**physical chemistry task problem**

## Project Description:

explicit expressions for the hydrogenic orbitals are given in your textbook and in various tables. using your favorite graphing software (such as microsoft excel, sigmaplot, or kaleidagraph), create plots of the radial distribution functions for the 3s, the 3px, and the 3dxy orbitals. you should plot r2|r(r)|2 versus r for values of r ranging up to about 1.5 nanometers, in increments of about five picometers. put all three plots on the same graph with each wavefunction clearly labeled. print out the graph and discuss the significance of the graphs for interpreting properties of many-electron atoms.

2. derive the ground-state term symbols for the following atoms or ions:

(a) h (b) f (c) f- (d) na (e) na+ (f) p (g) sc

3. ozone absorbs ultraviolet radiation in a part of the electromagnetic spectrum energetic enough to disrupt dna in biological organisms and that is absorbed by no other abundant atmospheric constituent. this spectral range, denoted uv-b, spans the wavelength of about 292 nm to 320 nm. the molar extinction coeffieient of ozone over this range was collected and reported in a journal article, the citation for which does not appear in this question but will be provided to any interested student after this exam has been submitted. the important results of this work are given in the table below:

λ/nm

292.0

296.3

300.8

305.4

310.0

315.0

320.0

ε/(l mol-1 cm-1)

1512

865

477

257

135.9

69.5

34.5

compute the integrated absorption coefficient, εtotal = ∫ ε(ν )dν, of ozone over the wavelength range 292-320 nm. (hint: ε( ν ) can be fit to an exponential function very well using a spreadsheet program such as microsoft excel. your answer should have the units of l mol-1 cm-2.)

4. relaxation dynamics of highly vibrationally excited (“hot”) molecules has been an active field of research for many years. collisions between atmospheric carbon dioxide and pyrazine involve substantial energy increases in co2. in some cases as much as 10000 cm-1 of energy is exchanged in a single collision. (see, for example, m. s. elioff, m. c. wall, a. s. lemoff, and a. s. mullin, journal of chemical physics, 110, 5578-5585, 1999.) gas-phase collisions of hot pyrazine and other azabenzenes (prepared with ~40000 cm-1 of vibrational energy) with co2 occur via either one of two modes: vibration-to-vibration (vv) or vibration-to-rotation and translation (vrt) collisions. in vv collisions, hot pyrazine molecules are collided with ground-state co2, and co2 scatters from the collision vibrationally excited, typically with one or more quantum of energy in the stretching or bending modes. in vrt collisions, hot pyrazine is collided with co2 and the result is co2

~

~

~

recoiling from the collision with no vibrational excitation, but with large amounts of translational and rotational excitation. (translational excitation accompanies rotational excitation in the co2 molecules, but it does not accompany vibrational excitation. large vvrt relaxation apparently does not occur. this intriguing result poses a number of questions beyond the scope of this problem.) we can determine a rotational temperature of the recoiling co2 molecules by scanning a high-resolution infrared laser over a doppler-broadened spectral feature corresponding to the antisymmetric stretching mode of co2. (this is the 001←000 transition.) these profiles can be fit to a gaussian function, and ascertaining a value for the full width at half-height profile of the lineshape.

suppose the observed δνfwhm linewidth of a particular rovibrational transition of co2 is 0.015 cm-1, and that the frequency of the transition is 2349 cm-1. from this linewidth, determine a rotational temperature for the recoiling carbon dioxide molecule, and from this temperature determine the molecule’s speed.

5. the spectrum of a star can be used to measure its radial velocity with respect to the sun. (the radial velocity with respect to the sun is the component of the star’s velocity vector that is parallel to a vector connecting the star’s center to the center of the sun.) the measurement relies on the doppler effect. you have probably noticed that the doppler effect leads to a broadening of a transition in a gas, but to a shift in the wavelength of a star. when a star emitting electromagnetic radiation of frequency ν moves with a speed s relative to an observer, the observer detects radiation of frequency νreceding or νapproaching,

where and

and c is the speed of light in the vacuum. since νreceding _ ν, a receding star is characterized by a red shift of its spectrum with respect to an identical, but stationary, source. similarly, an approaching star is characterized by a blue shift in its spectrum.

in a typical experiment, ν is the frequency of a spectral feature due to an element in the earthbound laboratory’s frame of reference. measurement of the same spectral line in the same element of a star will be shifted, as stated above, and the speed s of the star may be calculated. three fe lines from the star lmc5013b, which belongs to galaxy about 2×105 light years away from ours, occur at 438.882 nm, 441.000 nm, and 442.010 nm. the same lines occur at 438.392 nm, 440.510 nm, and 441.510 nm in the spectrum of an earth-bound iron arc.

answer the following two questions: (a) determine whether lmc5013b is receding from or approaching the earth and estimate the star’s radial speed with respect to the earth. (b) what additional information would you need to calculate the radial velocity of lmc5013b with respect to the sun?

