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F. Luo, G.c. McBane, G. Kim, cr. Giese, and W.R. Gentry (J. Chem. Phys. 98, 3564 (1993)) reported experimental observation of the He2 complex, a species that had escaped detection for a long time. The fact that the observation required temperatures in the neighbourhood of 1mK is consistent with computational studies that suggest that hcl); for He, is about 1.51 x 10-3 J, hc Do about 2 x 10-'6 J, and Re about 297 pm.
(a) Estimate the fundamental vibrational wave number, force constant, moment of inertia, and rotational constant based on the harmonic oscillator and rigid-rotor approximations.
(b) Such a weakly bound complex is hardly likely to be rigid. Estimate the vibrational wave number and anharmonicity constant based on the Morse potential.
Consider the molecule CH3Cl.
(a) To what point group does the molecule belong?
(b) How many normal modes of vibration does the molecule have?
(c) What are the symmetries of the normal modes of vibration for this molecule?
(d) Which of the vibrational modes of this molecule are infrared active?
(e) Which of the vibrational modes of this molecule is Raman active?
Show that the moment of inertia of a diatomic molecule composed of atoms of masses mA and mB and bond length R is equal to meffR2, where meff= mAmB/(mA + mB)'
In the group theoretical language developed in Chapter 12, a spherical rotor is a molecule that belongs to a cubic or icosahedral point group, a symmetric rotor is a molecule with at least a threefold axis of symmetry, and an asymmetric rotor is a molecule without a threefold (or higher) axis. Linear molecules are linear rotors. Classify each of the following molecules as a spherical, symmetric, linear, or asymmetric rotor and justify your answers with group theoretical arguments:
(a) CH4,
(b) CH3CN,
(c) CO2
(d) CH3OH,
(e) Benzene,
(f') Pyridine
The protein haemerythrin is responsible for binding and carrying 0, in some invertebrates. Each protein molecule has two Fe'+ ions that are in very close proximity and work together to bind one molecule of 0, The Fe, O, group of oxygenated haemerythrin is coloured and has an electronic absorption band at 500 nm. The resonance Raman spectrum of oxygenated haemerythrin obtained with laser excitation at 500 nm has a band at 844 cm-I that has been attributed to the 0-0 stretching mode of bound 160,
(a) Why is resonance Raman spectroscopy and not infrared spectroscopy the method of choice for the study of the binding of 0, to haemerythrin? (b) Proof that the 844 cm-1 band arises from a bound 0, species may be obtained by conducting experiments on samples of haemerythrin that have been mixed with 180" instead of 160,. Predict the fundamental vibrational wave number of the 180_180 stretching mode in a sample of haemerythrin that has been treated with 180.
(c) The fundamental vibrational wave numbers for the 0-0 stretching modes of02, 02 (super oxide anion), and O~- (peroxide anion) are 1555, 1107, and 878 cm-1, respectively. Explain this trend in terms of the electronic structures of 02' 02' and O2/2. Hint: Review Section 11.4. What are the bond orders of 02' 02", and O2/2? (d) Based on the data given above, which of the following species best describes the Fe2O2 group of haemerythrin: Fe2/2 +O2' Fe2+ Fe3+02, or Fe 3/2 + O 2/2? Explain your reasoning.
(e) The resonance Raman spectrum of haemerythrin mixed with 160180 has two bands that can be attributed to the 0-0 stretching mode of bound oxygen. Discuss how this observation may be used to exclude one or more of the four proposed schemes (5-8) for binding of O, to the Fe2 site of haemerythrin.

