Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-zm8ws Total loading time: 0.317 Render date: 2021-06-14T16:53:17.658Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Compressibility in a Variable Generalised Chaplygin Gas

Subject: Physics and Astronomy

Published online by Cambridge University Press:  10 August 2020

Manuel Malaver
Affiliation:
Bijective Physics Institute, Idrija, Slovenia Maritime University of the Caribbean, Department of Basic Sciences, Catia la Mar, Venezuela.
Corresponding
E-mail address:

Abstract

Considering the Panigrahi and Chatterjee model (2017) for variable generalised Chaplygin gas, in this paper we found for this kind of exotic matter an analytic expression for the adiabatic compressibility βs. It was analyzed the behaviour of the adiabatic compressibility in the limit of high and low pressure. The derived equation for βs was used to deduce the value of the heat capacity at constant pressure Cp for variable generalised Chaplygin gas.

Type
Research Article
Information
Result type: Novel result
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Introduction

The discovery of accelerating expansion of the universe (Amanullah et al., Reference Amanullah, Lidman, Rubin, Aldering, Astier, Barbary, Burns, Conley, Dawson, Deustua, Doi, Fabbro, Faccioli, Fakhouri, Folatelli, Fruchter, Furusawa, Garavini, Goldhaber and Hook2010; Kamenschick et al., Reference Kamenschick, Moschella and Pasquier2001; Reiss et al., Reference Reiss, Filippenko, Challis, Clocchiattia, Diercks, Garnavich, Gilliland, Hogan, Jha, Kirshner, Leibundgut, Phillips, Reiss, Schmidt, Schommer, Smith, Spyromilio, Stubbs, Suntzeff and Tonry1998) has allowed the opening new horizons in the field of physics and cosmology and the exotic matter model known as Chaplygin gas is one of the most explanations for this phenomena. Astronomical evidence (Spergel et al., Reference Spergel, Bean, Doré, Nolta, Bennett, Dunkley, Hinshaw, Jarosik, Komatsu, Page, Peiris, Verde, Halpern, Hill, Kogut, Limon, Meyer, Odegard, Tucker and Wright2007) has shown that the matter that makes up stars and galaxies is less than 5% of the universe’s total mass and that much of the universe’s total energy is in the form of dark energy and the rest as non-baryonic cold dark matter particles that has never been detected. The variable Chaplyging gas model was studied for Panigrahi (Reference Panigrahi2015) and Malaver (Reference Malaver2016). Panigrahi (Reference Panigrahi2015) demonstrated that this model satisfies the third law of thermodynamics and Malaver (Reference Malaver2016) derived an expression for the adiabatic compresibility in a variable Chaplygin gas model. More recently, Panigrahi and Chatterjee (Reference Panigrahi and Chatterjee2017) propose a variable generalized Chaplygin gas model and deduce some thermodynamic equations in terms of temperature and volume.

Objective

In this paper an expression has been deduced for the adiabatic compressibility of the variable generalised Chaplygin gas (VGCG) from the equation of state given for Panigrahi and Chatterjee (Reference Panigrahi and Chatterjee2017). With the equation obtained for the adiabatic compressibility we derived an expression for the heat capacity at constant pressure $$ {C}_P $$ in a VGCG model. We found that the adiabatic compressibility for this model only will depend on the pressure and $$ {C}_P $$ is always positive.

Compressibility in a Variable Generalised Chaplygin Gas

For a variable generalised Chaplygin gas (Panigrahi & Chatterjee, Reference Panigrahi and Chatterjee2017) the equation of state for the pressure is

(1)$$ P=-{\left({B}_0{V}^{-\frac{n}{3}}\right)}^{\frac{1}{1+\alpha }}{\left(\frac{N}{1+\alpha}\right)}^{\frac{\alpha }{1+\alpha }}{\left(1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right)}^{\frac{\alpha }{1+\alpha }} $$

where $$ {B}_0 $$ is a positive universal constant, $$ N=\frac{3\left(1+\alpha \right)-n}{3} $$, n is a constant,$$ \tau $$ is a universal constant with dimension of temperature and α is a parameter.

