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Physical Chemistry 8th Edition Peter Atkins, Julio De Paula - Solutions

The promotion of an electron from the valence band into the conduction band in pure TIO2 by light absorption requires a wavelength of less than 350 nm. Calculate the energy gap in electronvolts between the valence and conduction bands.
The magnetic moment of CrCl3 is 3.81µB. How many unpaired electrons does the Cr possess?
Calculate the molar susceptibility of benzene given that its volume susceptibility is −7.2 × 10−7 and its density 0.879 g cm−3 at 25°C.
According to Lewis theory, an O2 molecule should be diamagnetic. However, experimentally it is found that χm /(m3 mol−1) = (1.22 × 10−5 K)/T. Determine the number of unpaired spins in O2. How is the problem of the Lewis structure resolved?
Data on a single crystal of MnF2 give χm = 0.1463 cm3 mol−1 at 294.53 K. Determine the effective number of unpaired electrons in this compound and compare your result with the theoretical value.
Estimate the spin-only molar susceptibility of CuSO4·5H2O at 25°C.
Approximately how large must the magnetic induction, B, be for the orientational energy of an S = 1 system to be comparable to kT at 298 K?
A certain molecule fluoresces at a wavelength of 400 nm with a half-life of 1.0 ns. It phosphoresces at 500 nm. If the ratio of the transition probabilities for stimulated emission for the S* → S to the T → S transitions is 1.0 × 105, what is the half-life of the phosphorescent state?
The photoionization of H2 by 21 eV photons produces H2+. Explain why the intensity of the v = 2 ← 0 transition is stronger than that of the 0 ← 0 transition.
Light-induced degradation of molecules, also called photobleaching, is a serious problem in single-molecule spectroscopy. A molecule of a fluorescent dye commonly used to label biopolymers can withstand about 106 excitations by photons before light-induced reactions destroy its π system and the
The abundance of ozone is typically inferred from measurements of UV absorption and is often expressed in terms of Dobson units (DU): 1 DU is equivalent to a layer of pure ozone 10−3 cm thick at 1 atm and 0°C. Compute the absorbance of UV radiation at 300 nm expected for an ozone abundance of
Discuss in detail the effects of a 90° pulse and of a 180° pulse on a system of spin-1/2 nuclei in a static magnetic field.
33S has a nuclear spin of 3/2 and a nuclear g-factor of 0.4289. Calculate the energies of the nuclear spin states in a magnetic field of 7.500 T.
Calculate the frequency separation of the nuclear spin levels of a 13C nucleus in a magnetic field of 14.4 T given that the magnetogyric ratio is 6.73 × 107 T−1 s −1.
Discuss the origins of diagonal and cross peaks in the COSY spectrum of an AX system.
In which of the following systems is the energy level separation the largest?(a) A proton in a 600 MHz NMR spectrometer,(b) A deuteron in the same spectrometer.
Calculate the energy difference between the lowest and highest nuclear spin states of a 14N nucleus in a 15.00 T magnetic field.
Suggest how spin probes could be used to estimate the depth of a crevice in a biopolymer, such as the active site of an enzyme.
Calculate the relative population differences (δN/N) for protons in fields of(a) 0.30 T,(b) 1.5 T, and(c) 10 T at 25°C.
The first generally available NMR spectrometers operated at a frequency of 60 MHz; today it is not uncommon to use a spectrometer that operates at 800 MHz. What are the relative population differences of 13C spin states in these two spectrometers at 25°C?
The chemical shift of the CH3 protons in acetaldehyde (ethanal) is δ = 2.20 and that of the CHO proton is 9.80. What is the difference in local magnetic field between the two regions of the molecule when the applied field is(a) 1.5 T,(b) 15 T?
The hyperfine coupling constants observed in the radical anions (12), (13), and (14) are shown (in millitesla, mT). Use the value for the benzene radical anion to map the probability of finding the unpaired electron in the π orbital on each C atom. 0.011 0.172 NO
Sketch the appearance of the 1H-NMR spectrum of acetaldehyde (ethanal) using J = 2.90 Hz and the data in Exercise 15.9a in a spectrometer operating at(a) 250 MHz,(b) 500 MHz.Data in Exercise 15.9aThe chemical shift of the CH3 protons in acetaldehyde (ethanal) is δ = 2.20 and that of the CHO proton
The p-dinitrobenzene radical anion can be prepared by reduction of p-dinitrobenzene. The radical anion has two equivalent N nuclei (I = 1) and four equivalent protons. Predict the form of the EPR spectrum using a(N) = 0.148 mT and a(H) = 0.112 mT.
