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physical chemistry

Questions and Answers of Physical Chemistry

The oxidation of HSO3− by O2 in aqueous solution is a reaction of importance to the processes of acid rain formation and flue gas desulfurization. R.E. Connick et al. (Inorg. Chem. 34, 4543 (1995))
The first step in plant photosynthesis is absorption of light by chlorophyll molecules bound to proteins known as ‘light-harvesting complexes’, where the fluorescence of a chlorophyll molecule is
Show that t1/2 is given by eqn 20B.6 for a reaction that is nth order in A. Then deduce an expression for the time it takes for the concentration of a substance to fall to one-third the initial value
To determine the structures of macromolecules by NMR spectroscopy, chemists use spectrometers that operate at the highest available fields and frequencies. Justify this choice.
Given that g is a dimensionless number, what are the units of γN expressed in tesla and hertz?
A scientist investigates the possibility of neutron spin resonance, and has available a commercial NMR spectrometer operating at 300MHz. What field is required for resonance? What is the relative
Suppose that the FID in Fig. 14C.5 was recorded in a 400MHz spectrometer, and that the interval between maxima in the oscillations in the FID is 0.12s. What is the Larmor frequency of the nuclei and
Describe how the Fermi contact interaction and the polarization mechanism contribute to hyperfine interactions in EPR.
It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of 1.0kT? How
Describe the significance of the chemical shift in relation to the terms ‘high-field’ and ‘low-field’.
Consider the following series of molecules: benzene, methylbenzene, trifluoromethylbenzene, benzonitrile, and nitrobenzene in which the substituents para to the C atom of interest are H, CH3, CF3,
What are the relative values of the chemical shifts observed for nuclei in the spectrometers mentioned in Exercise 14A.9a in terms of (i) δ values, (ii) frequencies?Data in Exercise 14A.9aWhat
Compare the effects of magnetic fields on the energies of nuclei and the energies of electrons.
For a proton, what are the magnitude of the spin angular momentum and its allowed components along the z-axis? What are the possible orientations of the angular momentum in terms of the angle it
The relative sensitivity of NMR lines for equal numbers of different nuclei at constant temperature for a given frequency is Rν∝(I+1)μ3 whereas for a given field it is RB∝{(I+1)/I2}μ3.(a) From
What is the effective transverse relaxation time when the width of a resonance line is 1.5Hz?
Use mathematical software to construct the FID curve for a set of three nuclei with resonances at δ =3.2, 4.1, and 5.0 in a spectrometer operating at 800MHz. Suppose that T1=1.0s. Go on to plot FID
Explain how the EPR spectrum of an organic radical can be used to identify and map the molecular orbital occupied by the unpaired electron.
A radical containing two equivalent protons shows a three-line spectrum with an intensity distribution 1:2:1. The lines occur at 330.2mT, 332.5mT, and 334.8mT. What is the hyperfine coupling constant
The angular NO2 molecule has a single unpaired electron and can be trapped in a solid matrix or prepared inside a nitrite crystal by radiation damage of NO2− ions. When the applied field is
What is the Larmor frequency? What is its role in magnetic resonance?
To gain some appreciation for the numerical work done by computers interfaced to NMR spectrometers, perform the following calculations.(a) The total FID F(t) of a signal containing many frequencies,
Predict the maximum enhancement (as the value of η) that could be obtained in a NOE observation in which 31P is coupled to protons.
With special techniques, known collectively as magnetic resonance imaging (MRI), it is possible to obtain NMR spectra of entire organisms. A key to MRI is the application of a magnetic field that
The hyperfine coupling constant in ·CH3 is 2.3mT. Use the information in Table 14D.1 to predict the splitting between the hyperfine lines of the spectrum of ·CD3. What are the overall widths of the
Explain why groups of equivalent protons do not exhibit the spin–spin coupling that exists between them.
Discuss the origin of the nuclear Overhauser effect and how it can be used to measure distances between protons in a biopolymer.
14N has a nuclear spin of 1 and a nuclear g-factor of 0.404. Calculate the energies of the nuclear spin states in a magnetic field of 10.50T.
(a) In many instances it is possible to approximate the NMR lineshape by using a Lorentzian function of the formwhere I(ω) is the intensity as a function of the angular frequency ω=2πν, ω0 is
The proton chemical shifts for the NH, CαH, and CβH groups of alanine are 8.25ppm, 4.35ppm, and 1.39ppm, respectively. Sketch the COSY spectrum of alanine between 1.00 and 8.50ppm.
Explain the difference between magnetically equivalent and chemically equivalent nuclei, and give two examples of each.
Sketch the form of the 19F-NMR spectra of a natural sample of 10BF4− and 11BF4−.