Following Zemansky and Dittman (Reference Zemansky and Dittman1985), the adiabatic compressibility can be written as

(2)$$ {\beta}_S=-\frac{1}{V}{\left(\frac{\partial V}{\partial P}\right)}_S=-\frac{1}{V}{\left(\frac{\partial V}{\partial T}\right)}_S{\left(\frac{\partial T}{\partial P}\right)}_S=-\frac{\left(\partial v/\partial T \right)_S}{V\left(\partial v/\partial T \right)_S} $$

From the equation proposed by Malaver (Reference Malaver2015) for a VGCG model

(3)$$ V{\left[1-{\left(\frac{\tau }{T}\right)}^{\frac{1+\alpha }{\alpha }}\right]}^{-\frac{1}{N}}=const. $$

We obtain

(4)$$ {\left(\frac{\partial V}{\partial T}\right)}_S=\frac{\mathrm{const.}}{N}{\left[1-{\left(\frac{\tau }{T}\right)}^{\frac{1+\alpha }{\alpha }}\right]}^{\frac{1-N}{N}}\left(\frac{1+\alpha }{\alpha}\right){\left(\frac{\tau^2}{T^{2+\alpha }}\right)}^{\frac{1}{\alpha }} $$

Substituting (3) in (4) we have

(5)$$ {\left(\frac{\partial V}{\partial T}\right)}_S=\left(\frac{1+\alpha }{\alpha}\right)\frac{V}{N\left[1-{\left(\frac{\tau }{T}\right)}^{\frac{1+\alpha }{\alpha }}\right]}{\left(\frac{\tau^2}{T^{2+\alpha }}\right)}^{\frac{1}{\alpha }} $$

According with Malaver (Reference Malaver2017), the expression for an adiabatic reversible process for the VGCG model in terms of P and T is given by

(6)$$ \frac{PT^{\frac{N-\left(1+\alpha \right)}{N}}}{{\left[{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}-1\right]}^{\frac{N-1}{N}}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}. $$

With the eq. (6), $$ {\left(\frac{\partial P}{\partial T}\right)}_S $$ takes the form

(7)$$ {\left(\frac{\partial P}{\partial T}\right)}_S=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.{\left[{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}-1\right]}^{-\frac{1}{N}}{T}^{\frac{1+\alpha -N}{N}}\left\{\left(\frac{\alpha +1}{\alpha}\right)\left(\frac{N-1}{N}\right){\left(\frac{T}{\tau^{1+\alpha }}\right)}^{\frac{1}{\alpha }}-\left[\frac{N-\left(1+\alpha \right)}{N}\right]\left[{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}-1\right]\frac{1}{T}\right\} $$

Replacing (6) in (7) and rearranging terms

(8)$$ {\left(\frac{\partial P}{\partial T}\right)}_S=\frac{P\left\{\left(\frac{1+\alpha }{\alpha}\right)\left(\frac{N-1}{N}\right){\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}+\left[\frac{N-\left(1+\alpha \right)}{N}\right]\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]\right\}}{T\left[{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}-1\right]} $$

Substituting (5) and (8) in eq. (2) we have

(9)$$ {\beta}_S=\left(\frac{1+\alpha }{\alpha P}\right){\left(\frac{\tau }{T}\right)}^{\frac{2}{\alpha }}\frac{\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]}{\left[1-{\left(\frac{\tau }{T}\right)}^{\frac{1+\alpha }{\alpha }}\right]\left\{\left(1+\alpha \right)\left(N-1\right){\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}+\left[N-\left(1+\alpha \right)\right]\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]\right\}} $$

The expression for the adiabatic compressibility in the VGCG model (9) is an explicit function of the temperature and the pressure and can be used to calculate the heat capacity at constant pressure from the following equation (Zemansky & Dittman,Reference Zemansky and Dittman1985)

(10)$$ \frac{C_P}{C_V}=\frac{\beta_T}{\beta_S} $$

The isothermal compressibility $$ {\beta}_T $$ can be written as

(11)$$ {\beta}_T=-\frac{\left(1+\alpha \right)}{\left[N-\left(1+\alpha \right)\right]P} $$

With the equations (9) and (11) we obtain

(12)$$ \frac{\beta_T}{\beta_S}={\left(\frac{T}{\tau}\right)}^{\frac{2}{\alpha }}\frac{\left({\left(\frac{\tau }{T}\right)}^{\frac{1+\alpha }{\alpha }}-1\right)\left\{\left(1+\alpha \right)\left(N-1\right){\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}+\left[N-\left(1+\alpha \right)\right]\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]\right\}}{\left[N-\left(1+\alpha \right)\right]\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]} $$

The heat capacity at constant volume for a variable generalised Chaplygin gas (Panigrahi & Chatterjee,Reference Panigrahi and Chatterjee2017) is given by

(13)$$ {C}_V=\frac{{\left[\frac{B_0\left(1+\alpha \right){V}^N}{N}\right]}^{\frac{1}{1+\alpha }}{\left(\frac{T}{\tau^{1+\alpha }}\right)}^{\frac{1}{\alpha }}}{\alpha {\left\{1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right\}}^{\frac{2+\alpha }{1+\alpha }}} $$

