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
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
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
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