Adiabatic Compressibility of the Variable Chaplygin Gas

Transcription

1 AAI ommunications olume 3, Issue February 5, 6 online I: Adiabatic ompressibility of the ariable haplygin Gas Manuel Malaver Department of Basic ciences, Maritime University of the aribbean, argas tate, enezuela I Keywords Adiabatic ompressibility, haplygin Gas, osmological heories, Equation of tate, Exotic Matter n this paper, we have deduced an analytic expression for the adiabatic compressibility for a haplygin gas which is a function of the pressure and temperature. We have considered the work of anigrahi (5) for variable haplygin gas, B exotic matter used in some cosmological theories whose equation of state is of the form ρ with B B n 3 and n is a constant. he expression obtained for was used for the determination of the value of the thermal capacity at constant pressure for variable haplygin gas. It is predicted in this research the behaviour of the adiabatic compressibility of haplygin gas in the limit of high and low pressure. We have found that the adiabatic compressibility is function of the pressure and when and if as in the ideal gas and is always positive with n <. Introduction Recent observational evidences suggest that the present universe is accelerating [,, 3] and the cosmological model of haplygin gas (G) is one of the most reasonable explanations of recent phenomena [4, 5, 6, 7]. he form of the G equation B of state is the following where is the pressure of the fluid, ρ is the energy density of the fluid andb is a constant. ρ he thermodynamic behaviour of the G model was studied by antos et al. [8],Guo and Zhang [9], Myung [] and anigrahi []. antos et. al [8] have studied the thermodynamical stability in generalized haplygin gas model and determined that for an equation of state that depends only on the temperature the fluid presents thermodynamic stability during any expansion process. Guo and Zhang [9] proposed a new generalized haplygin gas model that includes the original haplygin gas model as a special case and found that the background evolution for the model is equivalent to that for a coupled dark energy model with dark matter. Myung [] found a new general equation of state that describes the haplygin gas completely and confirms that the haplygin gas could shows a unified picture of dark energy and energy which cools down through the universe expansion without any critical point. anigrahi [] analyzed the thermodynamical behaviour of the ariable haplygin gas and found that the third law of thermodynamics is satisfied in this model and that the volume increases when temperature falls during adiabatic expansions, which also is observed in an gas ideal []. onsidering the cosmological constant as a pressure, several authors [3, 4, 5, 6] have researched the thermodynamic behaviour of the black holes. Wang et. al [3] studied the first law of thermodynamics in black holes and Kerr-de itter spacetimes and found that by assuming the cosmological constant as a variable state parameter the mass formulas of the first law of black hole thermodynamics in the asymptotic flat spacetimes can be directly extended to rotating black holes. ekiwa [4] analyzed the thermodynamic properties and the cosmological horizon for black holes solutions and showed that cosmological constant must decrease when takes into account the quantum effect. Larrañaga [5] found that using the cosmological constant as a new thermodynamic state variable, the differential and integral mass formulas of the first law of thermodynamics for asymptotic flat spacetimes can be extended to be used in black holes. Dolan [6] has investigated the behaviour of the adiabatic compressibility and the speed of sound v of the black hole and obtain that vanishes for non- s rotating black holes but it increases when the angular momentum reaches a maximum value. Dolan [6] also found that v is s

2 65 6; 3(): 64-7 equal to c for a non-rotating black hole and decreases when the angular momentum is increased. An ideal gas is a gas composed of a group of randomly moving, non-interacting point particles. he ideal gas approximation is useful because it obeys the gases laws and represent the vapor phases of fluids at high temperatures for which the heat engines is constructed [7]. A heat engine that can work with an ideal gas as working substance is the arnot cycle. Malaver [8] found that the thermodynamic efficiency of arnot cycle for G model only depend on the limits of maximum and minimal temperature as in case of the ideal gas and the photons gas. In this paper an expression is deduced for the adiabatic compressibility of the variable haplygin gas from the thermal equation of state for the pressure given for anigrahi []. With the equation obtained for the adiabatic compressibility we have deduced an expression for the thermal capacity at constant pressure in a haplygin gas. We have found that the adiabatic compressibility for G model only will depend on the pressure as in case of the ideal gas and is always positive. he article is organized as follows: in ection, it presents the deduction of the compressibility for an ideal gas; in ection 3, we show the deduction for the adiabatic compressibility for a haplygin gas; in ection 4, we conclude. Adiabatic ompressibility in an Ideal Gas Following Dickerson [9] and ash [] an adiabatic process is one in which is no heat flow is or out of the system. In this case Q and in agreement with the first law UW. his implies that and for an ideal gas, Where is the thermal capacity at constant volume. du d d () d+ d () In a reversible expansion of an ideal gas R/. ubstituting this equality in () and dividing by, we obtain int Integrating (3) onverting to the exponential form ext d d R + (3) R + (4) ln ln R (5) ubstituting in (5) (6) and for a reversible adiabatic process with a ideal gas we have γ const (7) where γ