And from the eq. (12) and eq. (13), $$ {C}_P $$ can be written as

(14)$$ {C}_P={\left[\frac{B_0\left(1+\alpha \right)}{N}\right]}^{\frac{1}{1+\alpha }}{\left(\frac{T^3}{\tau^{3+\alpha }}\right)}^{\frac{1}{\alpha }}{V}^{\frac{N}{1+\alpha }}\frac{\left[{\left(\frac{\tau }{T}\right)}^{\frac{1+\alpha }{\alpha }}-1\right]\left\{\left(1+\alpha \right)\left(N-1\right){\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}+\left[N-\left(1+\alpha \right)\right]\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]\right\}}{\alpha \left[N-\left(1+\alpha \right)\right]{\left[1-{\left(\frac{T}{\tau}\right)}^{\frac{1+\alpha }{\alpha }}\right]}^{\frac{3+2\alpha }{1+\alpha }}} $$

$$ {C}_P $$ > 0, this implies that n < 0, α > 0 and τ > 0. From the thermodynamic stability considerations n must have a negative value (Panigrahi, Reference Panigrahi2015). With α = 1 is recovered the expression for $$ {C}_P $$ of variable Chaplygin gas obtained by Malaver (Reference Malaver2016) as a particular case of this work.

Conclusions

We obtained an expression for the adiabatic compressibility of a variable generalised Chaplygin gas in terms of the pressure, temperature and α parameter. Is predicted that for $$ {\beta}_S\to \infty $$ when $$ P\to 0 $$ and βs → 0 if P → ∞ as in the ideal gas. Furthermore, with the equation for $$ {\beta}_S $$ we found a new equation for the heat capacity at constant pressure $$ {C}_P $$ for VGCG model that depends only the temperature and parameter α and that always is positive for $$ n $$ < 0, α > 0 and 0 <$$ T $$ < $$ \tau $$.

Funding Information

This research received no specific grant from any funding agency, commercial or not-for-profit sectors.

Conflict of interest

The author declare that there is no conflict of interest regarding the publication of this article.

Data Availability

The data that support the findings of this study are openly available in https://arxiv.org/abs/1608.00244, Gen.Rel.Grav. 49 (2017), no.3, 35.

References

Amanullah, R., Lidman, C., Rubin, D., Aldering, G., Astier, P., Barbary, K., Burns, M. S., Conley, A., Dawson, K. S., Deustua, S. E., Doi, M., Fabbro, S., Faccioli, L., Fakhouri, H. K., Folatelli, G., Fruchter, A. S., Furusawa, H., Garavini, G., Goldhaber, G., … Hook, I. (2010). Spectra and light curves of six type Ia supernovae at 0.511 < z < 1.12 and the Union2 compilation. Astrophysical Journal, 716, 712738.CrossRefGoogle Scholar
Kamenschick, A., Moschella, U., & Pasquier, V. (2001). An alternative to quintessence. Physics Letters, B511, 265268.CrossRefGoogle Scholar
Malaver, M. (2015). Carnot engine model in a Chaplygin gas. Research Journal of Modeling and Simulation, 2, 4247.Google Scholar
Malaver, M. (2016). Adiabatic compressibility of the variable Chaplygin gas. AASCIT Communications, 3, 6470.Google Scholar
Malaver, M. (2017). Thermodynamical analysis for a variable generalized Chaplygin gas. World Scientific News, 66, 149162.Google Scholar
Panigrahi, D. (2015). Thermodynamical behaviour of the variable Chaplygin gas. International Journal of Modern Physics, D24, 1550030. doi:https://doi.org/10.1142/S0218271815500303.CrossRefGoogle Scholar
Panigrahi, D., & Chatterjee, S. (2017). Viability of variable generalised Chaplygin gas—A thermodynamical approach. GenRelGrav, 49, 35.Google Scholar
Reiss, A. G., Filippenko, A. V., Challis, P., Clocchiattia, A., Diercks, A., Garnavich, P. M., Gilliland, R. L., Hogan, C. J., Jha, S., Kirshner, R. P., Leibundgut, B., Phillips, M. M., Reiss, D., Schmidt, B. P., Schommer, R. A., Smith, R. C., Spyromilio, J., Stubbs, C., Suntzeff, N. B., … Tonry, J. (1998). Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astronomical Journal, 116, 10091038.CrossRefGoogle Scholar
Spergel, D. N., Bean, R., Doré, O., Nolta, M. R., Bennett, C. L., Dunkley, J., Hinshaw, G., Jarosik, N., Komatsu, E., Page, L., Peiris, H. V., Verde, L., Halpern, M., Hill, R. S., Kogut, A., Limon, M., Meyer, S. S., Odegard, N., Tucker, G. S., … Wright, E. L. (2007). Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implications for cosmology. Astrophysical Journal Supplement, 170, 377.CrossRefGoogle Scholar
Zemansky, M. W., & Dittman, R. H. (1985). McGraw-Hill Interamericana. ISBN: 968-451-631-2.Google Scholar
Reviewing editor:  Stefano Camera Universita degli Studi di Torino, Physics, Via Pietro Giuria, 1, Torino, Italy, 10124 University of the Western Cape, Physics & Astronomy, Bellville, South Africa, 7535
This article has been accepted because it is deemed to be scientifically sound, has the correct controls, has appropriate methodology and is statistically valid, and met required revisions.

