INFRARED ABSORPTION SPECTROSCOPY References: See relevant sections in undergraduate text Background: Learn from your instructor how to use the spectrometer. Know definitions of the following and their interrelations: Reduced mass, µ Frequency, ν (sec -1 ) Wave number, " (cm -1 ) Rotational constant, B e (cm -1 ) "Equilibrium internuclear separation," r e Anharmonicity constant, x e Objectives: Vibration-rotation interaction constant, α e Quantum numbers υ and J P and R branches Moment of inertia, I Measurement of: 1) Vibration-rotation spectrum of HCl gas 2) Frequency of rotational lines of HCl gas Computation of: 1) Vibration-rotation interaction constant, α e 2) Rotational constant, B e 3) Fundamental vibration frequency, " 0 (cm -1 ) 4) Moment of inertia of HCl 5) Bond length of HCl, r e 6) Force constant for the H-Cl bond
Physical Chemistry Laboratory Chemicals: Apparatus: Conc. H 2 SO 4, NaCl and anhydrous CaCl 2 IR gas cell and filling system Research grade IR spectrometer Remarks: Theory and Spectral Analysis The spectra of molecular systems can be quite complicated and the "sorting out" of the profusion of lines and their assignment to the appropriate transitions may be a formidable task. However a proper molecular model may be used towards this objective as well as the deduction of structural parameters of the molecule of interest. The vibrational-rotational band: An infrared, high resolution spectrum of a heteronuclear diatomic molecule, such as HCl gas, consists of a vibrational-rotational band due to transitions in which both vibrational and rotational energies change. Such a band can be conveniently considered to have three portions as shown in Figure (1). 1) The zero gap is the central portion and is signified by no absorption. 2) The P- branch is the low frequency (cm -1 ) portion to the right of the zero gap. It consists of several lines which diverge towards low frequency. 3) The R- branch is the high frequency (cm -1 ) portion to the left of the zero gap. It consists of several lines which converge towards high frequency. The vibrotor model: The spectral features of the vibrationalrotational band of HCl (gas) may be explained in terms of a model for a diatomic molecule simultaneously executing both vibrational and rotational motions. The simplest model is an approximate one in which the vibrational and rotational motions are treated independently. For the vibrational motion the molecule is assumed to vibrate as a simple harmonic oscillator. The energy of the allowed vibrational levels is given by:
E(υ) = hν(υ + 1/2) (1) 2,.. The vibrational quantum number υ is of integral values 0, 1, The rotational motion of the molecule is that of a rigid rotor with the energy of the allowed levels given by: E( J ) = h 2 8! 2 I J( J +1) (2) The rotational quantum number J is of integral values 0, 1, 2,. The moment of inertia I is related to the internuclear distance r, and the reduced mass by: I = µr 2 (3) As a first approximation the energy of a vibrating rotating molecule (vibrotor) is the sum of expressions (1) and (2): # E(!, J ) = h"! + 1 % $ 2& + h2 8' 2 I J( J +1) (4) Each vibrational energy level consists of closely spaced rotational energy levels. A more complete expression, but not necessarily the most exact one, should include: 1) The effect of anharmonicity which is accounted for by a term involving the anharmonicity constant, x e. 2) A term involving the distortion constant, B e, which accounts for the centrifugal stretching. This results from the stretching of the non-ridged chemical bond during rotation and is important at high rotational energies (high J). 3) A term involving the vibration-rotation interaction constant, α e, which accounts for changes in r during vibrations. The vibrational-rotation energy expression of a vibrotor may be represented by:
Physical Chemistry Laboratory ( ) = E (!, J) T!, J (4) hc T (!, J) = " #! + 1 % e $ 2 & ' x "! + 1 2 # % e e $ 2 & + B e J( J +1) + D e J 2 ( J +1 ) 2 ' ( #! + 1 % e $ 2 & J( J + 1) (5) where T (υ, J) is in cm -1. The frequency " e (cm -1 ) refers to the frequency of the molecule vibrating about its equilibrium internuclear separation r e. The rotational constant B e is defined by (c is the speed of light, cm/s): B e = h 8! 2 Ic (6) Selection Rules: The fine structure of the P- and R- branches represents transitions from a particular rotational level (J = J") in a given vibrational state (υ = υ") to a different rotational level (J = J') in an excited vibrational state (υ = υ'). The superscripts (') and (") refer to high and low vibrational states, respectively. A common pitfall for students first exposed to spectroscopy is to confuse energy levels T (υ, J) with transitions between energy levels Δ[T (υ, J)] where Δ [T(υ, J)] = T (υ', J') - T (υ", J") (7) It is the latter which is directly observed. For transitions to be allowed the selection rules for a diatomic molecule to be infrared active are: 1) Δυ = ± 1 2) ΔJ = ± 1 3) Change to dipole moment 0 during vibration. A homonuclear diatomic molecule, being nonpolar, has a dipole moment equal to zero, and invariant, unless the molecule is in a unique chemical environment. Thus, homonuclear diatomic molecules are, in general, infrared inactive. In the present experiment the transitions take place from