~

cscsreceding+−=11ννcscsgapproachin+−÷=11νν

6. a 2sp2 hybrid orbital that lies in the xy-plane and makes an angle of 120º to the x-axis has the form:

use hydrogenic atomic orbitals to write the explicit form of the hybrid orbital. show that it has a maximum amplitude in the direction specified.

7. what characteristics make laser light different than sunlight? also, describe and compare two-level, three-level, and four-level laser systems and tell why one of them is more efficient than the others.

8. the development of methods for preparing ultra-cold atoms has made possible the generation of bose-einstein condensates, the optical clock, and investigations of collisions at very low energies. although nitric oxide is not a boson, its spectrum is well characterized, it is a free radical important in combustion, and it has a lowest excited state lying only 3.8 ev below the ionization potential. for these reasons, as well as the fact that no is well-suited to laser-induced fluorescence, nitric oxide molecules are ideal for inelastic collisional cooling experiments.

no molecules have been cooled to temperatures of less than one kelvin in significant numbers (109 molecules per cubic centimeter per quantum state) via inelastic collisions with argon atoms. (see, for example, m. s. elioff, j. j. valentini, and d. w. chandler, science, 302, 1940-1943, 2003.)

write the term symbol for the the ground state of no and sketch out a molecular orbital energy diagram for no and place the electrons in the levels appropriate for the ground state. you will need to look up the ao ionization energies for o2s, o2p, n2s, and n2p. the mo energies, from lowest to highest, including core electrons, follow the sequence 1σg, 1σu, 2σg, 2σu, 3σg, 1πu, 1πg, 3σu,. (assume that the 1s orbitals do not mix with the valence orbitals on n and o, and define the 3σ orbital as originating primarily from the 2s aos on n and o. connect each mo level with the level of the major contributing ao on each atom.)

9. assume that the electronic states of the π electrons of a conjugated molecule can be approximated by the wavefunctions of a particle in a one-dimensional box, and that the dipole moment can be related to the displacement along this length by μ = -ex, where e is the fundamental unit of charge and x is the displacement. show that the transition probability for the transition 21 is nonzero, whereas that for the 31 transition is zero.

+−=ψyxpps2 232 21231

10. this question about photoelectron spectroscopy has two parts.

(a) in a simple model used to analyze uv photoelectron spectra, the orbital energies of the neutral molecule and the cation formed by ejection of an electron are assumed to be the same. in fact, some relaxation occurs to compensate for the reduction in the number of electrons by one. would you expect the orbital energies to increase or decrease in the relaxation?

(b) suppose you obtain the uv photoelectron spectrum shown below for a gas-phase molecule. each of the groups corresponds to a cation produced by ejecting an electron from a different mo. what can you conclude about the bond length of the cations in the three states formed relative to the ground-state neutral molecule? use the relative intensities of the individual vibrational peaks in each group to answer this question.

2. derive the ground-state term symbols for the following atoms or ions:

(a) h (b) f (c) f- (d) na (e) na+ (f) p (g) sc

3. ozone absorbs ultraviolet radiation in a part of the electromagnetic spectrum energetic enough to disrupt dna in biological organisms and that is absorbed by no other abundant atmospheric constituent. this spectral range, denoted uv-b, spans the wavelength of about 292 nm to 320 nm. the molar extinction coeffieient of ozone over this range was collected and reported in a journal article, the citation for which does not appear in this question but will be provided to any interested student after this exam has been submitted. the important results of this work are given in the table below:

λ/nm

292.0

296.3

300.8

305.4

310.0

315.0

320.0

ε/(l mol-1 cm-1)

1512

865

477

257

135.9

69.5

34.5

compute the integrated absorption coefficient, εtotal = ∫ ε(ν )dν, of ozone over the wavelength range 292-320 nm. (hint: ε( ν ) can be fit to an exponential function very well using a spreadsheet program such as microsoft excel. your answer should have the units of l mol-1 cm-2.)

4. relaxation dynamics of highly vibrationally excited (“hot”) molecules has been an active field of research for many years. collisions between atmospheric carbon dioxide and pyrazine involve substantial energy increases in co2. in some cases as much as 10000 cm-1 of energy is exchanged in a single collision. (see, for example, m. s. elioff, m. c. wall, a. s. lemoff, and a. s. mullin, journal of chemical physics, 110, 5578-5585, 1999.) gas-phase collisions of hot pyrazine and other azabenzenes (prepared with ~40000 cm-1 of vibrational energy) with co2 occur via either one of two modes: vibration-to-vibration (vv) or vibration-to-rotation and translation (vrt) collisions. in vv collisions, hot pyrazine molecules are collided with ground-state co2, and co2 scatters from the collision vibrationally excited, typically with one or more quantum of energy in the stretching or bending modes. in vrt collisions, hot pyrazine is collided with co2 and the result is co2

~

~

~

recoiling from the collision with no vibrational excitation, but with large amounts of translational and rotational excitation. (translational excitation accompanies rotational excitation in the co2 molecules, but it does not accompany vibrational excitation. large vvrt relaxation apparently does not occur. this intriguing result poses a number of questions beyond the scope of this problem.) we can determine a rotational temperature of the recoiling co2 molecules by scanning a high-resolution infrared laser over a doppler-broadened spectral feature corresponding to the antisymmetric stretching mode of co2. (this is the 001←000 transition.) these profiles can be fit to a gaussian function, and ascertaining a value for the full width at half-height profile of the lineshape.

suppose the observed δνfwhm linewidth of a particular rovibrational transition of co2 is 0.015 cm-1, and that the frequency of the transition is 2349 cm-1. from this linewidth, determine a rotational temperature for the recoiling carbon dioxide molecule, and from this temperature determine the molecule’s speed.

