4.3.5.1: Ideal Gas in Varying Gravity

Last updated
Save as PDF

Page ID: 673

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In physics, it was explained that the gravity is a function of the distance from the center of the plant/body. Assuming that the pressure is affected by this gravity/body force. The gravity force is reversely proportional to \(r^2\). The gravity force can be assumed that for infinity, \(r \rightarrow \infty\) the pressure is about zero. Again, equation (11) can be used (semi one directional situation) when \(r\) is used as direction and thus

\[ \dfrac{\partial P }{\partial r} = - \rho \dfrac {G}{r^2}
\label{static:eq:gravityGchange}
\]
where \(G\) denotes the general gravity constant. The regular method of separation is employed to obtain

\[ \int_{P_{b}}^{P} \dfrac{dP}{P} =
- \dfrac {G}{RT} \int_{r_{b}} ^r \dfrac {dr}{r^2}
\label{static:eq:gravityGchangeB}
\]
where the subscript \(b\) denotes the conditions at the body surface. The integration of equation (111) results in

\[ \ln \dfrac{P}{P_{b}} =
- \dfrac {G}{RT} \left( \dfrac{1}{r_{b}}-\dfrac{1}{r}\right)
\label{static:eq:gravityGchangeA}
\]
Or in a simplified form as

\[ \dfrac{\rho}{\rho_{b}} =
\dfrac{P}{P_{b}} =
\text{ e} ^{ -\dfrac{G}{RT}
\dfrac{r-r_{b}}{r\,r_{b}}
}
\label{static:eq:gravityGchangeF}
\]
Equation (113) demonstrates that the pressure is reduced with the distance. It can be noticed 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, equation (113) can be expanded in Taylor series as

\[ \dfrac{\rho}{\rho_{b}} =
\dfrac{P}{P_{b}} =
\overbrace{1 -
\dfrac{G\,\left( r-r_b\right) }{R\,T} }^{\text{standard}} -
\overbrace{\dfrac{\left( 2\,G\,{R\,T}+
{G}^{2}\,r_b\right) \,{\left( r-r_b\right)
}^{2}}{2\,r_b\,{(R\,T)}^{2}}
+\cdots}^{\text{correction factor}}
\label{static:eq:gravityGchangeFexpended}
\]

Notice that \(G\) isn't our beloved and familiar \(g\) and also that \(G\,r_b/RT\) is a dimensionless number (later in the Chapter 9 a discution about the definition of the dimensionless number and its meaning).

Contributors and Attributions

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.