Sketch the form of the 19F-NMR spectra of a natural sample of tetrafluoroborate ions, BF−4, allowing for the relative abundances of 10BF4− and 11BF4−.
Show that, for a perfect gas, (∂U/∂S)V = T and (∂U/∂V)S = −p.
The CH3Cl molecule belongs to the point group C3v. List the symmetry elements of the group and locate them in the molecule.
The group C2h consists of the elements E, C2, σh, i. Construct the group multiplication table and find an example of a molecule that belongs to the group.
List the symmetry operations and the corresponding symmetry elements of the point groups.
Which of the following molecules may be polar?(a) Pyridine (C2v),(b) Nitroethane (Cs),(c) Gas-phase HgBr2 (D∞h),(d) B3N3N6 (D3h).
The group D2h has a C2 axis perpendicular to the principal axis and a horizontal mirror plane. Show that the group must therefore have a centre of inversion.
Use symmetry properties to determine whether or not the integral ∫px zpzdτ is necessarily zero in a molecule with symmetry C4v.
Explain the symmetry criteria that allow a molecule to be optically active.
Show that the transition A1 → A2 is forbidden for electric dipole transitions in a C3v molecule.
Show that the function xy has symmetry species B2 in the group C4v.
The (one-dimensional) matrices D(C3) = 1 and D(C2) = 1, and D(C3) = 1 and D(C2) = −1 both represent the group multiplication C3C2 = C6 in the group C6v with D(C6) = +1 and −1, respectively. Use the character table to confirm these remarks. What are the representatives of σv and σd in each
Explain the construction and content of a character table.
Molecules belonging to the point groups D2h or C3h cannot be chiral. Which elements of these groups rule out chirality?
What irreducible representations do the four H1s orbitals of CH4 span? Are there s and p orbitals of the central C atom that may form molecular orbitals with them? Could d orbitals, even if they were present on the C atom, play a role in orbital formation in CH4?
B.A. Bovenzi and G.A. Pearse, Jr. (J. Chem. Soc. Dalton Trans., 2763 (1997)) synthesized coordination compounds of the tridentate ligand pyridine-2,6-diamidoxime (C7H9N5O2, 25). Reaction with NiSO4 produced a complex in which two of the essentially planar ligands are bonded at right angles to a
Outline how a direct product is expressed as a direct sum and how to decide whether the totally symmetric irreducible representation is present in the direct product.
Identify the point groups to which the following objects belong:(a) A sphere,(b) An isosceles triangle,(c) An equilateral triangle,(d) An unsharpened cylindrical pencil.
Which of the molecules in Exercises 12.9a and 12.10a can be (a) Polar,(b) Chiral?Data in Exercises 12.10aAssign (a) Cis-dichloroethene and
Consider the C2v molecule NO2. The combination px(A) − px(B) of the two O atoms (with x perpendicular to the plane) spans A2. Is there any orbital of the central N atom that can have a nonzero overlap with that combination of O orbitals? What would be the case in SO2, where 3d orbitals might be
The ground state of NO2 is A1 in the group C2v. To what excited states may it be excited by electric dipole transitions, and what polarization of light is it necessary to use?
Does the product xyz necessarily vanish when integrated over(a) A cube,(b) A tetrahedron,(c) A hexagonal prism, each centred on the origin?
Write f1 = sin θ and f2 = cos θ, and show by symmetry arguments using the group Cs that the integral of their product over a symmetrical range around θ = 0 is zero.
The chlorophylls that participate in photosynthesis (Impact I24.2) and the haem groups of cytochromes (Impact I7.2) are derived from the porphine dianion group (30), which belongs to the D4h point group. The ground electronic state is A1g and the lowest-lying excited state is Eu. Is a
The H3+ molecular ion has recently been found in the interstellar medium and in the atmospheres of Jupiter, Saturn, and Uranus. The H4 analogues have not yet been found, and the square planar structure is thought to be unstable with respect to vibration. Take as a basis for a representation of the
Calculate the ratio of the Einstein coefficients of spontaneous and stimulated emission, A and B, for transitions with the following characteristics:(a) 70.8 pm X-rays,(b) 500 nm visible light,(c) 3000 cm−1 infrared radiation.
Discuss the physical origins of the gross selection rules for microwave and infrared spectroscopy.
What is the Doppler-shifted wavelength of a red (660 nm) traffic light approached at 80 km h−1?