The benzene radical anion has g=2.0025. At what field should you search for resonance in a spectrometer operating at (i) 9.313GHz, (ii) 33.80GHz?
Show that the coupling constant as expressed by the Karplus equation passes through a minimum when cos ϕ=B/4C.
In a liquid, the dipolar magnetic field averages to zero: show this result by evaluating the average of the field given in eqn 14B.15. Data in eqn 14B.15.
Construct a version of Pascal’s triangle to show the fine structure that might arise from spin–spin coupling to a group of four spin- 3/2 nuclei.
What are the relative population differences of 19F spin states in spectrometers operating at 60MHz and 450MHz at 25 °C?
A proton jumps between two sites with δ=2.7 and δ=4.8. At what rate of interconversion will the two signals collapse to a single line in a spectrometer operating at 550MHz?
Use mathematical software, a spreadsheet, or the Living graphs on the web site for this book to:(a) Consider a three-level system with levels 0, ε, and 2ε. Plot the partition function against
Calculate (i) the thermal wavelength, (ii) the translational partition function at 300K and 3000K of a molecule of molar mass 150 gmol−1 in a container of volume 1.00 cm3.
Identify the conditions under which energies predicted from the equipartition theorem coincide with energies computed by using partition functions.
Compute the mean energy at 298K of a two-level system of energy separation equivalent to 500 cm−1.
Why is the concept of a canonical ensemble required?
Describe the molecular features that determine the magnitudes of the constant-volume molar heat capacity of a molecular substance.
Discuss the relationship between ‘population’, ‘configuration’, and ‘weight’. What is the significance of the most probable configuration?
Calculate the weight of the configuration in which 16 objects are distributed in the arrangement 0, 1, 2, 3, 8, 0, 0, 0, 0, 2.
Suggest a physical interpretation of the relation between pressure and the partition function.
Describe how the mean energy of a system composed of two levels varies with temperature.
Calculate the ratio of the translational partition functions of Ar and Ne at the same temperature and volume.
Evaluate 8! by using (i) the exact formula, (ii) Stirling’s approximation, eqn 15A.2b; (iii) the more accurate version of Stirling’s approximation, eqn 15A.2a.Data in eqn 15A.2a
Discuss and illustrate the proposition that 1/T is a more natural measurement of temperature than T itself.
Suggest a physical interpretation of the relation between equilibrium constant and the partition functions of the reactants and products in a reaction.
Use the information in Exercise 15E.3(a) to calculate the electronic contribution to the molar Gibbs energy of Cl atoms at (i) 500K and (ii) 900K.Data in Exercise 15E.3(a)The ground level of Cl is
What is the difference between a ‘state’ and an ‘energy level’? Why is it important to make this distinction?
The bond length of O2 is 120.75pm. Use the high-temperature approximation to calculate the rotational partition function of the molecule at 300K.
Evaluate, by explicit summation, the mean rotational energy of CH4 and plot its value as a function of temperature. At what temperature is the equipartition value within 5 per cent of the accurate
Under what circumstances may identical particles be regarded as distinguishable?
The first electronically excited state of O2 is 1Δg and lies 7918.1 cm−1 above the ground state, which is 3∑g− . Calculate the electronic contribution to the heat capacity of O2 at 400K.
What is temperature?
Use mathematical software to evaluate W for N=20 for a series of distributions over a uniform ladder of energy levels, ensuring that the total energy is constant. Identify the configuration of
How does a statistical analysis of the equilibrium constant account for the latter’s temperature dependence?
Why and when is it necessary to include a symmetry number in the calculation of a partition function?
The H2O molecule is an asymmetric rotor with rotational constants 27.877 cm−1, 14.512 cm−1, and 9.285 cm−1. Calculate the rotational partition function of the molecule at (i) 25 °C, (ii) 100
Evaluate, by explicit summation, the mean rotational energy of CH3Cl and plot its value as a function of temperature. At what temperature is the equipartition value within 5 per cent of the accurate
Deduce an expression for the root mean square energy, 〈ε2〉1/2, in terms of the partition function and hence an expression for the root mean square deviation from the mean, Δε=(〈ε2〉 –
What is meant by the ‘thermodynamic limit’?
Justify the differences between the partition-function expression for the entropy for distinguishable particles and the expression for indistinguishable particles.
Summarize the role of the Boltzmann distribution in chemistry.
What is the temperature of a two-level system of energy separation equivalent to 400 cm−1 when the population of the upper state is one-third that of the lower state?