The protein haemerythrin is responsible
The moments of inertia of the linear mercury (II) halides are very large, so the °and S branches of their vibrational Raman spectra show little rotational structure. Nevertheless, the peaks of both branches can be identified and have been used to measure the rotational constants of the molecules (R.J, H.
Clark and D.M. Rippon, f. Chem. Sac. Faraday Soc. II, 69, 1496 (1973)) Show, from a knowledge of the value of J corresponding to the intensity maximum, that the separation of the peaks of the °and S branches is given by the Placzek- Teller relation ∂v = (32BkT/ hc) 1/2 The following widths were obtained at the temperatures stated:
HgC12 HgBr2 HgI2
θ/oC 282 292 292
∂v/cm-1 23.8 15.2 11.4
In Problem 10.27, we saw that Doppler shifts of atomic spectral lines are used to estimate the speed of recession or approach of a star. From the discussion in Section 13.3a, it is easy to see that Doppler broadening of an atomic spectral line depends on the temperature of the star that emits the radiation. A spectral line of48Ti8+ (of mass 47.95 u) in a distant star was found to be shifted from 654.2 nm to 706.5 nm and to be broadened to 61.8 pm. What is the speed of recession and the surface temperature of the star?
There is a gaseous interstellar cloud in the constellation Ophiuchus that is illuminated from behind by the star ζ Ophiuci. Analysis of the electronic-vibrational-rotational absorption lines obtained by H.S. Uhler and RA Patterson (Astrophys], J. 42, 434 (1915)) shows the presence of CN molecules in the interstellar medium. A strong absorption line in the ultraviolet region at A= 387.5 nm was observed corresponding to the transition] = 0-1.
Unexpectedly, a second strong absorption line with 25 per cent of the intensity of the first was found at a slightly longer wavelength (∆λ = 0.061 nm) corresponding to the transition J = 1-1 (here allowed). Calculate the temperature of the CN molecules. Gerhard Hertzberg, who was later to receive the Nobel Prize for his contributions to spectroscopy, calculated the temperature as 2.3 K. Although puzzled by this result, he did not realize its full significance. If he had, his prize might have been for the discovery of the cosmic microwave background radiation.
The space immediately surrounding stars, also called the circumstellar space, is significantly warmer because stars are very intense black -body emitters with temperatures of several thousand kelvin. Discuss how such factors as cloud temperature, particle density, and particle velocity may affect the rotational spectrum of CO in an interstellar cloud. What new features in the spectrum of CO can be observed in gas ejected from and still near a star with temperatures of about 1000 K, relative to gas in a cloud with temperature of about 10 K? Explain how these features may be used to distinguish between circumstellar and interstellar material on the basis of the rotational spectrum of CO.
Repeat the calculation in Problem 11.4but plot the probability densities of the two orbitals. Then form the difference density, the difference between Ψ2 and ½| Ψ 2a + Ψ2b| Discuss.
Explain the origin of the term symbol 32:~for the ground state of Dioxygen.
How do the band heads in P and R branches arise? Could the Q branch show ahead?
Describe the mechanism of fluorescence. To what extent is a fluorescence spectrum not the exact mirror image of the corresponding absorption spectrum
Describe the principles of laser action, with actual examples.
The molar absorption coefficient of a substance dissolved in hexane is known to be 327 dm3 mol-1 cm-1 at 300 nm calculate the percentage reduction in intensity when light of that wavelength passes through 1.50 mm of a solution of concentration 2.22 mmol dm-3.
When light of wavelength 400 nm passes through 3.5 mm of a solution of an absorbing substance at a concentration 0.667 mmol dm-3, the transmission is 65.5 per cent. Calculate the molar absorption coefficient of the solute at this wavelength and express the answer in cm2 mol-1
The molar absorption coefficient of a solute at 440 nm is 323 dm2 mol-1 cm-1 when light of that wavelength passes through a 7.50 mm cell containing a solution of the solute 52.3 per cent of the light is absorbed what is the concentration of the solution?
The absorption associated with a certain transition begins at 199 nm, peaks sharply at 220 nm, and ends at 275 nm. The maximum value of the molar absorption coefficient is 2.25 x 104 dm3 mol-1 cm-1. Estimate the integrated absorption coefficient of the transition assuming an inverted parabolic line shape (Fig. 14.49; use eqn 13.5).
1, 3, 5-hexatriene (a kind of 'linear' benzene) was converted into benzene itself. On the basis of a free-electron molecular orbital model (in which hexatriene is treated as a linear box and benzene as a ring) would you expect the lowest energy absorption to rise or fall in energy?
The following data were obtained for the absorption by a dye dissolved in methylbenzene using a 2.50 mm cell. Calculate the molar absorption coefficient of the dye at the wavelength employed:
[dye]/mol dm-3) 0.0010 0.0050 0.0100 0.0500
TI(per cent) 73 21 4.2 1.33 x 10-5
A 2.50-mm cell was filled with a solution of a dye. The concentration of the dye was 15.5 mmol dm-3 Calculate the molar absorption coefficient of benzene at this wavelength given that the transmission was 32 per cent. What will the transmittance be in a 4.50-mm cell at the same wavelength?
Given that the maximum molar absorption coefficient of a molecule containing a carbonyl group is 30 dm3 mol-1 cm3 near 280 nm, calculate the thickness of a sample that will result in
(a) Half the initial intensity of radiation,
(b) One-tenth the initial intensity.
The electronic absorption band of a compound in solution had a Gaussian lineshape and a half-width at half-height of 4233 cm-1 and E max = 1.54 X 104 dm3 mol-1 cm-1. Estimate the integrated absorption coefficient.
The photo ionization of F2 by 21 eV photons produces Pr. would you expect the 2 f- 0 transition to be weaker or stronger than the 0 f- 0 transition? Justify your answer.
The vibrational wave number of the oxygen molecule in its electronic ground state is 1580 cm3, whereas that in the first excited state (B 3L), to which there is an allowed electronic transition, is 700 cm3, Given that the separation in energy between the minima in their respective potential energy curves of these two electronic states is 6.175 eV, what is the wave number of the lowest energy transition in the band of transitions originating from the v = 0 vibrational state of the electronic ground state to this excited state? Ignore any rotational structure or anharmonicity.
A lot of information about the energy levels and wave functions of small inorganic molecules can be obtained from their ultraviolet spectra. An example of a spectrum with considerable vibrational structure that of gaseous 502 at 25°C is shown in Fig. 14.6 Estimate the integrated absorption coefficient for the transition. What electronic states are accessible from the Al ground state of this Czv molecule by electric dipole transitions?
Aromatic hydrocarbons and 12 form complexes from which charge transfer electronic transitions are observed the hydrocarbon acts as an electron donor and 12 as an electron acceptor the energies hvmax of the charge-transfer transitions for a number of hydrocarbon-I, complexes are given below: Hydrocarbon benzene biphenyl naphthalene phenanthrene pyrene anthracene hvmax/eV 4.184 3.654 3.452 3.288 2.989 2.890 Investigate the hypothesis that there is a correlation between the energy of the HOMO of the hydrocarbon (from which the electron comes in the charge-transfer transition) and hvmax' Use one of the molecular electronic structure methods discussed in Chapter 11 to determine the energy of the HOMO of each hydrocarbon in the data set. 1