Estimate the lifetime of a state that gives rise to a line of width(a) 0.10 cm−1,(b) 1.0 cm−1.
Consider a diatomic molecule that is highly susceptible to centrifugal distortion in its ground vibrational state. Do you expect excitation to high rotational energy levels to change the equilibrium bond length of this molecule? Justify your answer.
The rotational constant of NH3 is equivalent to 298 GHz. Compute the separation of the pure rotational spectrum lines in GHz, cm−1, and mm, and show that the value of B is consistent with an N-H bond length of 101.4 pm and a bond angle of 106.78°.
Pure rotational Raman spectra of gaseous C6H6 and C6D6 yield the following rotational constants: B(C6H6) = 0.189 60 cm−1, B(C6D6) = 0.156 81 cm−1. The moments of inertia of the molecules about any axis perpendicular to the C6 axis were calculated from these data as I(C6H6) = 1.4759 × 10−45
Given that the spacing of lines in the microwave spectrum of 27Al1 H is constant at 12.604 cm−1, calculate the moment of inertia and bond length of the molecule (m(27Al) = 26.9815 u).
The rotational constant of 127I35Cl is 0.1142 cm−1. Calculate the ICl bond length (m(35Cl) = 34.9688 u, m(127I) = 126.9045 u).
Is the bond length in HCl the same as that in DCl? The wavenumbers of the J = 1 ← 0 rotational transitions for H35Cl and 2H35Cl are 20.8784 and 10.7840 cm−1, respectively. Accurate atomic masses are 1.007825 u and 2.0140 u for 1H and 2H, respectively. The mass of 35Cl is 34.96885 u. Based on
Determine the HC and CN bond lengths in HCN from the rotational constants B(1H12C14N) = 44.316 GHz and B(2 H12C14N) = 36.208 GHz.
The wavenumber of the incident radiation in a Raman spectrometer is 20 487 cm−1. What is the wavenumber of the scattered Stokes radiation for the J = 2 ← 0 transition of 14N2?
The rotational Raman spectrum of 35Cl2 (m(35Cl) = 34.9688 u) shows a series of Stokes lines separated by 0.9752 cm−1 and a similar series of antiStokes lines. Calculate the bond length of the molecule.
Which of the following molecules may show a pure rotational microwave absorption spectrum:(a) H2,(b) HCl,(c) CH4,(d) CH3Cl,(e) CH2Cl2?
Which of the following molecules may show a pure rotational Raman spectrum:(a) H2,(b) HCl,(c) CH4,(d) CH3Cl?
An object of mass 1.0 kg suspended from the end of a rubber band has a vibrational frequency of 2.0 Hz. Calculate the force constant of the rubber band.
At low resolution, the strongest absorption band in the infrared absorption spectrum of 12C16O is centred at 2150 cm−1. Upon closer examination at higher resolution, this band is observed to be split into two sets of closely spaced peaks, one on each side of the centre of the spectrum at 2143.26
Calculate the percentage difference in the fundamental vibration wavenumber of 23Na35Cl and 23Na37Cl on the assumption that their force constants are the same.
The wavenumber of the fundamental vibrational transition of 35Cl2 is 564.9 cm−1. Calculate the force constant of the bond (m(35Cl) = 34.9688 u).
The hydrogen halides have the following fundamental vibrational wavenumbers: 4141.3 cm−1 (HF); 2988.9 cm−1 (H35Cl); 2649.7 cm−1 (H81Br); 2309.5 cm−1 (H127I). Calculate the force constants of the hydrogen–halogen bonds.
For 16O2, ∆G values for the transitions v = 1 ← 0, 2 ← 0, and 3 ← 0 are, respectively, 1556.22, 3088.28, and 4596.21 cm−1. Calculate ∇ and xe. Assume ye to be zero.
The first five vibrational energy levels of HCl are at 1481.86, 4367.50, 7149.04, 9826.48, and 12 399.8 cm−1. Calculate the dissociation energy of the molecule in reciprocal centimetres and electronvolts.
Which of the following molecules may show infrared absorption spectra:(a) H2,(b) HCl,(c) CO2,(d) H2O?
How many normal modes of vibration are there for the following molecules:(a) H2O,(b) H2O2,(c) C2H4?
Which of the three vibrations of an AB2 molecule are infrared or Raman active when it is(a) Angular,(b) Linear?
