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CHEM 2500H 20FA Due. Mon Nov 16, 2020
Quantum Mechanics Lab 1:
Infrared Spectroscopy: Modelling Group Frequencies
Objective: In this exercise you will develop a simple model for predicting the wavenumber of the infrared absorption associated with the vibrational stretching modes of single, double and triple bonds.
Introduction: Polyatomic molecules exhibit complex vibrational spectra reflecting complicated polyatomic motions. A given non-linear molecule of “n” constituent atoms has 3n-6 degrees of freedom for vibrational motion, and accordingly may display up to this number of “normal modes of vibration”. Each of these normal modes involves movement of many and sometimes all of the atoms in the molecule. Thus a well-refined model for analysis of vibrational spectra of polyatomic molecules requires consideration of the masses of all atoms in the molecule, the force constants associated with each possible stretch, bend, twist and deformation of the molecular framework, and the interactions between these motions. The most popular of such models is the so-called “Wilson FG matrix method”, which is the standard approach to vibrational analysis. It is very complicated and, thankfully, it is possible to understand much about the basic nature of vibrational motion and spectroscopy without resorting to this method. This can be done by taking advantage of an important fact: Most bond stretching-type (not bending type) normal modes of molecules involve
predominantly motion of only two atoms. These two atoms are those that spectroscopists will most often use to label the type of motion involved. Thus, a so-called C=O bond stretch of acetone predominantly involves motion of the carbon and oxygen nuclei involved in the C=O bond, with minor contribution to the motion from the methyl groups attached to the carbonyl carbon. This is the reason for the so-called “group frequency” approach to interpretation of vibrational spectra, which states that absorptions due to a particular type of functional group, such as a carbonyl C=O, will always give rise to absorptions whose wavenumber falls within a certain narrow range of values. This is because all such vibrations involve essentially the same atoms held together with a bond of essentially the same strength. Learning the wavenumber ranges associated with common functional groups allows one to “eyeball” an infrared spectrum and make a quick evaluation of what the molecule is likely to be, or at least what important functional groups are in the molecule. The success of the group frequency approach suggests that we may consider stretching modes of molecules to be associated with pseudodiatomic species, that is, we may apply the quantum
mechanical harmonic oscillator (QMHO) model to any stretching mode of the molecule as if
it were comprised of only two atoms. Since the QMHO model depends only upon the strength CHEM 2500H 20FA Due. Mon Nov 16, 2020 of the bond and the mass of the atoms forming the bond, this exercise will allow you to see the influence of both of these variables on the wavenumber or “frequency” of the bond vibration, as it is sometimes referred to in older literature. Note in fact cm-1 is not a temporal frequency, although it is trivial to convert it to the true frequency in units of s-1 by multiplication by c, the speed of light. The approach you will take will be to treat the various stretching vibrations in a molecule as if they were associated with the stretching mode of diatomic molecules. Note that this model will not work for non-stretching vibrations, since bending and other more complicated motions inherently involve more than two atoms undergoing significant motion. Having recorded IR spectra of several key molecules, and establishing the wavenumber and identity of the bond or “carrier”, you will use the mass of the atoms involved, and simple force constants to develop a model for prediction of vibrational wavenumber values for that mode.
Theory: The quantum mechanical energy of a particular vibrational state of a diatomic molecule, when modeled as a harmonic oscillator, is given by:
(1) E n + n 0,1,2, h k 1 ( )
n 2 2  
   
= =    
    K Here En is the energy of vibrational state n, h is Planck’s constant, k is the force constant (N.m-1), μ is the reduced mass of the diatomic molecule in kg.molecule-1, (μ = m1 m2/(m1 + m2)) and n is the quantum number for the vibrational mode, equal to 0 for the ground state. The energy associated with a transition between vibrational energy levels is given by the Bohr frequency condition:
(2) E = h
 obs Given that transitions for vibrations of a harmonic oscillator are allowed only between adjacent energy levels, we must observe the condition Δn = +1 (for absorption), so that:
n 1 n
h k
(3) E E E
+ 2 
 
