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Entropy (astrophysics)

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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

d Q = d U − d W . {\displaystyle dQ=dU-dW.}

For an ideal gas in this special case, the internal energy, U, is a function of only 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

d Q = C v d T + P d V . {\displaystyle dQ=C_{\text{v}}dT+P\,dV.}

Further manipulation using the differential version of the ideal gas law, the previous equation, and assuming constant pressure, one finds

d Q = C p d T − V d P . {\displaystyle dQ=C_{\text{p}}dT-V\,dP.}

For an adiabatic process d Q = 0 {\displaystyle dQ=0\,} and recalling γ = C p / C v {\displaystyle \gamma ={C_{\text{p}}}/{C_{\text{v}}}\,} , one finds

V d P = C p d T P d V = − C v d T {\displaystyle {\frac {V\,dP=C_{\text{p}}dT}{P\,dV=-C_{\text{v}}dT}}}
d P P = − d V V γ . {\displaystyle {\frac {dP}{P}}=-{\frac {dV}{V}}\gamma .}

One can solve this simple differential equation to find

P V γ = constant = K {\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

P = ρ k B T μ m H , {\displaystyle P={\frac {\rho k_{\text{B}}T}{\mu m_{\text{H}}}},}

where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant. Substituting this into the above equation along with V = [ g ] / ρ {\displaystyle V=/\rho \,} and γ = 5 / 3 {\displaystyle \gamma =5/3\,} for an ideal monatomic gas one finds

K = k B T ( ρ / μ m H ) 2 / 3 , {\displaystyle K={\frac {k_{\text{B}}T}{(\rho /\mu m_{\text{H}})^{2/3}}},}

where μ {\displaystyle \mu \,} is the mean molecular weight of the gas or plasma; and m H {\displaystyle m_{\text{H}}} is the mass of the hydrogen atom, which is extremely close to the mass of the proton, m p {\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 . This quantity relates to the thermodynamic entropy as

Δ S = 3 / 2 ln ⁡ K . {\displaystyle \Delta S=3/2\ln K.}

References

  1. "Adiabatic Condition Development". hyperphysics.phy-astr.gsu.edu. Retrieved 2024-11-03.
  2. "m300l5". personal.ems.psu.edu. Retrieved 2024-11-03.
  3. "THERMAL PROPERTIES OF MATTER". www.sciencedirect.com. Retrieved 2024-11-03.
  4. "Mean molecular weight" (PDF).
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