Compressibility in a Variable Generalised Chaplygin Gas

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 β_s. It was analyzed the behaviour of the adiabatic compressibility in the limit of high and low pressure. The derived equation for β_s was used to deduce the value of the heat capacity at constant pressure C_p for variable generalised Chaplygin gas.

Introduction

The discovery of accelerating expansion of the universe (Amanullah et al., 2010; Kamenschick et al., 2001; Reiss et al., 1998) 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., 2007) 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 (2015) and Malaver (2016). Panigrahi (2015) demonstrated that this model satisfies the third law of thermodynamics and Malaver (2016) derived an expression for the adiabatic compresibility in a variable Chaplygin gas model. More recently, Panigrahi and Chatterjee (2017) 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 (2017). 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, 2017) 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 (1985), 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 (2015) 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 (2017), 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,1985)

(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,2017) 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, 2015). With α = 1 is recovered the expression for $$ {C}_P $$ of variable Chaplygin gas obtained by Malaver (2016) 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 <$$ T $$ < $$ \tau $$.