various J" - levels of the vibrational ground state (υ" = 0) to J' - levels of the first excited vibrational state (υ' = 1). Frequencies of the P- and R- lines: From the selection rules, and application of equation (7) with proper substitution from eq (5) the frequencies (cm -1 ) of the P- and R- lines are: P- branch: ΔJ = -1 ; which is J' - J" where J" = 1, 2, 3 R- branch: ΔJ = + 1 ; J' = J" + 1! P =! 0 " 2( B e " # e )J " # e J 2 ; (8) and! R =! 0 "( 2B e " 3# e ) + ( 2B e " 2# e )J " a e J 2 (9)! 0 =! e " 2x e! e (10) where " 0 is the frequency of the forbidden transition (ΔJ = 0). It corresponds to the missing line, somewhere near the midpoint in the zero gap. In the derivation of equations (8) and (9) from equation (7) the term in D e is neglected. The frequency of the P- and R- branches, in the observed spectrum can be represented by the empirical equation: where m is a running number which is: +1, +2, +3, for the R- branch " = c + dm + em 2 (11) and -1, -2, -3, for the P- branch. A single equation similar to equation (11) may be obtained from equations (8) and (7) by substituting for J as follows: for the P- branch (equation 8): J = -m and for the R- branch (equation 9): J = m - 1 This leads to:! =! 0 " 2( B e "# e )m "# e m 2 (12)
Physical Chemistry Laboratory The separation between adjacent lines in each branch is given by:! " ( m) = "( m +1) # "( m) = 2B e # 3$ e ( ) # 2$ e m (13) ( ) as ordinate and m as abscissa, a If we plot values of " # m fit of the best straight line through the points has its slope equal to -2α e, yielding the vibrotor interaction constant. If the points are from a sufficiently highly resolved spectrum and are plotted on a sufficiently large scale, the best fit will be a slightly curved line. The slope of the tangent to the curve at m = 0 should be used to obtain α e. The intercept of the curve with the " #( m) axis gives a value from which B e, the rotational constant, is obtained. The value of " 0 is obtained from equation (12) by using α e, B e and with data from an m that has a " # that falls on the least squares line. The moment of inertia, and hence the bond length, are obtained from the definition for B e. Procedure: Specific instructions for the operation of the infrared spectrometer to be used will be given in the laboratory. This is a precision research instrument -- use it carefully. If in doubt about anything after being instructed in its use, ask the instructor. Don't take chances with expensive equipment when you are unsure. The gas cell is constructed from a 10 cm length of large diameter PTFE tubing with stopcocks attached. Infraredtransparent windows are attached to the ends by an adhesive or by mechanical clamps or both. The windows are cut from large crystals of NaCl and are easily fogged by moisture from atmosphere or skin. Never touch the windows and store the cell in a desiccator when not in use. Connect the dropping funnel and flask (properly supported by a ring stand) to the drying tube and cell, with stopcocks open. This setup must be in a fume hood. To about 25 g NaCl, slowly add about 25 ml concentrated sulfuric acid to produce a steady evolution of gas, probably not all the acid will be needed. Check the open end of the cell for HCl evolution as evidenced by white fumes and/or blue litmus turning pink. Allow the gas to continue passing through the cell for a few minutes then disconnect and close the stopcocks. This usually provides
a reasonable partial pressure of HCl in the cell. Record the spectrum on a fresh piece of paper using the highest expansion of wavelength scale available with the instrument. This spectrum should run between values of 2700 cm -1 and 3000 cm -1 the frequencies as recorded by the FTIR are then used for the calculations. Results and Calculations: 1) In a single table, list the following information for each identifiable peak in your high resolution spectrum: ( ) J,! 0, P or R branch, m,! " m 2) Using the method given above, i.e. the plot of Δν~ (m) vs. m, from the tabulated data compute: a) α e b) c) B e! 0 3) Using the value of B e computed above, obtain a value for I, the moment of inertia of the HCl molecule. 4) Using 1.67379 x 10-27 kg and 5.80752 x 10-26 kg for the masses of individual atoms of hydrogen and chlorine, respectively, compute the reduced mass, µ, and bond length r e (in Angstroms and nm) for HCl from I = µ r e 2. 5) Calculate the force constant for the HCl bond. 6) Repeat steps 1-5 for the DCl spectrum given by the instructor.