Phys 132 Homework 19 Solutions
Professor Crystal Martin
TA: Ellie Hadjiyska
Problem 19 : (Phillips Problem 2.4) We are given a stellar atmosphere that is almost pure
hydrogen with 50% of the hydrogen molecules dissociated into hydrogen atoms and the pressure
P= 100 Pa. We want to find the temperature.
We will treat the hydrogen as being in one of two states: undissociated with energy ²1∼0 and
dissociated with energy ²2≈4.48 eV.
Since 50% dissociated, the number of undissociated H2molecules equals the number of H2molecules
that did dissociate. We get 2 H atoms for every H2molecule that dissociates, so n(H) = 2n(H2)
(Note: This is not the same as what was quoted in the hint). Thermodynamic equilibrium is
established if the sums of the chemical potentials on both sides are equal. Since photons have zero
chemical potential and
γ+H2H+H
then
µ(H2) = µ(H) + µ(H) = 2µ(H) (1)
where
µ(H2) = m(H2)c2−kT ln µg(H2)nQ(H2)
n(H2)¶(2)
and
µ(H) = m(H)c2−kT ln µg(H)nQ(H)
n(H)¶(3)
The mass of the hydrogen molecule is equal to the mass of the atoms making it up plus the binding
energy ²2=−4.48 eV. i.e.
m(H2)c2= 2m(H)c2+²2
Using this with Eqs. (??), (??), and (??) gives
²2=m(H2)c2−2m(H)c2=kT ln "g(H2)nQ(H2)
n(H2)µn(H)
g(H)nQ(H)¶2#(4)
or setting the statistical weights g(H2) = g(H) = 1, we get
exp (²2/kT ) = n(H)
nQ(H)µnQ(H2)
nQ(H)¶µn(H)
n(H2)¶
