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Astron. Astrophys. 334, 1000-1006 (1998)

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Appendix A: entropy for a mixture of perfect gases and radiation

From basic physics, the entropy of a mixture of perfect gases in molar fraction [FORMULA] is given by

[EQUATION]

with

[EQUATION]

where N is the number of moles, with

[EQUATION]

for the ions of species i and

[EQUATION]

for the electrons with the assumption of complete ionisation. We call µ the mean molecular weight, [FORMULA] the mass fraction of the ions of type i with atomic mass [FORMULA] and electron number [FORMULA].

Let us consider a mixture of hydrogen, helium and metals in respective mass fractions X, Y and Z, where Z is taken to be constant (case of H-burning).Then the entropy may be expressed in terms of µ only. Developing the various terms, we get

[EQUATION]

with

[EQUATION]

We identify the heavy elements with oxygen and the corresponding molar fraction is thus [FORMULA]. For the sake of consistency, we must also take

[EQUATION]

For a binary mixture with only H and He, we get

[EQUATION]

For a mixture of perfect gases and radiation, we are looking for an expression of the form

[EQUATION]

with [FORMULA]
and [FORMULA].

We have for one mole

[EQUATION]

At constant µ, the usual expression of TdS is (cf. Kippenhahn and Weigert 1990)

[EQUATION]

For variable µ, we get with [FORMULA]

[EQUATION]

We see that the entropy changes also depend on the changes of µ through the entropy of mixing, which is often not taken into account. Nevertheless, this term is not responsible for the term [FORMULA] in (4.38) as seen above.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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