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4.3.5.2: Real Gas in Varying Gravity

The regular assumption of constant compressibility, \(Z\), is employed. It has to remember when this assumption isn't accurate enough, numerical integration is a possible solution. Thus, equation (111) is transformed into  
\[
    \int_{P_{b}}^{P} \dfrac{dP}{P}  =
    - \dfrac {G}{Z\,R\,T} \int_{r_{b}} ^r \dfrac {dr}{r^2}
    \label{static:eq:gravityGchangeBZ} \tag{115}
\]  
With the same process as before for ideal gas case, one can obtain
\[
    \dfrac{\rho}{\rho_{b}} =
    \dfrac{P}{P_{b}} =
    \text{ e} ^{  -\dfrac{G}{Z\,R\,T}   
            \dfrac{r-r_{b}}{r\,r_{b}}
            }
    \label{static:eq:gravityGchangeFZ} \tag{116}
\]  
Equation (113) demonstrates that the pressure is reduced with the distance. It can be observed that for \(r \rightarrow r_{b}\) the pressure is approaching \(P \rightarrow P_{b}\). This equation confirms that the density in outer space is zero \(\rho(\infty) = 0\). As before Taylor series for equation (113) is
\[
    \dfrac{\rho}{\rho_{b}} =
    \dfrac{P}{P_{b}} =  
    \overbrace{1 -
    \dfrac{G\,\left( r-r_b\right) }{Z\,R\,T} }^{\text{standard}} -
            \overbrace{\dfrac{\left( 2\,G\,{Z\,R\,T}+ \\
        {G}^{2}\,r_b\right) \,{\left( r-r_b\right)
            }^{2}}{2\,r_b\,{(Z\,R\,T)}^{2}}
    +\cdots}^{\text{correction factor}}
    \label{static:eq:gravityGchangeFZexpended} \tag{117}
\]
It can be noted that compressibility factor can act as increase or decrease of the ideal gas model depending on whether it is above one or below one. This issue is related to Pushka equation that will be discussed later.

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.