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

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 is given by

with

where N is the number of moles, with

for the ions of species i and

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

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

with

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

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

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

with
and .

We have for one mole

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

For variable µ, we get with

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 in (4.38) as seen above.

© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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