Editing VE (section)

====Adiabatic Systems====
If, upon compression or expansion, the gaseous configuration evolves adiabatically, the pressure will vary with density as,
<div align="center">
<math>P = K \rho^{\gamma_g} \, ,</math> 
</div>
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,
<div align="center">
<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,
<div align="center">
<math>\mathfrak{w} = -  \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) \, ,</math>
</div>
and integration throughout the configuration gives (for the adiabatic case),
<div align="center">
<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.