For Problem 1:

Do not start with your knowledge of the value of C from another source
and then work from both ends of the problem to show that this is the
correct value of C.  I want you to derive the value of C directly as
part of the transformation.

For Problem2:

The algebraic manipulation is more insightful- I prefer it.  If you do
this, then you are best served to have a way of easily showing that
the value of x_{ij} is much less than unity or not.  Since x_{ij} is
the ratio of chi_{ij}/kT and units of eV are near unity for the
ionization potentials, then you will want to convert kT to eV
expressed as a scaled temperature T10 where T=(T10/10,000).  The
combination will give you a quick and easy expression for $x_{ij} that
can be eye balled as to its order of magnitude.

For Problems 3 and 4:

The routines you write here can eventually be directly inserted into
your stellar atmosphere model code if you write them in a general
fashion.  That means, do not hard wire the variables for hydrogen or
iron into your Boltzmann and Saha equations, but pass them to a
subroutine that the returns the Boltzmann and Saha results based upon
the variables passed to it.  Write a subroutine for Boltzmann and
write a subroutine for Saha (keep them seperate), then compute the
needed results in the driver routine.  Also, if you have any
interpolating to do, then also do that in the driver before you call
the subroutines (just send down the needed data for the calc)- keep
your subroutines very simple and focused on just the evaluation of the
function.  I highly recommend this for all your coding for all work
that you do.  As for this assignment, if you do this then you do not
need to rewrite a general subroutines for computing Boltzmann and Saha
ratios that you will need later for your model.  Just sayin.