Heres the question
User posted image: User posted image
I know that I'm supposed to derive these values from the fact that I know the absorptions of 2 different isotopes of the diatomic molecule but I don't know how to go about doing it. I can obtain the masses of hydrogen, deuterium and iodine from the periodic table but I don't know how I can use this data to get the internuclear distances.

Wikipedia has a nice article on moment of inertia where they spell out what the moment of inertia of a diatomic molecule.

http://en.wikipedia.org/wiki/Wavenumber#In_spectroscopy
http://en.wikipedia.org/wiki/Rotational_sp...ear_molecules_2
http://en.wikipedia.org/wiki/Moment_of_inertia#Examples

So from wavenumbers (in cgs!) you can get ΔE, from consecutive ΔE you can get B, from B you can get I, from I and the atomic masses you can get r.

And since J only changes by one, from B and a given spectral line, you can calculate J.

My experience with Wikipedia indicates they may have dropped a term of pi somewhere along the way. So I cross-checked my formula, then to be perverse used units of cm or cm^(-1) throughout. According to my chemistry sources, since 2000 the trend has been to use daltons (Da) instead of unified atomic mass units (amu or u).

mass(¹H) = 1.007825 Da = 7.57179742 × 10^(12) cm^(-1)
mass(²H) =2.01410178 Da = 1.5131963 × 10^(13) cm^(-1)
mass(¹²⁷I) = 126.90447 Da = 9.53434315 × 10^(14) cm^(-1)

http://en.wikipedia.org/wiki/Hydrogen_atom
http://en.wikipedia.org/wiki/Deuterium
http://en.wikipedia.org/wiki/Iodine
http://chemwiki.ucdavis.edu/Physical_Chemi...tomic_Molecules
(I think this source drops a term of pi in an unnumbered formula)

m̃₁ = m₁c/h
m̃₂ = m₂c/h
B̃= Δṽ/2
B = hcB̃
Ĩ = 1/(8 π² B̃) = 1/(4 π² Δṽ)
I = (h/c)Ĩ
r = √(Ĩ(m̃₁ + m̃₂)/(m̃₁m̃₂)) = √((m̃₁ + m̃₂)/(4 π² m̃₁ m̃₂ Δṽ))
J'' = ṽ/(2 B̃) - 1
J' = ṽ/(2 B̃)

HI
m̃₁ = 7.57179742 × 10^(12) cm^(-1)
m̃₂ = 9.53434315 × 10^(14) cm^(-1)
ṽ = 64.275 cm^(-1)
Δṽ = 12.855 cm^(-1)
B̃ = 6.4275 cm^(-1)
B = 7.9691 × 10^(-4) eV
Ĩ = 1.9705 × 10^(-3) cm
I = 26227 Da pm² (Anyone know what units are conventional?)
r = 1.6196× 10^(-8) cm = 0.16196 nm ( Cross-checks well with a reference )
J' = 5
J'' = 4