THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George Hardgrove 996. All rghts reserved. You are welcome to use ths document n your own classes but commercal use s not allowed wthout the permsson of the author. Ths document contans the nstructons for the Mathcad fle enttled corun.mcd. The equatons n ths document are btmap mages and not Mathcad actve. The Mathcad actve equatons are found n corun.mcd. Students should frst complete the exercses for the smple harmonc oscllator that are found n horun.mcd and ts accompanyng nstructonal fle hons.mcd. In ths lesson we wll examne the mathematcal bass for representng the vbratons of CO. We wll assume that the force constant for each C-O bond s k. The molecule s arranged along the x axs wth coordnates of the three atoms beng x, x, and x 3, the left oxygen, the carbon, and the rght oxygen respectvely. Note that m = m 3. Sketch the molecule. Goal: To explore classcal harmonc oscllator moton n a lnear tratomc molecule. Objectves:. To use matrx algebra to transform between coordnate systems used to represent a molecule. To wrte the Laplacan and equatons of moton for a lnear tratomc molecule. 3. To explan the moton of tratomc molecules n terms of the potental and knetc energy contrbutons from each atom. Prerequstes:. Successful completon of hons.mcd and horun.mcd exercses;. Mathcad 6.0+ s requred to to solve the dfferental equatons; 3. Dfferental calculus s needed n order to take the requred dervatves. Students should understand what "to solve a dfferental equaton" means. 4. Only elementary level skll wth Mathcad s requred to use the corun.mcd document. More advanced skll would be requred to author documents of ths type. Created: Sprng 996 Modfed: June 997 cons.mcd page
The knetc energy s the sum of the knetc energy contrbuted by each atom: T = ( mx + mx + mx 3 ) () The extent Q that the left bond s stretched or compressed compared to ts equlbrum bond length of r e s Q = x x r e () Smlarly for Q Q = x3 x r e (3) Where at equlbrum x - x = r e and x 3 - x = r e Takng the tme dervatves we get Q = x - x and Q = x - x (4) 3 The center of mass of the molecule s gven by C = m x m The tme dervatve of the center of mass yelds equaton 5. m x + m x + m x M 3 3 = C (5) Created: Sprng 996 Modfed: June 997 cons.mcd page
Next we wrte the potental energy n terms of the stretchng of each bond. Notce how Hooke's law s used here. V = (k Q + k Q ) (6) Above we descrbed the knetc energy T n terms of x coordnates, and the potental energy V n terms of Q coordnates. In order to obtan the equatons of moton we must descrbe T and V n the same coordnates. Snce we wll use the Q coordnates below, we must obtan both T and V n terms of the Q's. Ths can be done by usng a matrx transformaton. The matrx equaton to accomplsh ths s Q 0 x Q 0 x C = m m m 3 x 3 M M M (7) To obtan the x coordnates n terms of the Qs we nvert the 3x3 matrx and multply each sde of equaton 7 on the left by ths nverse to get: x x x = 3 m m m M Q m m M Q M m m + m M C (8) Where M = m + m. Recall that m = m 3. Queston. Wrte the exp resson for the x ' s as a functon of Q ' s and C. Created: Sprng 996 Modfed: June 997 cons.mcd page 3
The knetc energy as a functon of the Q 's can be obtaned by substtutng the x 's from the above matrx equaton nto equaton. After consderable algebra we get equaton 9. m m T = + M ( m ) ( Q Q ) m + + M Q Q + mc (9) Queston. Does the knetc energy depend on the center of mass? What coordnates are essental for descrbng ths system of three atoms? Complete the algebrac step leadng from equaton to equaton 9. Ths s not a trval algebrac process. Be careful. Collectng the constants nto B and B we can rewrte the knetc energy as T = B (Q +Q ) + B Q Q (0) whch wth the potental energy n terms of the Q's V = k (Q + Q ) () can be substtuted nto Lagrange's dfferental equaton (), frst for Q then for Q and then for Cdot. d dt L Q - L Q = 0 () to gve the followng equatons of moton B Q + B Q + kq = 0 (3) BQ + BQ + kq = 0 (4) mc = 0 (5) Equaton 5 does not contrbute to the equatons of moton descrbng the poston of the oxygens. We wll thus solve equatons 3 and 4 as a coupled par of dfferental equatons Created: Sprng 996 Modfed: June 997 cons.mcd page 4
Queston 3. Wrte the Lagrangan for the CO molecule. Queston 4. Complete the detals of obtanng equatons 3 and 4. Equatons 3 and 4 cannot be solved by MATHCAD n ths form shown above. We must frst solve them for each of the second dervatves n order to use the rkfxed Mathcad functon. The rkfxed functon gves us a numercal soluton to the set of dfferental equatons. Q = B k Q - k Q B B B - 4 B (6) Q = k B B Q - k Q B- B B (7) The values of k and the masses of the atoms are used wth these equatons n the computatonal document where the dfferental equatons are solved. Queston 5: Complete the detals of obtanng equatons 6 and 7 from equatons 3 and 4. EXERCISE The nput to the dfferental equatons routne has some adjustable constants. We must set values of the masses of the atoms m and m whch can be taken as 6 and for O (m and m 3 ) and C (m ) respectvely. The value of the force constant should be set to 00. The column matrx Q n corun.mcd gves the ntal values of the Q's and ther ntal veloctes: Q Q = Q Q Q Created: Sprng 996 Modfed: June 997 cons.mcd page 5
Set each ntal velocty equal to 3 and each ntal poston equal to zero. Calculate all four pages of the document. Whch vbratonal mode does the chosen ntal condtons smulate? For the anmaton part place the cursor (red cross) just to the rght of the top of the anmaton plot and clck on Graphcs and Anmaton and create. Then wth the mouse block out the porton of the pcture you wsh to anmate. If necessary press the lghtbulb button to actvate the automatc mode. The anmaton wll now be recorded and when t s complete a wndow wll appear at the left for the playback feature. Clckng on the lower left button n ths wndow wll actvate the playback. Note the Fourer transform of the moton of Q on the last page. Ths s a plot n frequency space of the vbratonal frequences contaned n ths moton. EXERCISE Go to the Q matrx and change one of the dervatves from 3 to -3. Repeat the calculatons. Has the frequency become larger or smaller? Whch vbratonal mode does ths represent? EXERCISE 3 Go agan to the Q matrx and change the ntal POSITION Q to and change the ntal VELOCITY for Q to 3. Repeat the calculaton. Note that the moton becomes a mxture of the two modes and appears to be haphazard. But the Fourer transform can dsplay two frequences (even though the phasng of the maxma s not great). Queston 6: What s the physcal meanng of havng ntal value of Q for partcle equal to 3 and the ntal velocty of partcle equal to zero and at the same tme have the ntal poston value of Q for partcle equal to zero and the correspondng velocty for ths coordnate equal to 3? Created: Sprng 996 Modfed: June 997 cons.mcd page 6