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

Suggest a reason why the spin–lattice relaxation time of benzene (a small molecule) in a mobile, deuterated hydrocarbon solvent increases with temperature whereas that of an oligonucleotide (a large molecule) decreases.
Sketch the form of the hyperfine structures of radicals XH2 and XD2, where the nucleus X has I = 5/2.
Calculate the percentage change in a given energy level of a particle in a cubic box when the length of the edge of the cube is decreased by 10 per cent in each direction.
Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 2.33 × 10−26 kg and force constant 155 N m−1.
For a harmonic oscillator of effective mass 1.33 × 10−25 kg, the difference in adjacent energy levels is 4.82 zJ. Calculate the force constant of the oscillator.
Calculate the wavelength of a photon needed to excite a transition between neighbouring energy levels of a harmonic oscillator of effective mass equal to that of a proton (1.0078 u) and force constant 855 N m−1.
Refer to Exercise 9.10a and calculate the wavelength that would result from doubling the effective mass of the oscillator. Data in Exercise 9.10aCalculate the wavelength of a photon needed to excite a transition between neighbouring energy levels of a harmonic oscillator of effective mass
Calculate the minimum excitation energies of(a) A pendulum of length 1.0 m on the surface of the Earth,(b) The balance-wheel of a clockwork watch (ν = 5 Hz)
Confirm that a function of the form e−gx2 is a solution of the Schrödinger equation for the ground state of a harmonic oscillator and find an expression for g in terms of the mass and force constant of the oscillator.
Locate the nodes of the harmonic oscillator wavefunction with v = 4.
Assuming that the vibrations of a 35Cl2 molecule are equivalent to those of a harmonic oscillator with a force constant k = 329 N m−1, what is the zeropoint energy of vibration of this molecule? The mass of a 35Cl atom is 34.9688 u.
The wavefunction, ψ(φ), for the motion of a particle in a ring is of the form ψ = Neimlφ. Determine the normalization constant, N.
Use mathematical software to construct a wavepacket of the form with coefficients c of your choice (for example, all equal). Explore how the wavepacket migrates on the ring but spreads with time. 1981 με Ψ(φ,t) = Σ cmei(mφ–Emt/h) = m=0 Em = mh?/2I
Show that, whatever superposition of harmonic oscillator states is used to construct a wavepacket, it is localized at the same place at the times 0, T, 2T, . . . , where T is the classical period of the oscillator.
Confirm that Y3,+3 is normalized to 1. (The integration required is over the surface of a sphere.)
Show that the function f = cos ax cos cos cz is an eigenfunction of ∇2, and determine its eigenvalue.
When ultraviolet radiation of wavelength 58.4 nm from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of 1.59 Mm s−1. Calculate the ionization energy of krypton.
A series of lines in the spectrum of atomic hydrogen lies at 656.46 nm, 486.27 nm, 434.17 nm, and 410.29 nm. What is the wavelength of the next line in the series? What is the ionization energy of the atom when it is in the lower state of the transitions?
List and describe the significance of the quantum numbers needed to specify the internal state of a hydrogenic atom.
By differentiation of the 2s radial wavefunction, show that it has two extrema in its amplitude, and locate them.
Locate the radial nodes in the 3s orbital of an H atom.
Explain the significance of(a) A boundary surface and(b) The radial distribution function for hydrogenic orbitals.
The wavefunction for the ground state of a hydrogen atom is Ne−r/a0. Determine the normalization constant N.
Calculate the average kinetic and potential energies of an electron in the ground state of a hydrogen atom.
The characteristic emission from K atoms when heated is purple and lies at 770 nm. On close inspection, the line is found to have two closely spaced components, one at 766.70 nm and the other at 770.11 nm. Account for this observation, and deduce what information you can.
Write down the expression for the radial distribution function of a 2p electron in a hydrogenic atom and determine the radius at which the electron is most likely to be found.
Positronium consists of an electron and a positron (same mass, opposite charge) orbiting round their common centre of mass. The broad features of the spectrum are therefore expected to be hydrogen-like, the differences arising largely from the mass differences. Predict the wavenumbers of the first
Explain the origin of spin–orbit coupling and how it affects the appearance of a spectrum.
What is the orbital angular momentum of an electron in the orbitals(a) 1s,(b) 3s,(c) 3d?Give the numbers of angular and radial nodes in each case.
Calculate the permitted values of j for(a) A d electron,(b) An f electron.
