Editing VE (section)

===Reservoir of Thermodynamic Energy===
<math>\mathfrak{S}_\mathrm{therm}</math> derives from the differential, "PdV" work that is often discussed in the context of thermodynamic systems.  It should be made clear that, here, "dV" refers to the differential volume ''per unit mass,''  so it should be written as "<math>~d(\rho^{-1})</math>", to be consistent with the notation used throughout this H_Book.  Therefore, the differential thermodynamic work is,
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<math>d\mathfrak{w} = Pd(1/\rho) = -  \biggl( \frac{P}{\rho^2} \biggr) d\rho \, .</math>
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After an ''evolutionary'' equation of state has been adopted, this differential relationship can be integrated to give an expression for the energy per unit mass, <math>\mathfrak{w}</math>, that is potentially available for work.  Then we define the thermodynamic energy reservoir as,
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<math>\mathfrak{S}_\mathrm{therm} \equiv - \int \mathfrak{w} ~dm \, .</math>
</div>

====Isothermal Systems====
If each element of gas maintains its temperature when the system undergoes compression or expansion &#8212; that is, if the compression/expansion is isothermal &#8212; then, the relevant evolutionary equation of state is,
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<math>P = c_s^2 \rho \, ,</math>
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where the constant, <math>c_s</math>, is the isothermal sound speed.  In this case, the expression for the differential thermodynamic work becomes,
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<math>d\mathfrak{w} =  -  \biggl( \frac{c_s^2}{\rho} \biggr) d\rho =  -  c_s^2 d\ln\rho \, .</math>
</div>
Hence, to within an additive constant, we have,
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<math>\mathfrak{w} = -  c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) \, ,</math>
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where, <math>\rho_0</math> is a (as yet unspecified) reference density, and integration throughout the configuration gives (for the isothermal case),
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<math>\mathfrak{S}_\mathrm{therm} = +  \int c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) dm \, .</math>
</div>

====Adiabatic Systems====
If, upon compression or expansion, the gaseous configuration evolves adiabatically, the pressure will vary with density as,
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<math>P = K \rho^{\gamma_g} \, ,</math> 
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where, <math>K</math> specifies the specific entropy of the gas and {{ Template:Math/MP_AdiabaticIndex }} is the ratio of specific heats that is relevant to the phase of compression/expansion.  In this case, the expression for the differential thermodynamic work becomes,
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<math>d\mathfrak{w} =  -  K \rho^{{\gamma_g}-2} d\rho =  -  \frac{K}{({\gamma_g}-1)} d\rho^{{\gamma_g}-1} \, .</math>
</div>
Hence, to within an additive constant, we have,
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<math>\mathfrak{w} = -  \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) \, ,</math>
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and integration throughout the configuration gives (for the adiabatic case),
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<math>\mathfrak{S}_\mathrm{therm} = +  \int \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr)  dm 
= \frac{2}{3({\gamma_g}-1)}  \int \frac{3}{2} \biggl( \frac{P}{\rho} \biggr)  dm
= \frac{2}{3({\gamma_g}-1)} S_\mathrm{therm} \, ,</math>
</div>
where, as introduced in our above discussion of the [[#Scalar_Virial_Theorem|scalar virial theorem]], <math>S_\mathrm{therm}</math> is the system's total thermal (''i.e.,'' random kinetic) energy.

====Relationship to the System's Internal Energy====
It is instructive to tie this introductory material to the classic discussion of thermodynamic systems, which relates a change in the system's internal energy per unit mass, <math>\Delta u_\mathrm{int}</math>, to the differential work, <math>\Delta \mathfrak{w}</math>, via the expression,
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<math>\Delta u_\mathrm{int} = \Delta Q - \Delta \mathfrak{w} \, ,</math>
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where, <math>\Delta Q</math> is the change in heat content of the system.  

'''Isothermal Evolutions''':  Because the internal energy is only a function of the temperature, we can set <math>\Delta u_\mathrm{int} = 0</math> for expansions or contractions that occur isothermally.  Hence, for isothermal evolutions the change in heat content can immediately be deduced from the expression derived for the differential work; specifically, <math>\Delta Q = \Delta \mathfrak{w}</math>.

'''Adiabatic Evolutions''':  By definition, <math>\Delta Q = 0</math> for adiabatic evolutions, in which case we find <math>\Delta u_\mathrm{int} = - \Delta \mathfrak{w}</math>.  The definition of the thermodynamic energy reservoir can therefore be rewritten as,
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<math>\mathfrak{S}_\mathrm{therm} = - \int \mathfrak{w} ~dm = + \int u_\mathrm{int} ~dm = U_\mathrm{int} \, .</math>
</div>
Quite generally, then &#8212; in sync with the above derivation &#8212; we can replace <math>\mathfrak{S}_\mathrm{therm}</math> by,
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<math>~U_\mathrm{int} = \frac{2}{3(\gamma_g-1)} S_\mathrm{therm} \, ,</math> 
</div>
in the expression for the free energy when analyzing adiabatic evolutions.