In astrophysics, what is referred to as "entropy" is actually the adiabatic constant derived as follows.
Using the first law of thermodynamics for a quasi-static, infinitesimal process for a hydrostatic system
\( {\displaystyle dQ=dU-dW.\,} \)
For an ideal gas in this special case, the internal energy, U, is only a function of the temperature T; therefore the partial derivative of heat capacity with respect to T is identically the same as the full derivative, yielding through some manipulation
\( {\displaystyle dQ=C_{V}dT+P\,dV.} \)
Further manipulation using the differential version of the ideal gas law, the previous equation, and assuming constant pressure, one finds
\( {\displaystyle dQ=C_{P}dT-V\,dP.} \)
For an adiabatic process \( {\displaystyle dQ=0\,} \) and recalling \( {\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}\,}, \) one finds
\( {\displaystyle {\frac {V\,dP=C_{P}dT}{P\,dV=-C_{V}dT}}} \)
\( {\displaystyle {\frac {dP}{P}}=-{\frac {dV}{V}}\gamma .} \)
One can solve this simple differential equation to find
\( {\displaystyle PV^{\gamma }={\text{constant}}=K\,} \)
This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows
\( {\displaystyle P={\frac {\rho k_{B}T}{\mu m_{H}}},} \)
where \( {\displaystyle k_{B}\,} \) is Boltzmann's constant. Substituting this into the above equation along with \( {\displaystyle V=[\mathrm {g} ]/\rho \,} \) and \( {\displaystyle \gamma =5/3\,} \) for an ideal monatomic gas one finds
\( {\displaystyle K={\frac {k_{B}T}{(\rho /\mu m_{H})^{2/3}}},} \)
where \( \mu \, \) is the mean molecular weight of the gas or plasma; and \( {\displaystyle m_{H}\,} \) is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, \( {\displaystyle m_{p}\,} \), the quantity more often used in astrophysical theory of galaxy clusters. This is what astrophysicists refer to as "entropy" and has units of [keV cm2]. This quantity relates to the thermodynamic entropy as
\( {\displaystyle \Delta S=3/2\ln K.} \)
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