Show that the sp2 hybrid orbital (s + 21/2p)/31/2 is normalized to 1 if the s and p orbitals are normalized to 1.
Consider a particle in a cubic box. What is the degeneracy of the level that has an energy three times that of the lowest level?
Describe the features that stem from nanometre-scale dimensions that are not found in macroscopic objects.
We normally think of the one-dimensional well as being horizontal. Suppose it is vertical; then the potential energy of the particle depends on x because of the presence of the gravitational field. Calculate the first-order correction to the zero-point energy, and evaluate it for an electron in a
What are the most likely locations of a particle in a box of length L in the state n = 3?
An electron is confined to a a square well of length L. What would be the length of the box such that the zero-point energy of the electron is equal to its rest mass energy, mec2 ? Express your answer in terms of the parameter λC = h/mec, the ‘Compton wavelength’ of the electron.
Discuss the physical origins of quantum mechanical tunnelling. Why is tunnelling more likely to contribute to the mechanisms of electron transfer and proton transfer processes than to mechanisms of group transfer reactions, such as AB + C → A + BC (where A, B, and C are large molecular groups)?
Calculate the energies of the first four rotational levels of 1H127I free to rotate in three dimensions, using for its moment of inertia I = µR2, with µ = mHmI/(mH + mI ) and R = 160 pm.
The mass to use in the expression for the vibrational frequency of a diatomic molecule is the effective mass µ = mAmB/(mA + mB), where mA and mB are the masses of the individual atoms. The following data on the infrared absorption wavenumbers (in cm−1) of molecules are taken from Spectra of
Calculate the expectation values of p and p2 for a particle in the state n = 1 in a square-well potential.
Calculate the probability that a particle will be found between 0.49L and 0.51L in a box of length L when it has(a) N = 1,(b) N = 2.Take the wavefunction to be a constant in this range.
Discuss the correspondence principle and provide two examples.
Solar energy strikes the top of the Earth’s atmosphere at a rate of 343 W m−2. About 30 per cent of this energy is reflected directly back into space by the Earth or the atmosphere. The Earth–atmosphere system absorbs the remaining energy and re-radiates it into space as black-body radiation.
The temperature of the Sun’s surface is approximately 5800 K. On the assumption that the human eye evolved to be most sensitive at the wavelength of light corresponding to the maximum in the Sun’s radiant energy distribution, determine the colour of light to which the eye is most sensitive.
Show that the expectation value of an operator that can be written as the square of an hermitian operator is positive.
Calculate(a) The mean potential energy and(b) The mean kinetic energy of an electron in the ground state of a hydrogenic atom.
Calculate the average linear momentum of a particle described by the following wavefunctions:(a) Eikx,(b) Cos kx,(c) E−αx2,where in each one x ranges from −∞ to +∞.
(a) Two (unnormalized) excited state wavefunctions of the H atom are Normalize both functions to 1.(b) Confirm that these two functions are mutually orthogonal. (i) u = 2 ਹੈ % -r/ao (ii) y=r sin cos e-r/2a0
Determine which of the following functions are eigenfunctions of the inversion operator î (which has the effect of making the replacement x → −x):(a) X3 − kx,(b) Cos kx,(c) X2 + 3x − 1.State the eigenvalue of î when relevant.
Normalize the following wavefunctions:(a) Sin(nπx/L) in the range 0 ≤ x ≤ L, where n = 1, 2, 3, . . . ,(b) A constant in the range −L ≤ x ≤ L,(c) E−r/a in three-dimensional space,(d) Xe−r/2a in three-dimensional space.
Derive Wien’s law, that λmaxT is a constant, where λmax is the wavelength corresponding to maximum in the Planck distribution at the temperature T, and deduce an expression for the constant as a multiple of the second radiation constant, c2 = hc/k.
A particle is in a state described by the wavefunction ψ(x) = a1/2e−ax, where a is a constant and 0 ≤ x ≤ ∞. Determine the expectation value of the commutator of the position and momentum operators.
A particle is in a state described by the wavefunction ψ(x) = (2a/π)1/4e−ax2, where a is a constant and −∞≤ x ≤ ∞. Verify that the value of the product ∆p∆x is consistent with the predictions from the uncertainty principle.
The ground-state wavefunction for a particle confined to a one-dimensional box of length L is Suppose the box is 10.0 nm long. Calculate the probability that the particle is(a) Between x = 4.95 nm and 5.05 nm,(b) Between x = 1.95 nm and 2.05 nm,(c) Between x = 9.90 nm and 10.00 nm,(d) In the
The normalized wavefunctions for a particle confined to move on a circle are ψ(φ) = (1/2π)1/2e−imφ, where m = 0, ±1, ±2, ±3,... and 0 ≤ φ ≤ 2π. Determine <φ>.
