# Species Distribution

To connect the details of atomic or molecular structure to the distribution of excited state populations, we presume that excited states are described by a Boltzmann distribution (population proportional to e^{-E/kT}, where E is the energy of the state, T the temperature of the region of space in which the atom or molecule resides, and k is the Boltzmann constant, 1.38×10^{-23} J K^{-1}) and that ionization is described by Saha equilibrium, the gas phase parallel to mass action equilibrium in solution.

\[\begin{align}

&\textbf{Solution Reaction}\hspace{20px} A\leftrightharpoons B+C

&&\textbf{Saha Equilibrium} \hspace{20px} A\leftrightharpoons A^+ + e\\

&K = \dfrac{[B][C]}{[A]}\textrm{ OR }\dfrac{[B]}{[A]} = \dfrac{K}{[C]}

&&\mathrm{\dfrac{[A^+]}{[A]}} = \dfrac{2\sum_{i=0}^{\infty}g_ie^{-\Large\frac{E_{i,ion}}{kT}}}{\sum_{i=0}^{\infty}g_ie^{-\Large\frac{E_{i,neutral}}{kT}}}\left(\dfrac{2\pi m_e kT}{h^2} \right )^{3/2}e^{-\Large\frac{E_{ionization}}{KT}}

\end{align}\]

Where this gets interesting is when there are multiple species A, A', A", etc. If an easily ionized substance generates lots of electrons at a given temperature, then less-easily ionized species will see a high n_{e}, changing the ratio of ions to neutrals and thus modifying the relative importance of ionic and neutral states. Because solutions and solids both introduce condensed phases that require substantial energy to vaporize and atomize, there are additional complications as analytes (regardless of ionization potential) cool flames, furnaces, and plasmas and so vary the local effective temperature. Rather than give an overly simplified simulation of interelement effects, we refer the interested student to work by J. Olesik, R. S. Houk, P. B. Farnsworth, and many others.