Consider some of the precautions that must be taken when conducting single-molecule spectroscopy experiments.

(a) What is the molar concentration of a solution in which there is, on average, one solute molecule in 1.0 m3 (1.0 £1) of solution?

(b) It is important to use pure solvents in single-molecule spectroscopy because optical signals from fluorescent impurities in the solvent may mask optical signals from the solute. Suppose that water containing a fluorescent impurity of molar mass 100 g mol-1 is used as solvent and that analysis indicates the presence of 0.10 mg of impurity per 1.0 kg of solvent. On average, how many impurity molecules will be present in 1.0 μm of solution? You may take the density of water as 1.0 g cm-3, Comment on the suitability of this solvent for single-molecule spectroscopy experiments.

Suppose that you are a colour chemist and had been asked to intensify the colour of a dye without changing the type of compound, and that the dye in question was a polyene. Would you choose to lengthen or to shorten the chain? Would the modification to the length shift the apparent colour of the dye towards the red or the blue?
Estimate the oscillator strength (see Problem 14.16) of a charge transfer transition modeled as the migration of an electron from an His orbital on one atom to another His orbital on an atom a distance R away. Approximate the transition moment by -eRS where 5 is the overlap integral of the two orbitals. Sketch the oscillator strength as a function of R using the curve for 5 given in Fig. 11.29. Why does the intensity fall to zero as R approaches 0 and infinity?
The fluorescence spectrum of anthracene vapour shows a series of peaks of increasing intensity with individual maxima at 440 nm, 410 nm, 390 nm, and 370 nm followed by a sharp cut-off at shorter wavelengths. The absorption spectrum rises sharply from zero to a maximum at 360 nm with a trail of peaks of lessening intensity at 345 nm, 330 nm, and 305 nm. Account for these observations.
Spin angular momentum is conserved when a molecule dissociates into atoms. What atom multiplicities are permitted when?
(a) An 02 molecule
(b) An N, molecule dissociates into atoms?
The flux of visible photons reaching Earth from the North Star is about 4 x 103 mm-2 S-I. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about 40 mm/ and the response time of the eye is about 0.1 s. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused on to the retina in 0.1 s? For a continuation of this story, see RW. Rodieck, The first steps in seeing, Sinauer, Sunderland (1998)

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 wavelengths of about 290 nm to 320 nm. The molar extinction coefficient of ozone over this range is given in the table below (W.B. DeMore, S.P. Sander, D.M. Golden, R.F. Hampson, M.J. Kurylo, e.J. Howard, A.R. Ravish Ankara, e.E. Kolb, and M.J. Molina, Chemical kinetics and photochemical data for use in stratospheric modeling: Evaluation Number 11, JPL Publication 94-26 (1994).

Compute the integrated absorption coefficient of ozone over the wavelength range 290-320 nm. (Hint £ (v) can be fitted to an exponential function quite well.)

