Notes on some basic quantities, units and constants: Part II

Mass fractions

The mass fraction of H, He and the metals is a quick way to specify the chemical composition of the stellar plasma. The mass fraction of hydrogen is defined as ratio of the mass of hydrogen particles and the total mass of all particles (in a given volume): $$ X = \frac{M_\mathrm{H}}{M}= \frac{\rho_\mathrm{H}}{\rho}, $$ where $\rho_\mathrm{H}$ and $\rho$ are the respective (mass) densities. The hydrogen mass density is equal to the hydrogen number density ($n_\mathrm{H}$) times the mass of one hydrogen particle $m_{\mathrm{H}} = A_{\mathrm{H}}\,m_{\mathscr{A}}$, thus: $$ X = \frac{M_\mathrm{H}}{M}= \frac{n_\mathrm{H}\,m_{\mathrm{H}}}{\rho}= \frac{n_\mathrm{H}\,A_{\mathrm{H}}\,m_{\mathscr{A}}}{\rho}. $$ Similar to that, for helium and the metals we define: $$ Y = \frac{M_\mathrm{He}}{M}= \frac{n_\mathrm{He}\,m_{\mathrm{He}}}{\rho}= \frac{n_\mathrm{He}\,A_{\mathrm{He}}\,m_{\mathscr{A}}}{\rho}, $$ $$ Z = \frac{M_\mathrm{metals}}{M}= \frac{n_\mathrm{metals}\,m_{\mathrm{metals}}}{\rho} = \frac{\sum_{i=3} n_\mathrm{i}\,A_{\mathrm{i}}\,m_{\mathscr{A}}}{\rho},. $$

Notes on some basic quantities, units and constants: Part I

There is a definition of mole as a unit that every student learns at very elementary level. Although mole is one of the seven base units of the International System, it is a bit specific and sometimes creates confusion. Here is my attempt to clarify the concept of mole and to derive in one place some useful relations used in the radiative transfer and atmospheric modeling. The first version of these notes I wrote for a group of students at the University of Belgrade many years ago. I still use them as a personal reminder. 

Counting "Elementary particles"

For the solar/stellar plasma, the "elementary" particles are atoms, ions (positive or negative), free electrons and molecules. In the very cool atmospheres there are dust particles as well. In the solar atmosphere dust can be completely neglected. Regarding the chemical composition, the atmospheric plasma is made out of hydrogen, helium and the metals (all other elements). There is no nuclear reactions and thus the total number density of nuclei per atomic specie is constant with time.

It is important to distinguish between the number of free atoms and the total number of atoms including those bound in the molecules. The former I denote as $N_{\mathrm{a}}^{\mathrm{free}}$, the latter as $N_{\mathrm{a}}^{\mathrm{tot}}$. The total number of atoms is identical to the number of atomic nuclei. The total number of molecules is $N_{\mathrm{m}}$.

The total number of particles $N$ is therefore:
$$N = N_\mathrm{e} + N_{\mathrm{a}}^{\mathrm{free}} + N_{\mathrm{m}},$$ or when there is no molecules $$N = N_\mathrm{e} + N_{\mathrm{a}}^{\mathrm{free}} = N_\mathrm{e} + N_\mathrm{H} + N_\mathrm{He} + N_\mathrm{\mathrm{metals}},$$ where $\mathrm{e}$, $\mathrm{m}$, $\mathrm{H}$ and $\mathrm{He}$ stand for the electrons, the molecules, hydrogen, helium and $\mathrm{metals}$ refer to all other elements together. The contribution of the metals can be further divided into the contributions of the individual elements.