**4.1. General considerations**

In almost all stellar environments, we may assume a strict thermodynamic equilibrium applies. Exceptions include optically thin regions and rapidly changing regions such as one might find in the initial stages of a supernova.

There are two dominant ways of considering how an assembly
of particles behaves.
One is called the *physical* picture, where one keeps
track of all elementary particles whether they are bound
together in atoms or not.
The other is called the *chemical* picture, where one
keeps track of 'chemical' species, eg. different ionization
states of elements or in some cases even different excitation
states.
For all intents and purposes, these chemical species are
treated as interacting, continuous fluids.
Virtually all stellar (and most other) astronomy uses the
chemical picture.
Only in non-thermodynamic conditions (plasma waves, etc.)
do astronomers sometimes use the physical picture.
There is no real reason except historical for this preference.

**4.2. Thermodynamic variables of interest**

At some point or another we will require knowing the values
of the following quantities:

:P | pressure (total or partial) in erg/cm^{3} |

:T | temperature in K |

:ρ | density in g/cm^{3}
(also V =1/_{ρ}ρ) |

:S | entropy in ergs/gK |

:E | internal energy in ergs/g |

:Q | heat in ergs/g |

:n_{i} | number density of species i
in /cm^{3} |

:N_{i} | number density of species i in /g. |

Of particular interest in the chemical picture of things is the
*chemical potential* of species *i*

* μ_{i}* ≡
(

e.g. how the internal energy per gram changes when the number of species
*i* per gram changes at constant entropy and volume.
Thermodynamic and chemical equilibrium demands that ∑_{i}
*μ _{i}*

Consider hydrogen: H^{+} + e^{−} ↔ H^{0} + γ,

which we may write generically as

∑_{i} *ν _{i} C_{i}* = 0

where *C _{i}* is the species label and

*dN _{i}* /

**Exercise:** show that the above relation holds true for the reaction 2H↔H_{2}.

Substituting the relation among differential changes into the thermodynamic equilibrium relation gives the chemical equilibrium equation

∑_{i} *ν _{i} μ_{i}* = 0.

Because photons are *not* conserved, *dN _{γ}* is arbitrary,
and as a result

In a perfect gas, the interaction process itself happens in a short time and is independent of the state variables. In a real gas, particularly one that is dense enough that internuclear distances approach electron or molecular binding distances, calculations are difficult and generally heuristic.

**4.3 Perfect gas statistical equilibrium equations**

Elementary statistical mechanics leads to the following expression for the
number of particles of species *i* in phase-space volume
*d*^{ 3}**r** *d*^{ 3}**p** :

* n_{i}*(

[+1: Fermi-Dirac; −1: Bose-Einstein; 0: distinguishable particles].

*ε*(*p*) = (*p*^{2}*c*^{2} +
*m*^{2}*c*^{4})^{1/2} −
*m**c*^{2},
velocity *v* = *∂ε*/*∂p*, and

*P _{i}* = (1/3)
∫

for isotropic momenta.

**Note:**

Hansen and Kawaler are rather cavalier with their units and notation.
I have specifically included the *d*^{ 3}**r** *d*^{ 3}**p**
on both sides to show that so far, the units of * n_{i}*(

When the authors start talking about blackbody and other distributions, they
suddenly shift units for *n _{i}*(

*n _{i}*(

H&K make no notational difference and never say that one *n* is not the same
as another.
Since we are assuming isotropy, the internal integration is straightforward.
I don't think the 6D distribution will ever come up again.

**4.4 Special cases**

A. Blackbody radiation

B. Monatomic gas

C. The Saha equation

D. Fermi-Dirac degeneracy