G.C.G. Wachewsky, R. Horansky, and V. Vaida (J. Phys. Chem. 100, 11559 (1996)) examined the UV absorption spectrum ofCH3I, a species of interest in connection with stratospheric ozone chemistry. They found the integrated absorption coefficient to be dependent on temperature and pressure to an extent inconsistent with internal structural changes in isolated CH3I molecules; they explained the changes as due to dimerization of a substantial fraction of the CH), a process that would naturally be pressure and temperature dependent.
(a) Compute the integrated absorption coefficient over a triangular lineshape in the range 31250 to 34483 cm-1 and a maximal molar absorption coefficient of 150 dm3 mol-l cm3 at 31 250 cm-1.
(b) Suppose 1 per cent of the CH) units in a sample at 2.4 Torr and 373 K exist as dimers. Compute the absorbance expected at 31250 cm3 in a sample cell of length 12.0 cm.
(c) Suppose 18 per cent of the CH3I units in a sample at 100 Torr and 373 K exists as dimers. Compute the absorbance expected at 31250 cm-1 in a sample cell of length 12.0 cm; compute the molar absorption coefficient that would be inferred from this absorbance if dimerization was not considered.
One of the principal methods for obtaining the electronic spectra of unstable radicals is to study the spectra of comets, which are almost entirely due to radicals. Many radical spectra have been found in cornets, including that due to CN. These radicals are produced in comets by the absorption of far ultraviolet solar radiation by their parent compounds. Subsequently, their fluorescence is excited by sunlight of longer wavelength. The spectra of comet Hale-Bopp (C/1995 01) have been the subject of many recent studies. One such study is that of the fluorescence spectrum of CN in the cornet at large heliocentric distances by R.M. Wagner and D.G. Schleicher (Science 275, 1918 (1997)), in which the authors determine the spatial distribution and rate of production of CN in the coma. The (0-0) vibrational band is centred on 387.6 nm and the weaker (1- 1) band with relative intensity 0.1 is centred on 386.4 nm. The band heads for (0-0) and (O-I) are known to be 388.3 and 421.6 nm, respectively. From these data, calculate the energy of the excited 51 state relative to the ground So state, the vibrational wave numbers and the difference in the vibrational wave numbers of the two states, and the relative populations of the v = 0 and v = 1 vibrational levels of the SI state. Also estimate the effective temperature of the molecule in the excited 51 state. Only eight rotational levels of the SI state are thought to be populated. Is that observation consistent with the effective temperature of the 51 state?
Discuss in detail the origins of the local, neighbouring group, and solvent contributions to the shielding constant.
Suggest a reason why the relaxation times of BC nuclei are typically much longer than those of IH nuclei.
Discuss how the Fermi contact interaction and the polarization mechanism contribute to spin-spin couplings in NMR and hyperfine interactions in EPR.
Suppose a uniform disk-shaped organ is in a linear field gradient, and that the MRI signal is proportional to the number of protons in a slice of width Oxat each horizontal distance x from the centre of the disk. Sketch the shape of the absorption intensity for the MRI image of the disk before any computer manipulation has been carried out.
What is the resonance frequency of a proton in a magnetic field of 14.1 T?
33S has a nuclear spin of% and a nuclear g-factor of 0.4289 calculate the energies of the nuclear spin states in a magnetic field of7.500 T.
Calculate the frequency separation of the nuclear spin levels of a 14N nucleus in a magnetic field of 15.4 T given that the magnetogyricrati ratio is 1.93 X 107 T-1 s-l
In which of the following systems is the energy level separation the largest?
(a) A 14Nnucleus in (for protons) a 600 MHz NMR spectrometer,
(b) An electron in a radical in a field of 0.300 T
Calculate the magnetic field needed to satisfy the resonance condition for unshielded protons in a 150.0 MHz radiofrequency field.
Use Table 15.2 to predict the magnetic fields at which