 = − =  
  Thus, the frequency (s-1) of the transition is:
1 k

 
= CHEM 2500H 20FA Due. Mon Nov 16, 2020 and the wavenumber (cm-1) is:
1 k
2 c

 
= The last equation, equation (5), illustrates that the vibrational wavenumber is a function of a constant, the reduced mass of the oscillating bond, and the force constant for the bond. Please
note: Use of this equation will require proper units for the various constants and variables
involved. Recommended units are c in cm s-1; K in Nm-1; μ in kg molecule -1. Thus, within this simple model, the vibrational wavenumber of a particular diatomic molecule is dependent only on the mass of the atoms involved and the strength of the bond. When working with a given family of bond types, in this case stretching modes between two atoms, it is possible to further simplify the situation by invoking “Pauling’s Rule” which states that: “The force constant for a given type of bond is proportional to the bond order for that bond” This rule predicts that the force constant for a given type of single bond stretching mode will be double that of the same type of bond when the bond order is two, or triple when the bond order is three. (i.e. k for C≡O is 3 times that for C-O, and k for C=O is 2 times that for C-O, or the force constant for a general bond = ks, where ks is the single bond force constant, and  is the bond order.) With this in mind we may write a “master equation” for prediction of the wavenumbers of all stretching modes of diatomic molecules, involving one “master” force constant, ks:
1 k
2 c