(a) Derive a general relation for (∂E/∂p)T,n for electrochemical cells employing reactants in any state of matter.(b) E. Cohen and K. Piepenbroek (Z. Physik. Chem. 167A, 365 (1933)) calculated the change in volume for the reaction TlCl(s) + CNS−(aq) → TlCNS(s) + Cl−(aq) at 30°C from
If the mitochondrial electric potential between matrix and the intermembrane space were 70 mV, as is common for other membranes, how much ATP could be synthesized from the transport of 4 mol H+, assuming the pH difference remains the same?
The standard potential of the AgCl/Ag,Cl− couple has been measured very carefully over a range of temperature (R.G. Bates and V.E. Bowers, J. Res. Nat. Bur. Stand. 53, 283 (1954)) and the results were found to fit the expressionCalculate the standard Gibbs energy and enthalpy of formation of
The emf of the cell Ag|AgI(s)|AgI(aq)|Ag is +0.9509 V at 25°C. Calculate(a) The solubility product of AgI and(b) Its solubility.
Use the Debye–Hückel limiting law and the Nernst equation to estimate the potential of the cell Ag|AgBr(s)|KBr(aq, 0.050 mol kg−1)||Cd(NO3)2(aq, 0.010 mol kg−1)|Cd at 25°C.
A fuel cell develops an electric potential from the chemical reaction between reagents supplied from an outside source. What is the emf of a cell fuelled by(a) Hydrogen and oxygen,(b) The combustion of butane at 1.0 bar and 298 K?
Write the cell reaction and electrode half-reactions and calculate the standard emf of each of the following cells: (a) Zn | ZnSO4(aq)||AgNO3(aq)|Ag(b) Cd |CdCl2(aq)||HNO3(aq)|H2(g)|Pt(c) Pt|K3[Fe(CN)6](aq),K4[Fe(CN)6](aq)||CrCl3(aq)|Cr
Fuel cells provide electrical power for spacecraft (as in the NASA space shuttles) and also show promise as power sources for automobiles. Hydrogen and carbon monoxide have been investigated for use in fuel cells, so their solubilities in molten salts are of interest. Their solubilities in a molten
For CaF2(s) ⇌ Ca2+(aq) + 2 F−(aq), K = 3.9 ×10−11 at 25°C and the standard Gibbs energy of formation of CaF2(s) is −1167 kJ mol−1. Calculate the standard Gibbs energy of formation of CaF2(aq).
Estimate the temperature at which CaCO3(calcite) decomposes.
The dissociation of I2 can be monitored by measuring the total pressure, and three sets of results are as follows:where nI is the amount of I atoms per mole of I2 molecules in the mixture, which occupied 342.68 cm3. Calculate the equilibrium constants of the dissociation and the standard enthalpy
The standard Gibbs energy of formation of NH3(g) is −16.5 kJ mol−1 at 298 K. What is the reaction Gibbs energy when the partial pressures of the N2, H2, and NH3 (treated as perfect gases) are 3.0 bar, 1.0 bar, and 4.0 bar, respectively? What is the spontaneous direction of the reaction in this
What is the standard enthalpy of a reaction for which the equilibrium constant is(a) Doubled,(b) Halved when the temperature is increased by 10 K at 298 K?
The equilibrium constant for the gas-phase isomerization of borneol (C10H17OH) to isoborneol at 503 K is 0.106. A mixture consisting of 7.50 g of borneol and 14.0 g of isoborneol in a container of volume 5.0 dm3 is heated to 503 K and allowed to come to equilibrium. Calculate the mole fractions of
A sealed container was filled with 0.300 mol H2(g), 0.400 mol I2(g), and 0.200 mol HI(g) at 870 K and total pressure 1.00 bar. Calculate the amounts of the components in the mixture at equilibrium given that K = 870 for the reaction H2(g) + I2(g) ⇌ 2 HI(g).
Calculate the percentage change in Kx for the reaction H2CO(g) ⇌ CO(g) + H2(g) when the total pressure is increased from 1.0 bar to 2.0 bar at constant temperature.
Describe a method for the determination of a standard potential of a redox couple.
The standard reaction Gibbs energy of the isomerization of borneol (C10H17OH) to isoborneol in the gas phase at 503 K is +9.4 kJ mol−1. Calculate the reaction Gibbs energy in a mixture consisting of 0.15 mol of borneol and 0.30 mol of isoborneol when the total pressure is 600 Torr.
The equilibrium constant of the reaction 2 C3H6(g) ⇌ C2H4(g) + C4H8(g) is found to fit the expression ln K = A + B/T + C/T2 between 300 K and 600 K, with A = −1.04, B = −1088 K, and C = 1.51 × 105 K2. Calculate the standard reaction enthalpy and standard reaction entropy at 400 K.
Distinguish between cell potential and electromotive force and explain why the latter is related to thermodynamic quantities.
The standard reaction enthalpy of Zn(s) + H2O(g) → ZnO(s) + H2(g) is approximately constant at +224 kJ mol−1 from 920 K up to 1280 K. The standard reaction Gibbs energy is +33 kJ mol−1 at 1280 K. Estimate the temperature at which the equilibrium constant becomes greater than 1.
Explain the molecular basis of the van ’t Hoff equation for the temperature dependence of K.
Suggest how the thermodynamic equilibrium constant may respond differently to changes in pressure and temperature from the equilibrium constant expressed in terms of partial pressures.
The vapour pressure of pure liquid A at 300 K is 76.7 kPa and that of pure liquid B is 52.0 kPa. These two compounds form ideal liquid and gaseous mixtures. Consider the equilibrium composition of a mixture in which the mole fraction of A in the vapour is 0.350. Calculate the total pressure of the
Calculate the masses of(a) Ca(NO3)2 and, separately,(b) NaCl to add to a 0.150 mol kg−1 solution of KNO3(aq) containing 500 g of solvent to raise its ionic strength to 0.250.
An electron in two different states of an atom is known to have j = 3/2 and 1/2. What is its orbital angular momentum quantum number in each case?
In 1976 it was mistakenly believed that the first of the ‘superheavy’ elements had been discovered in a sample of mica. Its atomic number was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements,
At what radius does the probability of finding an electron at a point in the H atom fall to 50 per cent of its maximum value?
Show by explicit integration that(a) Hydrogenic 1s and 2s orbitals,(b) 2px and 2py orbitals are mutually orthogonal.
What information does the term symbol 1D2 provide about the angular momentum of an atom?
Distinguish between semi-empirical, ab initio, and density functional theory methods of electronic structure determination.
Distinguish between delocalization energy, π-electron binding energy, and π-bond formation energy.
Calculate the total amplitude of the normalized bonding and antibonding LCAO-MOs that may be formed from two H1s orbitals at a separation of 106 pm. Plot the two amplitudes for positions along the molecular axis both inside and outside the internuclear region.
The ‘size’ of an atom is sometimes considered to be measured by the radius of a sphere that contains 90 per cent of the charge density of the electrons in the outermost occupied orbital. Calculate the ‘size’ of a hydrogen atom in its ground state according to this definition. Go on to
(a) Write the electronic configuration of the Ni2+ ion.(b) What are the possible values of the total spin quantum numbers S and MS for this ion?
Determine whether the px and py orbitals are eigenfunctions of lz. If not, does a linear combination exist that is an eigenfunction of lz?
From the ground-state electron configurations of B2 and C2, predict which molecule should have the greater bond dissociation energy.
Sketch the molecular orbital energy level diagram for XeF and deduce its ground-state electron configurations. Is XeF likely to have a shorter bond length than XeF+?
What is the energy of an electron that has been ejected from an orbital of ionization energy 11.0 eV by a photon of radiation of wavelength 100 nm?
What atomic terms are possible for the electron configuration ns1 nd1? Which term is likely to lie lowest in energy?
What values of J may occur in the terms(a) 1S,(b) 2P,(c) 3P?How many states (distinguished by the quantum number MJ) belong to each level?
Give the possible term symbols for(a) Li [He]2s1,(b) Na [Ne]3p1.
Hydrogen is the most abundant element in all stars. However, neither absorption nor emission lines due to neutral hydrogen are found in the spectra of stars with effective temperatures higher than 25 000 K. Account for this observation.
Discuss the steps involved in the calculation of the energy of a system by using the variation principle.
The d-metals iron, copper, and manganese form cations with different oxidation states. For this reason, they are found in many oxidoreductases and in several proteins of oxidative phosphorylation and photosynthesis (Impact I7.2 and I23.2). Explain why many d-metals form cations with different

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