Appendix B: Derivation of the Saha equation

This derivation follows the one provided in [41], and is included here for completeness. Consider an equilibrium of ideal gases $C \rightleftharpoons A + B$. We compute the number of particles $n_{A}$, $n_{B}$, and $n_{C}$ by maximizing the total number of available states:

$$\frac{Z_{A}^{n_{A}}}{n_{A}!} \cdot \frac{Z_{B}^{n_{B}}}{n_{B}!} \cdot \frac{Z_{C}^{n_{C}}}{n_{C}!}$$ (4.14)

where $Z_{A}$, $Z_{B}$, and $Z_{C}$ are the partition functions of A, B, and C. If we take $f$ to be the fraction of dissociated C, we then have

$$\frac{f}{1-f} = \frac{n_{A} n_{B}}{n_{C}} = \frac{Z_{A} Z_{B}}{Z_{C}}$$

$$= \frac{(V/\Lambda_{A}^{3})Z_{A}^{\mathrm{vre}} \cdot (V/\Lambda_{B... ...m{vre}} }{(V/\Lambda_{C}^{3})Z_{C}^{\mathrm{vre}} \cdot \exp(\Delta E_{d}/k T)}$$

where we have factored out the translational partition functions explicitly to leave $Z^{vre}$, the vibrational-rotational-electronic partition functions. We compute the partition functions from the ``bottom of the well,'' which incurs a factor of $\exp(\Delta E_{d}/k T)$, where $E_{d}$ is the dissociation energy of $C$. Substituting the de Broglie wavelength $\Lambda = (h^{2}/2 \pi m k T)^{1/2}$ and the volume per atom $V = 4 \pi r_{s}^{3} / 3$ we are left with the expression

$$\frac{f}{1-f} = \frac{Z_{A}^{vre} Z_{B}^{vre}}{Z_{c}^{vre}} \frac... ...{C}}\right) \left(\frac{2 \pi k T}{h^{2}}\right)^{3/2} \exp(-\Delta E_{d}/k T).$$ (4.15)

For hydrogen atom, electron, and proton, we take $Z^{vre} = 1$; for hydrogen molecule we use the expression for an ideal diatomic

$$Z^{vre} = \frac{T}{2 \Theta_{r}} \cdot \frac{\exp(-\Theta_{v}/2 T)}{1-\exp(-\Theta_{v}/T)}$$ (4.16)

where $\Theta_{r}$ = 85.3 K and $\Theta_{v}$ = 6215 K.