 
= where α is the bond order for the bond in question. Observe that this consideration works for all bond orders in terms of a single bond strength parameter, as the value of k = αks doubles for double bonds and triples for triple bonds, as required by Pauling’s Rule. CHEM 2500H 20FA Due. Mon Nov 16, 2020
This procedure has been left unchanged as if you are doing the actual
experiment. Please watch the Power Point Presentation provided and
download the 5 spectra and Excel table template provided. 1) Samples of organic species have been provided for you – You should have at least 2- pentanone, propanol, acetonitrile, cyclohexane, and n-butylamine. These are neat samples of each compound. Use the NaCl windows provided in the desiccator with one of the metal cells to take the infrared spectra of the solutions.
NOTE: Do not use or clean these cells with water, as the windows are made of
pure salt!! Your TA will demonstrate to you how to prepare the sample, and additionally there is a set of instructions that can provide some guidance. If you are unsure, wait for the TA as this experiment does not take your full period. A set of pictorial instructions is provided to demonstrate cell preparation, and how to use the IR software. Some of these instructions are also stored on the desktop of the lab computer, so you do not need to print them out. Create a folder in the CHEMISTRY 2500 folder on the C drive of the computer folder labelled with you and your partner’s names or initials – this is not the default
save location, so make sure you are saving in the right place. 2) For each of the provided molecules, draw a Lewis diagram of the molecule. You will need this to identify each of the two-atom combinations of the molecule. For example, in 2-pentanol, there are C-C, C-H, C-O, and O-H single bonds. Using Table 1, identify the wavenumber position associated with each stretching vibrations of your molecules. An Excel template is provided in LS for transforming your data in the report. It also lists the stretching modes for which you should be identifying associated wavenumber values. Record your wavenumber values, which you can then directly enter into the template to prepare your laboratory report. Using excel with DRASTICALLY facilitate this process, as the calculations would be quite laborious by hand. The template also has columns that for the calculations you are performing in the following steps. 3) For each vibration, assume that it is associated with a diatomic atom, and enter the masses of the two atoms involved in the correct columns. Using these masses, and the conversion from AMU to kg/molecule, calculate the reduced mass of each pair. Complete the bond order column for each pair of atoms. Using Equation (5), calculate the force constant that would be predicted by the QMHO model for your “diatomic” molecule. WATCH YOUR UNITS! If properly followed, your k value should have units of N.m-1. N = kg.m.s-2. Typical single bond force constants have an order of magnitude of 102. CHEM 2500H 20FA Due. Mon Nov 16, 2020 4) Assume that Pauling’s Rule applies, and divide all the force constants by the bond order of the corresponding bond, to convert all of your calculated force constants into effective single-bond force constants. Take an average of all these single-bond force constants as your first guess for the value of ks, the master force constant for a single covalent bond. Your final ks value should have an order of magnitude of 102 with units of N m-1. 5) Using this average k value as a guess for the value of ks, use Equation (6) to generate predicted vibrational wavenumbers for all the observed modes that you have recorded data for. For each prediction, feed in the values of α, ks (the average value obtained), and μ for the bond in question to generate a predicted value for . If you use the Excel template provided in LS, a plot of the predicted wavenumber vs the observed wavenumber, with a best fit line. If this does not appear, or if you have done this on your own, you will need to plot the predicted wavenumber values versus the observed wavenumber values for each of the associated bonds. 6) If your guess for ks is the best, your best fit line should have a slope of 1.00, indicating that, on average, the value of the predicted wavenumber is the same as that of the observed wavenumber. It is unlikely that the initial value of ks will be the “best” value, so it must be adjusted. To do this easily, in a new cell in the Excel template enter the average ks value and change the equation for the predicted wavenumber so that it refers to this new cell. You can now adjust the value in the cell and change the slope of your graph until it is equal to 1.00 ± 0.01. When you reach this point, you will have a “refined” model, which can be used to predict the vibrational wavenumber associated with the stretching mode of any reasonable stretching mode of your choice. Note that, since your model was “calibrated” or “validated” with molecules having conventional covalent bonds, you should not expect this model to work with stretching modes associated with unusual, strained or partially ionic bonds. 7) Use your refined force constant value (ks) in Equation 6 to predict the vibrational wavenumber associated with the: a. S-H stretch of mercaptans b. N-F stretch of a fluorine-nitrogen bond c. C-D stretch of deuterium-substituted hydrocarbons d. C=S bond of carbonyl sulphide e. You should compare these calculated values with literature values, which you may need to use library resources to find. This comparison should appear in your report. CHEM 2500H 20FA Due. Mon Nov 16, 2020
Instructions for the reporting:
No formal lab report is needed. In a single .pdf file named,
please include the following: 1) Please include a concise Objective for the experiment. 2) Please include the Lewis structures of the compounds from Step 2. 3) Include sample calculations for each step, for 2-Pentanone (only!) 4) Provide a table with all of your data – the table has been provided as an Excel File. 5) Provide your graph of predicted wavenumber vs observed wavenumber. 6) Provide any correction factor if needed 7) Provide your final force constant with proper units. 8) A tabulated comparison of your calculated force constants for the stretches in Step 7, as well as a comparison to the actual values. 9) Provide a concise conclusion about whether the quantum harmonic oscillator is a good model for explaining these vibrational spectra. How do the masses of the atoms effect the wavenumber? Are there any limitations to this model, or alternatively, when can we use this model? 10) Any references you used. CHEM 2500H 20FA Due. Mon Nov 16, 2020
Type of Vibration Frequency (cm-1) Intensity
C−H Alkanes (stretch) 3000-2850 s
−CH3 (bend) 1450 and 1375 m
−CH2− (bend) 1465 m
Alkenes (stretch)
(out-of-plane bend)
m s
Aromatics (stretch)
(out-of-plane bend)
s s
Alkyne (stretch) ca. 3300 s
Aldehyde 2900-2800
w w
C−C Alkane not useful, transitions are very weak
C=C Alkene
1600 and 1475
C≡C Alkyne 2250-2100 m-w
C=O Aldehyde
Carboxylic Acid
Acid Chloride
1810 and 1760
s s s s s s s
C−O Alcohols
Carboxylic Acids
s s s s
O−H Alcohols, Phenols Free
Alcohols, Phenols H−Bonded
Carboxylic Acids
m m m
N−H Primary and Secondary Amines
and Amides (stretch)
m m
C−N Amines 1350-1000 m-s
C=N Imines and Oximes 1690-1640 w-s
C≡N Nitriles 2260-2240 m
X=C=Y Allenes, Ketenes, Isocyanates, Isothiocyanates 2270-1950 m-s
N=O Nitro (R-NO2) 1550 and 1350 s
S−H Mercaptans 2550 w
S=O Sulfoxides
Sulfones, Sulfonyl Chlorides,
Sulfates, Sulfonamides
1375-1300 and
s s s
C−X Fluoride
Bromide, Iodide
s s s



Quantum Mechanics Lab 1:

Infrared Spectroscopy: Modelling Group Frequencies



Report Template: Please upload the completed template to blackboard as a pdf.




