Idealized greenhouse model: Difference between revisions

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{{Unreferenced|date=December 2008}}
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{{Orphan|date=December 2008}}
 
In [[astrophysics]], what is referred to as "[[entropy]]" is actually the [[adiabatic invariant|adiabatic constant]] derived as follows.
 
Using the first law of [[thermodynamics]] for a [[quasistatic process|quasi-static]], infinitesimal process for a [[hydrostatics|hydrostatic system]]
:<math>dQ = dU-dW.\,</math>
 
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
:<math>
dQ = C_{V} dT+P\,dV.
</math>
 
Further manipulation using the differential version of the [[ideal gas law]], the previous equation, and assuming constant pressure, one finds
:<math>
dQ = C_{P} dT-V\,dP.
</math>
 
For an [[adiabatic process]] <math>dQ=0\,</math> and recalling <math>\gamma = \frac{C_{P}}{C_{V}}\,</math>, one finds
:{|
|<math>\frac{V\,dP = C_{P} dT}{P\,dV = -C_{V} dT}</math>
|-
|<math>\frac{dP}{P} = -\frac{dV}{V}\gamma.</math>
|}
 
One can solve this simple differential equation to find
:<math>
PV^{\gamma} = \text{constant} = K\,
</math>
 
This equation is known as an expression for the adiabatic constant, K, also called the adiabat. From the ideal gas equation one also knows
:<math>
P=\frac{\rho k_{B}T}{\mu m_{H}},
</math>
where <math>k_{B}\,</math> is [[Boltzmann's constant]].
Substituting this into the above equation along with <math>V=[grams]/\rho\,</math> and <math>\gamma = 5/3\,</math> for an ideal [[monatomic gas]] one finds
:<math>
K = \frac{k_{B}T}{\mu m_{H} \rho^{2/3}},
</math>
where <math>\mu\,</math> is the mean [[molecular weight]] of the gas or [[plasma (physics)|plasma]]; and <math>m_{H}\,</math> is the mass of the [[Hydrogen atom]], which is extremely close to the mass of the [[proton]], <math>m_{p}\,</math>, the quantity more often used in astrophysical theory of [[galaxy cluster]]s.
This is what [[astrophysicists]] refer to as "entropy" and has units of [keV cm<sup>2</sup>]. This quantity relates to the thermodynamic entropy as
:<math>
S = k_{B} \ln \Omega + S_{0}\,
</math>
where <math>\Omega\,</math>, the density of states in statistical theory, takes on the value of K as defined above.
 
[[Category:Astrophysics]]
[[Category:Entropy]]

Latest revision as of 02:33, 4 November 2014

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