Editing Appendix/Ramblings/PatrickMotl (section)

===Tying Expressions into H_Book Context===

In our wiki-based chapter titled, "[[PGE/FirstLawOfThermodynamics#First_Law_of_Thermodynamics|First Law of Thermodynamics]]," we have introduced the concept of an ''entropy tracer,'' <math>~\tau</math>.  In the subsubsection of this chapter that is titled, "[[PGE/FirstLawOfThermodynamics#Substantiation|Substantiation]]," we show that an expression for the specific entropy of a fluid element is,
<div align="center">
<math>~s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, .</math>
</div>
In addition, from our wiki-based chapter titled, "[[SR/IdealGas#Ideal_Gas_Equation_of_State|Ideal Gas Equation of State]]," we find the relations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c_P - c_V </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Re}{\bar\mu} </math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;
  <td align="right">
<math>~\gamma_g </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{c_P}{c_V} </math>
  </td>
  <td align="center">&nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp;
  <td align="right">
<math>~c_P </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g}{(\gamma_g-1)} \biggl( \frac{\Re}{\bar\mu} \biggr) \, .</math>
  </td>
</tr>
</table>
Hence this expression for the entropy may be rewritten as,
<div align="center">
<math>~s = \frac{\gamma_g}{\gamma_g-1} \biggl( \frac{\Re}{\bar\mu} \biggr) \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant}  \, .</math>
</div>
Aside from the factor of <math>~({\bar\mu})^{-1}</math> that appears on the RHS &#8212; more on this [[#Pick_a_Different_Molecular-Weight_Ratio|below]] &#8212; these are the expressions that Patrick has used to generate the <font color="red">'''s.ps'''</font> plot, where the (unlabeled) ordinate is the normalized specific entropy, <math>~s/\Re</math>.

At the end of another subsubsection titled, "[[PGE/FirstLawOfThermodynamics#Initial_Recognition|Initial Recognition]]," we also find a relevant expression, namely,
<div align="center">
<math>~\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, .</math>
</div>
Hence, ignoring the additive constant, in general we may write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{s}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{(\gamma_g-1)}\ln \biggl(\frac{\tau}{\rho}\biggr)^{\gamma_g} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P}{(\gamma_g-1)\rho^{\gamma_g}} \biggr] \, .
</math>
  </td>
</tr>
</table>