Solution

For any nonrelativistic species

j

, part (a) gives

n_{j} = g_{j} (\frac{2 π m _{j} k _{B} T}{h ^{2}})^{3/2} e^{(μ_{j} - m_{j} c^{2}) / (k_{B} T)} .

(99)

Chemical equilibrium and zero photon chemical potential imply

μ_{H^{+}} + μ_{e} = μ_{H^{0}}, μ_{H e^{+}} + μ_{e} = μ_{H e^{0}} .

(100)

Taking the appropriate ratios of the number-density formula, approximating the ion and atom masses as equal in the translational prefactors, and using the given degeneracies yields

\frac{n _{e} n _{H^{+}}}{n _{H^{0}}} = (\frac{2 π m _{e} k _{B} T}{h ^{2}})^{3/2} e^{- I_{H} / (k_{B} T)}

(101)

and

\frac{n _{e} n _{H e^{+}}}{n _{H e^{0}}} = 2 (\frac{2 π m _{e} k _{B} T}{h ^{2}})^{3/2} e^{- I_{He} / (k_{B} T)} .

(102)

The factors are

g_{e} g_{H^{+}} / g_{H^{0}} = 1

and

g_{e} g_{H e^{+}} / g_{H e^{0}} = 2

Solved by gpt-5.6-sol high.

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