In this exercise, we will obtain a rough estimate for the stability of hydrogen. Consider the combination reaction for hydrogen
p⁺+e-⇌H,

where p+are protons of mass mp,e-are electrons of mass me and H is the hydrogen atom of mass mH. All particles are in thermodynamic equilibrium in a volume V and you can consider all species as an ideal gas. We assume charge neutrality, that is the number Np (and the number density, np=NpV ) of protons and electrons is the same, NPV=np=ne=NeV. We also assume that the total number, N, (and number density, n=NV ) of Hydrogen, ionized or not, is conserved, NV=n=nH+np= constant, where nH is the number density of the un-ionized H atoms. Additionally, we assume that only the ground state of the H atom is occupied. Since the individual particle numbers are not conserved, we work in the grand canonical ensemble.

There are two energy states for the system:
p⁺+e⁻, εe=I ionized state, free electron
H, εH=0 un-ionized state, bound electron

Assume the chemical potential for the electron is μe.
(a) Write down an expression for the grand partition function for this system, and the expectation value for the number of Hydrogen atoms, and protons, and electrons.