(a) 14N,
(b) 19F, and
(c) 31p comes into resonance at (i) 300 MHz, (ii) 750 MHz.
Calculate the relative population differences (oN/N) for 13Cnuclei in fields of
(a) 0.50 T,
(b) 2.5 T, and
(c) 15.5 T at 25°C.
What are the relative values of the chemical shifts observed for nuclei in the spectrometers mentioned in Exercise 15.8a in terms of?
(a) Values,
(b) Frequencies?
The chemical shift of the CH3 protons in diethyl ether is, = 1.16 and that of the CH2 protons is 3.36. What is the difference in local magnetic field between the two regions of the molecule when the applied field is
(a) 1.9 T,
(b) 16.5 T?
Sketch the appearance of the IH-NMR spectrum of diethyl ether using J = 6.97 Hz and the data in Exercise 15.9b in a spectrometer operating at
(a) 350 MHz,
(b) 650 MHz.
Two groups of protons are made equivalent by the isomerization of a fluxional molecule. At low temperatures, where the interconversion is slow, one group has 0=5.5 and the other has 0=6.8. At what rate of interconversion will the two signals merge in a spectra meter operating at 350MHz?
From the data in Table 15.2, predict the frequency needed for 31p_ NMR in an NMR spectrometer designed to observe proton resonance at 500 MHz. Sketch the proton and 31presonances in the NMR spectrum of PH;.
Sketch the form of an A2M2XS spectrum, where A, M, and X are protons with distinctly different chemical shifts and JhM > JAX > JMX'
Which of the following molecules have sets of nuclei that are chemically but not magnetically equivalent?
(a) CHZ=C=CP1,
(b) cis- and trans-[Mo (CO) 4(PHJ) 2]
The duration of a 90° or 180° pulse depends on the strength of the 'El field. If a 180° pulse requires 12.5 us, what is the strength of the 'El field? How long would the corresponding 90° pulse require?
Some commercial EPR spectrometers use 8 mm microwave radiation (the Q band). What magnetic field is needed to satisfy the resonance condition?
The centre of the EPR spectrum of atomic deuterium lies at 330.02 mT in a spectrometer operating at 9.2482 GHz. What is the g-value of the electron in the atom?
A radical containing three equivalent protons shows a four-line spectrum with an intensity distribution 1:3:3:1. The lines occur at 331.4 mT, 333.6 mT, 335.8 mT, and 338.0 mT. What is the hyperfine coupling constant for each proton? What is the g-value of the radical given that the spectrometer is operating at 9.332 GHz?
A radical containing three in equivalent protons with hyperfine constants 2.11 mT, 2.87 m'T and 2.89 mT gives a spectrum centred on 332.8 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) ·CH1H3,
(b) ·CD2CD3.
The naphthalene radical anion has g = 2.0024. At what field should you search for resonance in a spectrometer operating at?
(a) 9.312 GHz,
(b) 33.88 GHz?
The EPR spectrum of a radical with two equivalent nuclei of a particular kind is split into five lines of intensity ratio 1:2:3:2: 1. What is the spin of the nuclei?
Sketch the form of the hyperfine structures of radicals XHJ and XDJ, where the nucleus X has I =3/2
A scientist investigates the possibility of neutron spin resonance, and has available a commercial NMR spectrometer operating at 300 MHz. What field is required for resonance? What is the relative population difference at room temperature? Which is the lower energy spin state of the neutron?
Suppose that the PID in Pig. 15.31 was recorded in a 300 MHz spectrometer, and that the interval between maxima in the oscillations in the PID is 0.10 s. What is the Larmor frequency of the nuclei and the spin-spin relaxation time?
Various versions of the Karplus equation (eqn 15.27) have been used to correlate data on vicinal proton coupling constants in systems of the type RjR2CHCHR3R4. The original version, (M. Karplus, f. Am. Chem. Soc. 85, 2870 (1963», is 3JHH=A cos'1>HH+ B. When R3= R4 = H, 3JHH= 7.3 Hz; when R3 = CH3 and R4 = H, 3JHH= 8.0 Hz; when R3= R4 =CH3, 3JHH = 11.2 Hz. Assume that only staggered conformations are important and determine which version of the Karplus equation fits the data better.
Figure 15.64 shows the proton COSY spectrum of l-nitro propane. Account for the appearance of off-diagonal peaks in the spectrum

