Question

7. Saha-Langmuir Equation for Hydrogen: A reaction of great relevance in astrophysics is the ion- ization/recombination of atomic hydrogen: Hp+ +e-, which can be treated like other chemical reactions -or equivalently, in effect as the adsorption of an electron onto a proton Assume: (i) thermodynamic equilibrium conditions prevail; and that (ii) all species can be treated as ideal gases (meaning the particles are sufficiently dilute so that Coulombic interactions amongst the free charged species can be neglected); (iii) the system is electrically neutral overall; and also that (iv) the internal partition function of the atomic hydrogen is dominated by the ground state. (This last assumption makes the calculation tractable, but is difficult to justify at temperatures high enough to ionize hydrogen with appreciable probability, and has engendered much discussion about how to handle the internal partition functions of atoms-a naive summation over electronic bound states will diverge) (a) Show that under these assumptions, the equilibrium concentrations of reactants/products will satisfy the relation ne nH nH where no. (T) is the quantum concentration of the free electrons, and E, is the ionization energy of atomic hydrogen (b) Argue why including or ignoring spin degeneracy will not affect the final answer (c) Find the relative fraction NH/(NH Np) of protons which are bound in hydrogen atoms, as ia function of temperature T, the binding energy εο--Ei-一13.6 eV of (ground-state) hydrogen, and any other needed quantities

Answer 1