Name and Student Number:


  • Please include a concise objective that includes why and how this experiment will be done. (2 marks)











  • Why is the quantum mechanical harmonic oscillator (QMHO) used to model stretching modes of a polyatomic molecule? Can the QMHO be used to model bending modes of a polyatomic molecule? (3 marks)




















  • Please insert a picture of the Lewis Structures for 2-pentanone, n-propanol, acetonitrile, cyclohexane, and n- Clearly label each structure and calculate the normal modes of vibration for each molecule. You may draw the structures by hand or use chemical drawing software.  (Recall that a non-linear molecule of “n” constituent atoms has 3n-6 degrees of freedom for vibrational motion.) (10 marks)






















  • Include sample calculations for the stretching modes of 2‑Pentanone.  Fill in the missing values with units as required in the following layout.


  1. What are the observed wavenumbers from this experiment for the stretching vibrations of 2‑Pentanone? (1 marks)








  1. Calculate the reduced mass of the stretching mode of 2‑Pentanone. (Recall that the unified atomic mass unit (u) is equivalent to the dalton (Da) and that molar mass is numerically equivalent to the dalton ).  (3 marks)

















  1. How do the masses of the atoms affect the wavenumber? (2 marks)



  1. Rearrange equation (6) from the lab manual to find the force constant . Use the equation editor to show your rearranged equation. In this equation μ is the reduced mass in, α is the bond order, c is the speed of light in , and  is the wavenumber in .  (1 marks)







  1. Calculate the force constant (using the above rearranged equation for the    stretching modes of 2-Pentanone.  Use the equation editor to show your calculations.

(4 marks)



  1. Show why the units of the force constant (are . (Hint:) (1 mark)



  • Use the excel file provided to calculate an average single bond force constant (). (1 mark)




  • What is Pauling’s rule? Why does bond order affect the wavenumber (or frequency) of the stretching mode? (3 marks)







  • Use your averaged force constant to calculate the predicted vibrational wavenumber of the stretching mode of 2‑Pentanone.  Use the equation editor to show your calculations. (4 marks)



  • Insert the appropriately labeled table with the data used to calculate the average force constant and predicted wavenumbers of the stretching modes of 2-pentanone, n-propanol, acetonitrile, cyclohexane, and n- (29 marks)










  • Show the equations used in excel to calculate the: predicted wavenumbers, force constant and reduced mass of the stretching modes of 2-pentanone, n-propanol, acetonitrile, cyclohexane, and n- (You may copy and paste your equation from excel here) (6 marks)













  • Insert the appropriately labeled graph of predicted wavenumber vs observed wavenumber. Provide any correction factor if needed. (10 marks)






























  • Use the experimentally determined force constant value  and excel to calculate predicted vibrational wavenumbers associated with the: S-H stretch of mercaptans, N-F stretch of a fluorine-nitrogen bond, C-D stretch of deuterium-substituted hydrocarbons, C=S bond of carbonyl sulphide.  Insert the appropriately labeled table comparing the calculated vibrational wavenumbers to the experimentally observed wavenumbers for these stretching modes.     (10 marks)

















  • Is the quantum harmonic oscillator a good model for explaining these vibrational spectra?  Are there any limitations to this model, or alternatively, when can we use this model?          (5 marks)














  • List approved references below using an appropriate citation style. (5 marks)







Quantum Mechanics Lab 1:  Infrared Spectroscopy: Modelling Group Frequencies
Bond Name Bond Order Mass (1) Mass (2) Reduced Mass Observed Wavenumber Force Constant Single-Bond FC Predicted Wavenumber
(kg/mol (kg/mol) (kg/molecule) (/cm) (N/m) (N/m) (/cm)
C-H (alkane)
C-H (alkene)
1st Average Single-Bond FC
k = force constant

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