The hyperfine coupling constant in ·CH3 is 2.3 mT. Use the information in Table 15.3 to predict the splitting between the hyperfine lines of the spectrum of ·CD what are the overall widths of the hyperfine spectra in each case?

When an electron occupies a 2s orbital on an N atom it has a hyper fine interaction of 55.2 mT with the nucleus. The spectrum of N02 shows an isotropic hyper fine interaction of 5.7 mT. For what proportion of its time is the unpaired electron of NO, occupying a 25 orbital? The hyper fine coupling constant for an electron in a 2p orbital of an N atom is 3.4 mT. In NO, the anisotropic part of the hyper fine coupling is 1.3 mT. What proportion of its time does the unpaired electron spend in the 2p orbital of the N atom in N0z? What is the total probability that the electron will be found on

(a) The N atoms,

(b) The °atoms' what is the hybridization ratio of the N atom? Does the hybridization support the view that NO, is angular?

When an electron occupies a 2s orbital on an N atom it has a hyperfine interaction of 55.2 mT with the nucleus. The spectrum ofN02 shows an isotropic hyperfine interaction of 5.7 mT. For what proportion of its time is the unpaired electron of NO, occupying a 25 orbital? The hyperfine coupling constant for an electron in a 2p orbital of an N atom is 3.4 mT. In NO, the anisotropic part of the hyperfine coupling is 1.3 mT. What proportion of its time does the unpaired electron spend in the 2p orbital of the N atom in N0z? What is the total probability that the electron will be found on
(a) The N atoms,
(b) The °atoms' what is the hybridization ratio of the N atom? Does the hybridization support the view that NO, is angular?
The z-component of the magnetic field at a distance R from a magnetic moment parallel to the z-axis is given by eqn 15.28. In a solid, a proton at a distance R from another can experience such a field and the measurement of the splitting it causes in the spectrum can be used to calculate R. In gypsum, for instance, the splitting in the H20 resonance can be interpreted in terms of a magnetic field of 0.715 mT generated by one proton and experienced by the other. What is the separation of the protons in the H20 molecule?
The shape of a spectral line, J(w), is related to the free induction decay signal G(t) by where a is a constant and 'Re' means take the real part of what follows. Calculate the lineshape corresponding to an oscillating, decaying function G(t) = cos wot e-t/r

