2. Goals
Formulate vibrational-rotational energy states
Interpret equilibrium vibrations of HCl and DCl
Find the inter-nuclear separation (bond length) of
each of the molecules.
Calculate the harmonic oscillator force constant
for HCl and DCl
Calculate constant volume heat capacities
Observe the isotope effect in diatomic molecules
3. 100mm glass sample cell placed in IR
spectrometer, which was purged with N2 to
produce a background scan.
Sample cell was filled with HCl gas
generated from the reaction of NaCl and
D20 with Sulfuric Acid
The spectrometer was
purged again, and a reading
of the sample was taken
4. Four types of peaks are seen in the spectra
1H35Cl
Interested in 35Cl in
this experiment
2H35Cl
1H37Cl
2H is Deuterium
2H37Cl
6. The anharmonic oscillator and rigid rotor
energy expressions are as follows:
These can be combined to form the expression:
T(v,J) = E(v,J)/hc = ṽe(v + ½ ) - ṽexe(v + ½ )2 + BeJ(J+1) –
DeJ2(J+1)2 – αe(v + ½ )J(J+1)
Where the final term accounts for vibrational-rotational
interaction due to centrifugal stretching.
7. Selection rules allow for
ΔJ = ±1
Δv = ±1
Substitutinginto the previous energy
equation lets us find expressions for the P
and R branch energies
ṽR = ṽ0 + (2Be - 2αe) + (2Be - 4αe)J’’ - αeJ’’2
ṽP = ṽ0 + (2Be - 2αe)J’’ - αeJ’’2
8. Thesecan be combined to form the
expression
ṽ(m) = ṽ0 + (2Be - 2αe)m – αem2 – 4Dem3
This
equation can be used to analyze the
spectra
9. m = 0 gives the “forbidden” transition from
ṽ(m). Directly assigning m=0 to the Q-
branch and indirectly assigning values of
m to all of the other peaks
We can then produce a plot of (m) vs.
wavenumber (ṽ) and use a third-order
polynomial curve fit to determine the
coefficients of ṽ(m) for both HCl and DCl
14. y = -0.0102x3 - 0.0715x2 + 11.043x + 2090 DCL
y = -0.002x3 - 0.3035x2 + 20.572x + 2885.7 HCL
ṽ(m) = ṽ0 + (2Be - 2αe)m – αem2 – 4Dem3
These equations give the following coefficients for ṽ(m)
HCl DCl
ṽ0 = 2885.7 cm-1 ṽ0 = 2090.0 cm-1
Be= 10.590 cm-1 Be= 5.593cm-1
αe= 0.3035 cm-1 αe= 0.0715 cm-1
De= 5 E-4 cm-1 De= 2.55 E-3 cm-1
15. The isotope effect 4,7
• The difference in mass between atoms effects the
vibrational and rotational energies (2 peaks)
Relationsbetween two diatomic molecules
with an isotopic substitution are
HCland DCl can be related using these
expressions.
16. ṽ0 = ṽe - 2ṽexe
Usingthe above with the isotope effect
allows us to solve for ṽe and ṽexe for HCl.
HCl DCl
ṽe = 2986.9 cm-1 ṽe = 2142.2 cm-1
ṽexe= 50.602 cm-1 ṽexe= 26.029 cm-1
17. Harmonic oscillator expression
Allows us to find the “spring” force
constant for HCl and DCl
k = 514.96 N/m
18. Using the rotational constants from the
polynomial curve fit with the definition of B
gives the moment of inertia
19. Finding r is trivial once the moment of
inertia is known
HCl DCl
R = 1.2699 E -10 meters R = 1.2580 E -10 meters
20. Heatcapacity expression and vibronic
terms for HCl and DCl give constant
volume heat capacities at different
temperatures.
21. H35Cl µ ῦe ῦ ex e Be αe De Re K
x10-27kg cm-1 cm-1 cm-1 cm-1 cm-1 A N/m
Calculated 1.6267 2986.9 50.602 10.590 0.3035 5. E -4 1.2699 514.96
Literature 1.6267 2990.95 52.8186 10.5934 0.30718 5.319E-4 1.2746 516.35
% Error 0 0.135 4.197 0.032 1.198 5.997 0.369 0.269
D35Cl µ ῦe ῦ ex e Be αe De Re K
x10-27kg cm-1 cm-1 cm-1 cm-1 cm-1 A N/m
Calculated 3.1624 2142.2 26.029 5.593 0.0715 2.55 E -3 1.2580 514.96
Literature 3.1624 2145.16 27.1825 5.44879 0.11329 1.39 E-4 1.2746 516.35
% Error 0.137 4.243 2.646 36.887 1734.532 1.302 0.269