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.


When there is no molecules formed in the plasma, the number of the free atoms (neutral and ions) is equal to the total number of atoms $N_{\mathrm{a}} \equiv N_{\mathrm{a}}^{\mathrm{free}} = N_{\mathrm{a}}^{\mathrm{tot}}$. Whenever this is the case I will simply use $N_{\mathrm{a}}$. In addition I will use $N_{\mathrm{a}}$ instead of $N_{\mathrm{a}}^{\mathrm{tot}}$ when the presence of molecules is not relevant (e.g. the total mass is the same if all the atoms are bound in the molecules as if they were all free).

With small $n$ I will denote all respective number densities ($V$ is the unit density): $$n = \frac{N}{V}.$$ 
A brief but important reminder

Any quantity in physics is characterized by its value, its dimension and its unit. For example, dimension of the force in $m \times l \times t^{-2}$, where $m$, $l$ and $t$ are mass, length and time. The standard unit for the force in the SI system is therefore $\mathrm{kg} \times \mathrm{m} \times \mathrm{s}^{-2}$ and it has its own name - Newton. There can be several different units for the same quantity. Different units correspond to different basic systems of units (e.g. CGS and SI units for the force, dyne and Newton) or they just represent smaller or larger versions of the basic units (e.g. in the SI system, mega-meter, pico-meter and so on). However, there are also dimensionless quantities (or quantities with dimension 1), either because they are defined as a ratio of two quantities with the same dimension (e.g. Reynolds number) or because they count discrete features (e.g. number of particles, number of dogs). Dimensionless quantities are, in many cases, also unitless. In principle, we can assign units to them (e.g. 'particles', 'dogs'), but these are not physical units and they should be ignored in the dimensional analysis.

If a quantity $q$ can be expressed in units $[U_1]$ and $[U_2]$ and if, in these units, it takes values $v_1$ and $v_2$, respectively, then: $$q = v_1\,[U_1] = v_2\,[U_2].$$Since both sides of the equation have the same dimension, there must be a dimensionless number $c_{12}$ such that $[U_1]  = c_{12}\,[U_2]$. It is the conversion factor between the two units. Formally, its unit is $[U_1]/[U_2]$. For example, length can be expressed in miles and in kilometers:
$$l = 3\, \mathrm{miles} = 4.82803\,\mathrm{km}$$.
The conversion factor is
$$c = 1.60934 \,\frac{\mathrm{km}}{\mathrm{mile}}.$$
Avogadro's number

The simplest definition of a mole says that mole stands for a number. Not for any number, but for a very specific one - Avogadro's number: $$N_{\mathscr{A}} = 6.022140857(74) \times 10^{23}.$$ One mole of hydrogen atoms contains $N_{\mathscr{A}}$ particles of hydrogen atoms. One mole of electrons contains $N_{\mathscr{A}}$ electrons. One mole of dogs contains $N_{\mathscr{A}}$ dogs, etc. The number of moles (also known as the amount of substance) $\widetilde{n}$ is thus defined as the number of free particles divided by Avogadro's number:
$$ \widetilde{n} = \frac{N}{N_{\mathscr{A}}} = \frac{N_{\mathrm{e}} + N_{\mathrm{a}}^{\mathrm{free}} + N_{\mathrm{m}}}{N_{\mathscr{A}}}. $$ The unit of $\widetilde{n}$ is mole. To make it consistent, the unit of Avogadro's number is formally mole^-1. Since $\widetilde{n}$ is the ratio of two dimensionless numbers, the mole is dimensionless as well. However, note that although both quantities are dimensionless, the amount of substance has a unit, while number of particles is unitless. In principle, we could introduce "particle" as a formal unit of the number of particles. If we do so, the unit of $N_{\mathscr{A}}$ would become particle/mole. In the context of the previous section, $N_{\mathscr{A}}$ is the conversion factor between the number of particles expressed in moles and the number of particles expressed in particles.

The main purpose of  $N_{\mathscr{A}}$ is to help us work with the large numbers that are typical when we solve equations with quantities of both micro and macro world.

Atomic mass unit

The reciprocal value of $N_{\mathscr{A}}$ is $$m_{\mathscr{A}} = \frac{1}{N_{\mathscr{A}}} = 1.660538921(73)\times 10^{-24}.$$ It gives us the fraction of one mole that corresponds to one particle. For example, if the mass of one mole of particles is 3 g, than the mass of one particle is obviously $m_{\mathscr{A}} \times 3\,\mathrm{g}$. This leads us to the concept of the atomic mass unit (a.m.u.). The a.m.u. is a unit for the mass used in atomic physics. It is not essentially different from the pound used in the imperial system of units. One pound is 453.59237 grams. One a.m.u. is $m_{\mathscr{A}}$ grams:$$1\,\mathrm{a.m.u.} = 1.660538921(73)\times 10^{-24}\,\mathrm{g}.$$ The quantity measured in a.m.u.'s is known as atomic weight, while the quantity measured in grams is the mass itself. The two are simply related as $$m = A\, m_{\mathscr{A}}.$$Therefore, $m_{\mathscr{A}}$ is the conversion factor between mass and atomic weight. It is a dimensionless number; its formal unit is a.m.u./gram.

Both $m_{\mathscr{A}}$ and $N_{\mathscr{A}}$ are conversion factors but between two quantities of different dimension (mass and number of particles, respectively), and thus the formal dimensions of $m_{\mathscr{A}}$ and $N_{\mathscr{A}}$ differ as well.

Some additional notes:

1. In chemistry, atomic weight is often used as it is unitless.

2. Atomic mass unit (with label a.m.u.) had been replaced in 1961 by unified atomic mass unit (with label u). However, the difference between the two is only in the value of $m_{\mathscr{A}}$ with the rest of the concept unchanged.

3. Another name for the unified atomic mass unit is Dalton (Da).

4. When the numbers are dimensionless but still have units, one has to be very careful with dimensional analysis. For example, all the following equations are numerically correct: $$1\,\mathrm{a.m.u.} = m_{\mathscr{A}} \times 1\, \mathrm{g} = \frac{1\,\mathrm{g}}{N_{\mathscr{A}}} = \frac{1\,\mathrm{g}}{1\,\mathrm{mol}}.$$For example for hydrogen all the following forms are common and correct:$$A_\mathrm{H} = 1.008 = 1.008\,\mathrm{a.m.u.} = 1.008\,\frac{\mathrm{g}}{\mathrm{mol}}.$$

The formal definition

The lecture-book definition of the mole says that: 1 mole is the amount of substance in 12 grams of ¹²C (the isotope of carbon with the atomic mass $A_{\mathrm{C}} = 12\;\mathrm{a.m.u.}$).

This definition specifies the standard particle (one ¹²C atom) which mass is the unified atomic mass unit (pre-1961 atomic mass unit was defined based on oxygen) and which number in 12 grams is exactly Avogadro's number. In other words, in 12 grams of ¹²C there is exactly $N_{\mathscr{A}}$ particles with atomic mass $12 \times \mathrm{a.m.u.} =$ $12 \times (1 / N_{\mathscr{A}})$ grams.