5. the spectrum of a star can be used to measure its radial velocity with respect to the sun. (the radial velocity with respect to the sun is the component of the star’s velocity vector that is parallel to a vector connecting the star’s center to the center of the sun.) the measurement relies on the doppler effect. you have probably noticed that the doppler effect leads to a broadening of a transition in a gas, but to a shift in the wavelength of a star. when a star emitting electromagnetic radiation of frequency ν moves with a speed s relative to an observer, the observer detects radiation of frequency νreceding or νapproaching,

where and

and c is the speed of light in the vacuum. since νreceding _ ν, a receding star is characterized by a red shift of its spectrum with respect to an identical, but stationary, source. similarly, an approaching star is characterized by a blue shift in its spectrum.

in a typical experiment, ν is the frequency of a spectral feature due to an element in the earthbound laboratory’s frame of reference. measurement of the same spectral line in the same element of a star will be shifted, as stated above, and the speed s of the star may be calculated. three fe lines from the star lmc5013b, which belongs to galaxy about 2×105 light years away from ours, occur at 438.882 nm, 441.000 nm, and 442.010 nm. the same lines occur at 438.392 nm, 440.510 nm, and 441.510 nm in the spectrum of an earth-bound iron arc.

answer the following two questions: (a) determine whether lmc5013b is receding from or approaching the earth and estimate the star’s radial speed with respect to the earth. (b) what additional information would you need to calculate the radial velocity of lmc5013b with respect to the sun?

~

cscsreceding+−=11ννcscsgapproachin+−÷=11νν

6. a 2sp2 hybrid orbital that lies in the xy-plane and makes an angle of 120º to the x-axis has the form:

use hydrogenic atomic orbitals to write the explicit form of the hybrid orbital. show that it has a maximum amplitude in the direction specified.

7. what characteristics make laser light different than sunlight? also, describe and compare two-level, three-level, and four-level laser systems and tell why one of them is more efficient than the others.

8. the development of methods for preparing ultra-cold atoms has made possible the generation of bose-einstein condensates, the optical clock, and investigations of collisions at very low energies. although nitric oxide is not a boson, its spectrum is well characterized, it is a free radical important in combustion, and it has a lowest excited state lying only 3.8 ev below the ionization potential. for these reasons, as well as the fact that no is well-suited to laser-induced fluorescence, nitric oxide molecules are ideal for inelastic collisional cooling experiments.

no molecules have been cooled to temperatures of less than one kelvin in significant numbers (109 molecules per cubic centimeter per quantum state) via inelastic collisions with argon atoms. (see, for example, m. s. elioff, j. j. valentini, and d. w. chandler, science, 302, 1940-1943, 2003.)

write the term symbol for the the ground state of no and sketch out a molecular orbital energy diagram for no and place the electrons in the levels appropriate for the ground state. you will need to look up the ao ionization energies for o2s, o2p, n2s, and n2p. the mo energies, from lowest to highest, including core electrons, follow the sequence 1σg, 1σu, 2σg, 2σu, 3σg, 1πu, 1πg, 3σu,. (assume that the 1s orbitals do not mix with the valence orbitals on n and o, and define the 3σ orbital as originating primarily from the 2s aos on n and o. connect each mo level with the level of the major contributing ao on each atom.)

9. assume that the electronic states of the π electrons of a conjugated molecule can be approximated by the wavefunctions of a particle in a one-dimensional box, and that the dipole moment can be related to the displacement along this length by μ = -ex, where e is the fundamental unit of charge and x is the displacement. show that the transition probability for the transition 21 is nonzero, whereas that for the 31 transition is zero.

+−=ψyxpps2 232 21231

10. this question about photoelectron spectroscopy has two parts.

(a) in a simple model used to analyze uv photoelectron spectra, the orbital energies of the neutral molecule and the cation formed by ejection of an electron are assumed to be the same. in fact, some relaxation occurs to compensate for the reduction in the number of electrons by one. would you expect the orbital energies to increase or decrease in the relaxation?

(b) suppose you obtain the uv photoelectron spectrum shown below for a gas-phase molecule. each of the groups corresponds to a cation produced by ejecting an electron from a different mo. what can you conclude about the bond length of the cations in the three states formed relative to the ground-state neutral molecule? use the relative intensities of the individual vibrational peaks in each group to answer this question.

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