EPR spectra are commonly discussed in terms of the parameters that occur in the spin-Hamiltonian, a Hamiltonian operator that incorporates various effects involving spatial operators (like the orbital angular momentum) into operators that depend on the spin alone. Show that, if you use H = -ge Ye Bo sz - Ye' Bo lz as the true Hamiltonian, then from second-order perturbation theory (and specifically eqn 9.65), the eigen values of the spin are the same as those of the spin-Hamiltonian H spin = -gYe'130sz (note the g in place of gel and find an expression for g.
When interacting with a large biopolymer or even larger organelle, a small molecule might not rotate freely in all directions and the dipolar interaction might not average to zero. Suppose a molecule is bound so that, although the vector separating two protons may rotate freely around the zaxis, the colatitudes may vary only between 0 and 8'. Average the dipolar field over this restricted range of orientations and confirm that the average vanishes when 8' =11: (corresponding to rotation over an entire sphere). What is the average value of the local dipolar field for the H20 molecule in Problem 15.15 if it is bound to a biopolymer that enables it to rotate up to 8' = 30°?
Sketch the EPR spectra of the di-tert-butyl nitroxide radical (10) at 292 K in the limits of very low concentration (at which electron exchange is negligible), moderate concentration (at which electron exchange effects begin to be observed), and high concentration (at which electron exchange effects predominate). Discuss how the observation of electron exchange between nitroxide spin probes can inform the study of lateral mobility of lipids in a biological membrane.
Describe the physical significance of the partition function.
Enumerate the ways by which the parameter f3 may be identified with 1/kT.
Explain what is meant by an ensemble and why it is useful in statistical thermodynamics.
What is the temperature of a two-level system of energy separation equivalent to 300 cm-I when the population of the upper state is one-half that of the lower state?
Calculate the ratio of the translational partition functions of xenon and helium at the same temperature and volume.
A certain atom has a doubly degenerate ground level, a triply degenerate electronically excited level at 1250 cm3, and a doubly degenerate level at 1300 cm3. Calculate the partition function of these electronic states at 2000 K.
Calculate the electronic contribution to the molar internal energy at 2000 K for a sample composed of the atoms specified in Exercise 16.4b.
A certain molecule has a doubly degenerate excited state lying at 360 cm3 above the non-degenerate ground state. At what temperature will is per cent of the molecules be in the upper state?
A nitrogen nucleus spin can adopt any of three orientations in a magnetic field, and its energies are 0, ±YNn'13, where YN is the magnetogyricrati ratio of the nucleus. Deduce an expression for the partition function and mean energy of the nucleus and sketch the variation of the functions with 13. Calculate the relative populations of the spin states at
(a) 1.0 K,
(b) 298 K when13=20.0 T.
Consider a system of distinguishable particles having only three no degenerate energy levels separated by an energy which is equal to the value of kTat 25.0 K. Calculate
(a) The ratio of populations in the states at (1) 1.00 K, (2) 25.0 K, and (3) 100 K,
(b) The molecular partition function at 25.0 K,
(c) The molar energy at 25.0 K,
(d) The molar heat capacity at 25.0 K,
(e) The molar entropy at 25.0 K.
At what temperature would the population of the first excited rotational level ofHCl are lie times its population of the ground state?
Calculate the standard molar entropy of xenon gas at
(a) 100 K,
(b) 298.15 K.
Calculate the vibrational contribution to the entropy of Br, at 600 K given that the wave number of the vibration is 321 cm3,
Identify the systems for which it is essential to include a factor of 1/N on going from Q to q:
(a) A sample of carbon dioxide gas,
(b) A sample of graphite,
(c) A sample of diamond,
(d) Ice.
Consider a system A consisting of subsystems Al and A2, for which WI = 1 X 1020 and W2 = 2 X 1020 what is the number of configurations available to the combined system? Also, compute the entropies 5, Sp and 52 what is the significance of this result?
By what factor does the number of available configurations increase when 100 J of energy is added to a system containing 1.00 mol of particles at constant volume at 298 K?
Explore the conditions under which the 'integral' approximation for the translational partition function is not valid by considering the translational partition function of an Ar atom in a cubic box of side 1.00 cm. Estimate the temperature at which, according to the integral approximation, q = 10 and evaluate the exact partition function at that temperature.
(a) Calculate the electronic partition function of a tellurium atom at (i) 298 K, (ii) 5000 K by direct summation using the following data:
(b) What proportion of the Te atoms are in the ground term and in the term labelled 2 at the two temperatures?
(c) Calculate the electronic contribution to the standard molar entropy of gaseous Te atoms.