A certain atom has a doubly degenerate ground state and an upper level of four degenerate states at 450 cm−1 above the ground state. In an atomic beam study of the atoms it was observed that 30 per
J. Sugar and A. Musgrove (J. Phys. Chem. Ref. Data 22, 1213 (1993)) have published tables of energy levels for germanium atoms and cations from Ge+ to Ge+31. The lowest-lying energy levels in neutral
The rotational constant of CO is 1.931 cm−1. Evaluate the rotational partition function explicitly (without approximation) and plot its value as a function of temperature. At what temperature is
Account for the temperature and volume dependence of the entropy of a perfect gas in terms of the Boltzmann distribution.
Calculate the standard molar entropy at 298K of (i) gaseous helium, (ii) gaseous xenon.
Explore the consequences of using the full version of Stirling’s approximation, x!≈(2π)1/2xx+1/2e−x, in the development of the expression for the configuration of greatest weight. Does the
The rotational constant of CH4 is 5.241 cm−1. Evaluate the rotational partition function explicitly (without approximation but ignoring the role of nuclear statistics) and plot its value as a
Evaluate, by explicit summation, the mean vibrational energy of CS2 and plot its value as a function of temperature. At what temperature is the equipartition value within 5 per cent of the accurate
At what temperature is the standard molar entropy of helium equal to that of xenon at 298K?
Calculate the rotational partition function of H2O at 298K from its rotational constants 27.878 cm−1, 14.509 cm−1, and 9.287 cm−1 and use your result to calculate the rotational contribution to
The variation of the atmospheric pressure p with altitude h is predicted by the barometric formula to be p=p0e−h/H where p0 is the pressure at sea level and H=RT/Mg with M the average molar mass of
Give the symmetry number for each of the following molecules: (i) CO, (ii) O2, (iii) H2S, (iv) SiH4, and (v) CHCl3.
Consider the electronic partition function of a perfect atomic hydrogen gas at a density of 1.99×10−4 kgm−3 and 5780K. These are the mean conditions within the Sun’s photosphere, the surface
Calculate the mean contribution to the electronic energy at 1900K for a sample composed of the atoms specified in Exercise 15B.13(a).Data in Exercise 15B.13(a).A certain atom has a fourfold
Predict the standard molar entropy of methanoic acid (formic acid, HCOOH) at (i) 298K, (ii) 500K. The normal modes occur at wavenumbers 3570, 2943, 1770, 1387, 1229, 1105, 625, 1033, 638 cm−1.
The vibrational wavenumber of Br2 is 323.2 cm−1. Evaluate the vibrational partition function explicitly (without approximation) and plot its value as a function of temperature. At what temperature
Use the accurate expression for the rotational partition function calculated in Problem 15B.6 for HCl(g) to calculate the rotational contribution to the molar entropy over a range of temperature and
Calculate the vibrational partition function of CS2 at 500K given the wavenumbers 658 cm−1 (symmetric stretch), 397 cm−1 (bend; two modes), 1535 cm−1 (asymmetric stretch).
It is possible to write an approximate expression for the partition function of a protein molecule by including contributions from only two states: the native and denatured forms of the polymer.
Calculate the vibrational partition function of CCl4 at 500K given the wavenumbers 459 cm−1 (symmetric stretch, A), 217 cm−1 (deformation, E), 776 cm−1 (deformation, T), 314 cm−1
Explore how the entropy of a collection of two-level systems behaves when the temperature is formally allowed to become negative. You should also construct a graph in which the temperature is
Derive the Sackur–Tetrode equation for a monatomic gas confined to a two-dimensional surface, and hence derive an expression for the standard molar entropy of condensation to form a mobile surface
Which of the following molecules may be polar: CIF3, O3, H2O2?
Calculate the molar energy required to reverse the direction of an H2O molecule located 100pm from a Li+ ion. Take the magnitude of the dipole moment of water as 1.85 D.
The surface tensions of a series of aqueous solutions of a surfactant A were measured at 20 °C, with the following results: Calculate the surface excess concentration. [A]/(mol dm³) 7/(mN
Describe the process of condensation.
Plot the magnitude of the electric dipole moment of hydrogen peroxide as the HeOeOeH (azimuthal) angle φ changes from 0 to 2π. Use the dimensions shown in 1. H 97 pm O 149 pm 1
Acetic acid vapour contains a proportion of planar, hydrogen bonded dimers (2). The apparent dipole moment of molecules in pure gaseous acetic acid has a magnitude that increases with increasing
Draw examples of the arrangements of electrical charges that correspond to monopoles, dipoles, quadrupoles, and octupoles. Suggest a reason for the different distance dependencies of their electric
Phenylalanine (Phe, 4) is a naturally occurring amino acid. What is the energy of interaction between its phenyl group and the electric dipole moment of a neighbouring peptide group? Take the
Estimate the energy of the dispersion interaction (use the London formula) for two He atoms separated by 1.0nm. Relevant data can be found in the Resource section.

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