Review 1: Compressibility in a Variable Generalised Chaplygin Gas

Conflict of interest statement

Reviewer declares none

Comments

Comments to the Author: This paper has been written very well. My vote to this paper is minor revision. I will accept the paper after doing modifications. The reason is: The paper does not include results and discussion part. Also, it needs to be investigated that there is needed to provide some graphs and tables to show results in tabular and illustrative format. Also, it needs to be explained more about applications of Generalized Chaplygin Gas model in this paper and also its difference with its classic format to analyze results. I hope these comments help the author about improvement of this paper.

Presentation

Overall score 5 out of 5
Is the article written in clear and proper English? (30%)
5out of5
Is the data presented in the most useful manner? (40%)
5out of5
Does the paper cite relevant and related articles appropriately? (30%)
5out of5

Context

Overall score 5 out of 5
Does the title suitably represent the article? (25%)
5out of5
Does the abstract correctly embody the content of the article? (25%)
5out of5
Does the introduction give appropriate context? (25%)
5out of5
Is the objective of the experiment clearly defined? (25%)
5out of5

Analysis

Overall score 5 out of 5
Does the discussion adequately interpret the results presented? (40%)
5out of5
Is the conclusion consistent with the results and discussion? (40%)
5out of5
Are the limitations of the experiment as well as the contributions of the experiment clearly outlined? (20%)
5out of5

Review 2: Compressibility in a Variable Generalised Chaplygin Gas

Conflict of interest statement

Reviewer declares none

Comments

Comments to the Author: Cosmology progress is based on experimental data not on theoretical speculations. “Chaplygin gas” is a theoretical exotic idea, it is not experimental data. Experimental data is that universal space has Euclidean shape which is measured by NASA in 2014. This means the volume of the universal space is infinite. Results of Barbour, Fiscaletti, Sorli confirm that time has no physical existence, time is the numerical sequential order of events running in the universal space, which is the primordial energy of the universe we call today “superfluid quantum vacuum”, see work of Sbitnev, Fedi, Fiscaletti, Sorli. The energy of the vacuum itself is 95% of the missing energy of the universe, see the article published in Scientific Reports in August 2019: “Mass-energy Equivalence Extention on Superfluid Quantum Vacum”. Universal space is time-invariant (as time is the numerical sequential order of change in space). Cosmological principle is time-invariant. The idea of the beginning of the universe from some singularity as proposed by Hawking and Hartle seems is not the best idea because nobody knows how a mathematical point can turn into infinite Euclidean space. Big Bang cosmology calculated the age of the universe is controversial with a measured diameter of the observable universe. According to BB cosmology universe to reach today size should expand with 3,34 of light speed. This does not make sense and I think this article somehow could be accepted 20 years ago. But not today. According to the measured observed universe, CMB could not reach us yet. See the recent article A THREE-DIMENSIONAL NON-LOCAL QUANTUM VACUUM AS THE ORIGIN OF PHOTONS in the Ukrainian Journal of physics (Fiscaletti, Sorli). CMB is the radiation of the existent quantum vacuum. And the gravitational redshift of the light coming from distant galaxies can be seen as a “tired light effect” proposed by Zwicky. Direct reading of data is not in favor of BB cosmology.

Presentation

Overall score 4 out of 5
Is the article written in clear and proper English? (30%)
4out of5
Is the data presented in the most useful manner? (40%)
4out of5
Does the paper cite relevant and related articles appropriately? (30%)
4out of5

Context

Overall score 4 out of 5
Does the title suitably represent the article? (25%)
5out of5
Does the abstract correctly embody the content of the article? (25%)
5out of5
Does the introduction give appropriate context? (25%)
4out of5
Is the objective of the experiment clearly defined? (25%)
2out of5

Analysis

Overall score 2 out of 5
Does the discussion adequately interpret the results presented? (40%)
2out of5
Is the conclusion consistent with the results and discussion? (40%)
2out of5
Are the limitations of the experiment as well as the contributions of the experiment clearly outlined? (20%)
2out of5
You have Access
Open access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Compressibility in a Variable Generalised Chaplygin Gas
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Compressibility in a Variable Generalised Chaplygin Gas
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Compressibility in a Variable Generalised Chaplygin Gas
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *