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Physical Chemistry Thermodynamics And Kinetics 10th Edition Peter Atkins, Julio De Paula - Solutions

Distinguish between kinetic and thermodynamic control of a reaction. Suggest criteria for expecting one rather than the other.
An amino acid on the surface of an enzyme was labelled covalently with 1.5-I AEDANS and it is known that the active site contains a tryptophan residue. The fluorescence quantum yield of tryptophan decreased by 15 per cent due to quenching by 1.5-I AEDANS. What is the distance between the active
The Förster theory of resonance energy transfer and the basis for the FRET technique can be tested by performing fluorescence measurements on a series of compounds in which an energy donor and an energy acceptor are covalently linked by a rigid molecular linker of variable and known length. L.
Polypeptides are polymers of amino acids. Suppose that a long polypeptide chain can undergo a transition from a helical conformation to a random coil. Consider a mechanism for a helix–coil transition that begins in the middle of the chain: In which h and c label, respectively, an amino acid
Conventional equilibrium considerations do not apply when a reaction is being driven by light absorption. Thus the steady-state concentration of products and reactants might differ significantly from equilibrium values. For instance, suppose the reaction A→B is driven by light absorption, and
The composition of a liquid phase reaction 2 A→B was followed by a spectrophotometric method with the following results: Determine the order of the reaction and its rate constant. t/min [B]/(mol dm-³) 0 0 10 0.089 20 30 0.153 0.200 40 0.230 w 0.312
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)) report that the reaction 2 HSO3−(aq)+O2(g)→2 SO42− (aq)+2 H+(aq) follows the rate law v=kr
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 quenched by nearby chlorophyll molecules. Given that for a pair of chlorophyll a molecules
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 in an nth-order reaction.Data in eqn 20B.6 t1/2 = 2-1-1 (n-1)k, [A]-¹ nth-order
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 population difference at room temperature? Which is the lower energy spin state of the neutron?
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 the spin–spin relaxation time?Data in Fig. 14C.5 Signal AAA Time, t
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 does this frequency compare to typical molecular rotational, vibrational, and electronic
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, CN, and NO2, respectively.(a) Use the computational method of your choice to calculate the net charge
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 magnetic field would be required in order to use an EPR X-band spectrometer (9GHz) to observe 1H-NMR and
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 makes with the z-axis?
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 the data in Table 14A.2, calculate these sensitivities for the deuteron, 13C, 14N, 19F,
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 curves that show how they vary as the magnetic field of the spectrometer is changed.
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 for each proton? What is the g value of the radical given that the spectrometer is operating at
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 parallel to the OO direction the centre of the spectrum lies at 333.64mT in a spectrometer operating at
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, each corresponding to a different nucleus, is given by where, for each nucleus j, S0j is the
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 varies linearly across the specimen. Consider a flask of water held in a field that varies in the
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 hyperfine spectra in each case?Data in Table 14D.1 Nuclide 1Η 2H 14N 19F Isotropic
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 resonance frequency, S0 is a constant, and T2 is the spin–spin relaxation time. Confirm that for
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. Bu nuc Yићно 4ntR -(1-3cos²0)m (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 kT/ε.(b) Plot the function dS/dT for a twolevel system, the temperature coefficient of its entropy,
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 Inx!=(x+1)Inx-x+In 2 X>1 Stirling's approxi- mation (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 2P3/2 and a 2P1/2 level lies 881 cm−1 above it. Calculate the electronic contribution to the heat
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 value? ∼B(CH4)=0.0527 cm-1.
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 greatest weight and compare it to the distribution predicted by the Boltzmann expression. Explore what
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 °C.
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 value? ∼A=5.097cm-1. and ∼B =0.443cm−1.
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〉 – 〈ε〉2)1/2. Evaluate the resulting expression for a harmonic oscillator.
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 cent of the atoms were in the upper level, and the translational temperature of the beam was 300K.
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 Ge are as follows:Calculate the electronic partition function at 298K and 1000K by direct
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 the value within 5 per cent of the value calculated from the approximate formula?
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 more accurate approximation have a significant effect on the form of the Boltzmann distribution?
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 function of temperature. At what temperature is the value within 5 per cent of the value calculated from
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 value? Use the wavenumbers 658 cm−1 (symmetric stretch), 397 cm−1 (bend; two modes), 1535 cm−1
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 molar entropy of gaseous water at 25 °C.
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 air and T the average temperature. Obtain the barometric formula from the Boltzmann distribution.
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 layer of the Sun that is about 190 km thick.(a) Show that this partition function, which involves a
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 degenerate ground level, a nondegenerate electronically excited level at 2500 cm−1, and a twofold
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 is the value within 5 per cent of the value calculated from the approximate formula?
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 plot the contribution as a function of temperature.Data in Problem 15B.6The pure rotational
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. Proceeding with this crude model gives us insight into the contribution of denaturation to the heat
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 (deformation, T).
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 replaced by the variable β =1/kT. Account for the appearance of the graphs physically.
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 film.
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 m-¹) 0 0.10 72.8 70.2 0.20 0.30 67.7 65.1 0.40 62.8 0.50 59.8

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