3 I: is the thermal capacity at constant pressure. he adiabatic compressibility is defined as [, ] (8) According Zemansky and Dittman [] of the eq.(7) we obtain γ γ c o n st γ (9) Replacing (9) in (8) we have that for an ideal gas () γ Adiabatic ompressibility in an haplygin Gas For a haplygin gas, the thermal equation of state for the pressure [] is given by n 6 B () where B is a positive universal constant, n is a constant, temperature. 6 n and is a universal constant with dimension of 3 Following Zemansky and Dittman [], the adiabatic compressibility (8) can be written as ( ) ( ) () For a reversible adiabatic process in a haplygin gas [8] const (3) he eq.(3), we obtain const 3 (4) In this paper, we have deduced an expression for an adiabatic reversible process in terms of and (see Appendix): co nst (5) With the eq.(5), takes the form

4 67 6; 3(): 64-7 co n st (6) Replacing (5) in (6) and rearranging terms ( ) ( ) + (7) ubstituting (4) and (7) in eq.(), we have ( ) + ( ) (8) he expression for adiabatic compressibility (8) is an explicit function of the temperature and the pressure. he equation (8) can be used to calculate the thermal capacity at constant pressure in a haplygin gas from the following expression [] (9) where is the isothermal compressibility defined as [, ] ( ) () with (), for a haplygin gas we have B () ubstituting () in (), we obtain () Replacing () in (), can be written as ( ) (3) and from the equations (8) and (3), we obtain ( ) ( ) ( ) + (4) he thermal capacity at constant volume for a haplygin gas is given by [] / B 3 (5) with eq.(4) and eq.(5) can be written as

5 I: B 3 / ( ) ( ) ( ) (6) >, this implies that n < and >. Form the thermodynamic stability considerations should n always have negative value []. onclusions We have deduced an expression for the adiabatic compressibility of a haplygin gas, which is a function of the pressure and the temperature. Is predicted for that when and if as in the ideal gas. Furthermore, with the equation for we found a new formula for the thermal capacity at constant pressure for G model that depends only the temperature and that always is positive for n < and < <. he study of haplygin gas can enrich the courses of thermodynamics, which contributes to a better compression of the thermal phenomena. he thermodynamic equations that describe the behavior of the haplygin gas are tractable mathematically and offer a wide comprehension of the accelerated universe expansion and of the basic ideas of the modern cosmology. Appendix In this Appendix, we have deduced an expression for a reversible adiabatic process in terms of and in a haplygin gas. With the eq. () we obtain B n 6 A. and B n 6 A. Dividing A. with A., we have A.3 Following Malaver [8], for a reversible adiabatic process in terms of and const A.4 his implies that A.5 ubstituting A.5 in A.3

6 69 6; 3(): 64-7 A.6 rearranging A.6 we have A.7 const A.8 References Manuel Malaver Manuel Malaver graduated in entral University of enezuela, who held Doctor in heoretical and Applied Mechanics. He has been rofessor in General hysics, hermodynamics and Mechanics of Fluids from the Maritime University of the aribbean since January 7. His areas of research are osmology and Gravitation, General Relativity, Mathematical hysics and Applied Mathematics. [] ushkov,. (5). Wormholes supported by a phantom energy. hys. Rev, D7, 435. [] Lobo, F... (5). tability of phantom wormholes. hys. Rev, D7, 4. [3] Lobo, F... (6). table dark energy stars. lass. Quant.Grav, 3, [4] Fabris, J.., Goncalves..B. and de ouza,.e. (). Gen. Relativ. Grav. 34, 53. [5] Lima, J.A.., unha, J.. and Alcaniz, J.. (8). A simplified approach for haplygin-type cosmologies. Astropart. hys. 3, [6] Bento, M.., Bertolami, O. and en, A.A. (). hys. Rev. D66, [7] anigrahi, D. and haterjee,. (). JA. [8] antos, F.., Bedran, M.L. and oares,. (6). hys. Lett. B636, 86. [9] Guo, Z.K. and Zhang, Y.Z. (7). hys. Lett. B645, 36. [] Myung, Y.. (). hermodynamics of haplygin gas. Astrophys. pace ci.335, [] anigrahi, D. (5). hermodynamical behaviour of the variable haplygin gas. Int. J. Mod. hys. D4, 553. [] Leff, H.. (). eaching the photon gas in introductory physics. Am. J. hys, 7, [3] Wang,., Wu, -Q., Xie, F. and Dan, L. (6). hin. hys. Lett. 3, 96. [4] ekiwa, Y. (6). hys. Rev. D73, 849. [5] Larrañaga, E.A. (7). [6] Dolan, B.. (). ompressibility of rotating black holes.hys.rev, D84,753. [7] Lee, H. (). arnot cycle for photon gas? Am. J. hys, 69, [8] Malaver, M. (5). arnot engine model in a haplygin gas. Research Journal ofmodeling and imulation, (), [9] Dickerson, R. (969). Molecular hermodynamics, W. A. Benjamin, Inc, Menlo ark, alifornia, IB:

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