The No molecule has a doubly degenerate excited electronic level 121.1 cm-1 above the doubly degenerate electronic ground term. Calculate and plot the electronic partition function of NO from T= 0 to 1000 K. Evaluate
(a) The term populations and
(b) The electronic contribution to the molar internal energy at 300 K. Calculate the electronic contribution to the molar entropy of the NO molecule at 300 K and 500 K.
Calculate, by explicit summation, the vibrational partition function and the vibrational contribution to the molar internal energy of is molecules at
(a) 100 K,
(b) 298 K given that its vibrational energy levels lie at the following wave numbers above the zero-point energy level: 0,213.30,425.39, 636.27,845.93 cm3. What proportion of, molecules are in the ground and first two excited levels at the two temperatures? Calculate the vibrational contribution to the molar entropy of 12 at the two temperatures.
A sample consisting of five molecules has a total energy SE. Each molecule is able to occupy states of energy jE, with j = 0, 1,2, ....
(a) Calculate the weight of the configuration in which the molecules are distributed evenly over the available states.
(b) Draw up a table with columns headed by the energy of the states and write beneath them all configurations that are consistent with the total energy. Calculate the weights of each configuration and identify the most probable configurations.
The most probable configuration is characterized by a parameter we know as the 'temperature'. The temperatures of the system specified in Problems 16.13 and 16.14 must be such as to give a mean value of E forth energy of each molecule and a total energy NE for the system.
(a) Show that the temperature can be obtained by plotting against j, where Pi is the (most probable) fraction of molecules in the state with energy i.e. Apply the procedure to the system in Problem 16.14. What is the temperature of the system when E corresponds to 50 cm-3?
(b) Choose configurations other than the most probable, and show that the same procedure gives a worse straight line, indicating that a temperature is not well-defined for them.
Consider a system with energy levels Ej =jE and N molecules.
(a) Show that if the mean energy per molecule is ae, then the temperature is given by
Evaluate the temperature for a system in which the mean energy is E, taking E equivalent to 50 cm-I
(b) Calculate the molecular partition function q for the system when its mean energy is ae.
(c) Show that the entropy of the system is Slk= (1 +a) In(1 + a) - a In a and evaluate this expression for a mean energy E.

Here you will use the zipper model discussed in Impact I16.1 to explore the helix-coil transition in polypeptides.

(a) Investigate the effect of the parameter s on the distribution of random coil segments in a polypeptide with n = 20 by plotting p., the fraction of molecules with a number of amino acids in a coil region, against i for s = 0.8, 1.0, and 1.5, with O'= 5.0 x 10-2. Discuss the significance of any effects you discover.

(b) The average value of i given by (i) = Ijipj. Use the results of the zipper model to calculate (i) for all the combinations of sand oused in Fig. 16.10 and part (a).

For gases, the canonical partition function, Q, is related to the molecular partition function q by Q = q NIN!. Use the expression for q and general thermodynamic relations to derive the perfect gas law P V = I1RT.
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