Derive an expression for the value of J corresponding to the most highly populated rotational energy level of a diatomic rotor at a temperature T remembering that the degeneracy of each level is 2J + 1. Evaluate the expression for ICl (for which B = 0.1142 cm−1) at 25°C. Repeat the problem for
The molecule CH2Cl2 belongs to the point group C2v. The displacements of the atoms span 5A1 + 2A2 + 4B1 + 4B2. What are the symmetries of the normal modes of vibration?
Confirm that a Morse oscillator has a finite number of bound states, the states with V < hcDe. Determine the value of νmax for the highest bound state.
A. Dalgarno, in Chemistry in the interstellar medium, Frontiers of Astrophysics, E.H. Avrett (ed.), Harvard University Press, Cambridge (1976), notes that although both CH and CN spectra show up strongly in the interstellar medium in the constellation Ophiuchus, the CN spectrum has become the
Explain the origin of the term symbol 3Σg− for the ground state of dioxygen.
The term symbol for the ground state of N2+ is 2Σg+. What is the total spin and total orbital angular momentum of the molecule? Show that the term symbol agrees with the electron configuration that would be predicted using the building-up principle.
The molar absorption coefficient of a substance dissolved in hexane is known to be 855 dm3 mol−1 cm−1 at 270 nm. Calculate the percentage reduction in intensity when light of that wavelength passes through 2.5 mm of a solution of concentration 3.25 mmol dm−3.
The molar absorption coefficient of a solute at 540 nm is 286 dm3 mol−1 cm−1. When light of that wavelength passes through a 6.5 mm cell containing a solution of the solute, 46.5 per cent of the light is absorbed. What is the concentration of the solution?
Explain how colour can arise from molecules.
A solution of an unknown component of a biological sample when placed in an absorption cell of path length 1.00 cm transmits 20.1 per cent of light of 340 nm incident upon it. If the concentration of the component is 0.111 mmol dm−3, what is the molar absorption coefficient?
What is the evidence for the correctness of the mechanism of fluorescence?
The two compounds, 2,3-dimethyl-2-butene and 2,5-dimethyl-2,4- hexadiene, are to be distinguished by their ultraviolet absorption spectra. The maximum absorption in one compound occurs at 192 nm and in the other at 243 nm. Match the maxima to the compounds and justify the assignment.
What features of laser radiation are applied in chemistry? Discuss two applications of lasers in chemistry.
A 2.0-mm cell was filled with a solution of benzene in a non-absorbing solvent. The concentration of the benzene was 0.010 mol dm−3 and the wavelength of the radiation was 256 nm (where there is a maximum in the absorption). Calculate the molar absorption coefficient of benzene at this
The compound CH3CH=CHCHO has a strong absorption in the ultraviolet at 46 950 cm−1 and a weak absorption at 30 000 cm−1. Justify these features in terms of the structure of the molecule.
A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of sea water in the visible region is 6.2 × 10−3 dm3 mol−1 cm−1, calculate the depth at which a diver will experience(a) Half the surface intensity of light,(b) One
The electronic absorption bands of many molecules in solution have half-widths at half-height of about 5000 cm−1. Estimate the integrated absorption coefficients of bands for which(a) εmax ≈ 1 × 104 dm3 mol−1 cm−1,(b) εmax ≈ 5 × 102 dm3 mol−1 cm−1.
Sketch the form of an A3M2X4 spectrum, where A, M, and X are protons with distinctly different chemical shifts and JAM > JAX > JMX.
Which of the following molecules have sets of nuclei that are chemically but not magnetically equivalent?(a) CH3CH3,(b) CH2=CH2.
The duration of a 90° or 180° pulse depends on the strength of the B1 field. If a 90° pulse requires 10 µs, what is the strength of the B1 field? How long would the corresponding 180° pulse require?
What magnetic field would be required in order to use an EPR X–band spectrometer (9 GHz) to observe 1H-NMR and a 300 MHz spectrometer to observe EPR?
The centre of the EPR spectrum of atomic hydrogen lies at 329.12 mT in a spectrometer operating at 9.2231 GHz. What is the g-value of the electron in the atom?
A radical containing two inequivalent protons with hyperfine constants 2.0 mT and 2.6 mT gives a spectrum centred on 332.5 mT. At what fields do the hyperfine lines occur and what are their relative intensities?
Predict the intensity distribution in the hyperfine lines of the EPR spectra of(a) ·CH3,(b) ·CD3.
The benzene radical anion has g = 2.0025. At what field should you search for resonance in a spectrometer operating at(a) 9.302 GHz,(b) 33.67 GHz?
The EPR spectrum of a radical with a single magnetic nucleus is split into four lines of equal intensity. What is the nuclear spin of the nucleus?

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