## 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

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

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

We obtain

Substituting (3) in (4) we have

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

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

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)

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

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

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 *β _{s}* → 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.

## 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.