Rotational energy levels of diatomic molecules are well described by the formula `E_(J)=BJ(J+1)`, where `J` is the rotational quantum number of the molecule and `B` its rotational constant. `B` is related to the reduced
mass `mu` and the bond length `R` of the molecule through the equation `B=h^(2)/(8pi^(2)muR^(2))`.
In general, spectroscopic transitions appear at photon energies which are equal to the energy difference between appropriate states of a molecule `(hv=DeltaE)`. The observed rotational transitions occur between adjacent rotational levels, hence `DeltaE=E_(J+1)-E_(J)=2B(J+1)`. Consequently, successive rotational transitions that appear on the spectrum (such as the one shown here) follow the equation `h(Deltaupsilon)=2B`.

By inspecting the spectrum provided, determine the following quantities for `^(2) C ^(p)C` with appropriate units
(a) `Deltaupsilon` (b) `B` (c) `R`

Answer

{:a) For example : `DeltaV=1150-1035=115 GHz`
{:b) `B=(hDeltav)/(2)=(6.63xx10^(-34)xx115'xx10^(9))/(2)=3.81xx10^(-23) J`
{:c) `mu=(m(C)xxm(O))/(m(CO))=(12xx16)/(28)=6.86 a.u.=1,14 xx10^(-26) kg`
For interatomic distance R:
`R=(h)/(2pisqrt(2muB))=(3.63xx10^(-34))/(2xx3.14sqrt(2xx1.14xx10^(-26)xx3.81xx10^(-23)))=1.13xx10^(-10) m =1.13 Å`