) () ad for the iteral eergy () With () ad () we obtai the followig expressios: () () (4) Agai, replacig (), () ad (4) i eq. (), we obtai

7 World Scietific ews 66 (7) 49-6 J (5) ad the iversio temperature i is give by i (6) For α = i the equatios () ad () we have the expressios for ad of work of aigrahi [7] with the G model ad the values of µ J ad i of (8) ad (9). 4. ROESSES I A ALYGI GAS We will cosider ow some thermodyamical processes i the GG model ad we have derived useful expressios for the study of these processes. A. Reversible Isothermal rocesses Followig Dickerso [], i a reversible isothermal process, there is o temperature chage ad ext = gas =. he thermodyamical variable iteral eergy (,) ca be expressed by d d d (7) he d = ad d takes the form d d (8) With () ad cosiderig = ((+α)-)/ i accordig to aigrahi ad haterjee [], the eq. (8) ca be itegrated ad we obtai -55-

8 World Scietific ews 66 (7) (9) For the first law of the thermodyamics dw dq d () Substitutig (9) e (), we have for dq d dq () With the well kow thermodyamical relatio () he eq. () ca be writte as d dq () ad with the expressio () for for the GG model, we have (4) for replacemet of (4) i () ad itegratig we obtai

9 World Scietific ews 66 (7) q (5) We have used the covetio of Wark ad Richards [4] that defies the work durig a reversible process as W d (6) he work for a isothermal reversible process i a GG model is give by W (7) Agai, with α = i (9), (5) ad (7), we ca recover the expressios for, q ad W deduced for Malaver [8] with the G model.. Reversible Adiabatic rocesses A adiabatic process is oe i which there is o heat flow i or out of the system []. I this case Q = ad for the eq. () d d dw d (8) is the thermal heat capacity at the costat volume With (7), () ad (8) we obtai d d (9) I the GG model []

10 World Scietific ews 66 (7) (4) Replacig eq. (4) ad eq. (4) i (9) we have d d (4) Itegratig l l (4) he for a reversible adiabatic process i a GG model we obtai cost (4) We ca deduce also a expressio for a adiabatic process i terms of ad. With the eq. (), (44)

11 World Scietific ews 66 (7) ad (45) Dividig (44) ad (45) we have (46) y substitutig of (4) i (46), we obtai a equivalete expressio i fuctio of ad (47) he eq. (47) implies that cost (48) whe α = we obtai the expressios for adiabatic process i the G model [8,9].. Reversible Isochoric rocesses he isochoric process is oe that is carried out to costat volume. For this case d = ad dw =. For d we have

12 World Scietific ews 66 (7) d dq d (49) ut we kow that (5) Replacig eq.(4) i (49) ad itegratig we obtai Q (5) with the expressio for of G model [7] (5) we have deduced Q (5) that is equivalet the case whe α = i the eq. (5)

13 World Scietific ews 66 (7) OLSIOS I this paper, we studied some thermodyamical aspects of GG model. We deduced a equatio for Joule-homso coefficiet for this model ad we have derived expressios for W, ad Q i some thermodyamical process. We have determied the iversio temperature of Joule-homso expasio which is fuctio of parameter α. I all the studied cases, we obtai the G model whe α =. As i the case of the ideal gas, we have used the thermodyamic relatio () i order to obtai expressios for reversible adiabatic process for GG model as fuctio of temperature, volume, pressure ad the parameter α. he study of the differet models of haplygi gas allows erich the courses of thermodyamics, which cotributes to a better compressio of the thermal pheomea. he equatios that describe the behavior of the haplygi gas are tractable mathematically ad offer a wide comprehesio of the accelerated uiverse expasio ad of the basic ideas of the moder cosmology. Refereces [] Sushkov, S. (5). Wormholes supported by a phatom eergy. hys. Rev. D7, 45. [] Lobo, F.S.. (5). Stability of phatom wormholes. hys. Rev. D7, 4. [] Lobo, F.S.. (6). Stable dark eergy stars. lass. Quat. Grav., [4] Satos, F.., edra, M. L. ad Soares,. (6). hys. Lett. 66, 86. [5] Guo, Z.K. ad Zhag, Y. Z. (7). hys. Lett. 645, 6. [6] Myug, Y.S. (). hermodyamics of haplygi gas. Astrophys. Space Sci. 5, [7] aigrahi, D. (5). hermodyamical behaviour of the variable haplygi gas. It. J. Mod. hys. D4, 55. [8] Malaver, M. (5). arot egie model i a haplygi gas. Research Joural of Modelig ad Simulatio, (), [9] Malaver, M. (6). Adiabatic ompressibility of the ariable haplygi gas. AASI ommuicatios, (), [] Leff,. S. (). eachig the photo gas i itroductory physics. Am. J. hys. 7, [] aigrahi, D. ad hatterjee, S. iability of variable geeralised haplygi gas - a thermodyamical approach. arxiv: gr-qc/68.44v. [] Ökcü, Ö. ad Aydier, E. Joule-homso expasio of charged Ads lack oles. arxiv: gr-qc/6.67v [] Dickerso, R. (969). Molecular hermodyamics, W. A. ejami, Ic, Melo ark, aliforia, IS:

14 World Scietific ews 66 (7) 49-6 [4] Wark, K. & Richards, D. (). ermodiámica, McGraw-ill Iteramericaa, Sexta Edició, IS: X. ( Received 6 Jauary 7; accepted Jauary 7 ) -6-