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Physical Chemistry Thermodynamics And Kinetics 10th Edition Peter Atkins, Julio De Paula - Solutions
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 temperature. Suggest an interpretation of this observation. H 2
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 fields.
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 distance between the groups as 4.0nm and treat the phenyl group as a benzene molecule. The dipole moment of
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.
D.D. Nelson et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the H atoms in NH3 formed hydrogen bonds, but found none. For example, they found that the complex of NH3 and CO2 has the carbon atom nearest the nitrogen (299pm
Given that F=−dV/dr, calculate the distance dependence of the force acting between two non-bonded groups of atoms in a polymer chain that have a London dispersion interaction with each other.
This problem gives a simple example of a quantitative structure– activity relation (QSAR). The binding of nonpolar groups of amino acid to hydrophobic sites in the interior of proteins is governed largely by hydrophobic interactions. (a) Consider a family of hydrocarbons ReH. The
Some polymers have unusual properties. For example, Kevlar (3) is strong enough to be the material of choice for bulletproof vests and is stable at temperatures up to 600K. What molecular interactions contribute to the formation and thermal stability of this polymer? H-N- H -NH 3 Kevlar OTH n
The magnitude of the electric field at a distance r from a point charge Q is equal to Q/4πε0r2. How close to a water molecule (of polarizability volume 1.48×10−30m3) must a proton approach before the dipole moment it induces has a magnitude equal to that of the permanent dipole moment of the
Consider the arrangement shown in 5 for a system consisting of an O-H group and an O atom, and then use the electrostatic model of the hydrogen bond to calculate the dependence of the molar potential energy of interaction on the angle θ. -0.83e 200 pm 0 95.7 pm 5 H +0.45e -0.83e
Suppose you distrusted the Lennard-Jones (12,6) potential for assessing a particular polypeptide conformation, and replaced the repulsive term by an exponential function of the form e−r r/0 .(a) Sketch the form of the potential energy and locate the distance at which it is a minimum.(b)
Acetic acid vapour contains a proportion of planar, hydrogen-bonded dimers. The relative permittivity of pure liquid acetic acid is 7.14 at 290K and increases with increasing temperature. Suggest an interpretation of the latter observation. What effect should isothermal dilution have on the
In the square planar XeF4 molecule, consider the symmetry-adapted linear combination p1= pA −pB +pC −pD where pA, pB, pC, and pD are 2pz atomic orbitals on the fluorine atoms (clockwise labelling of the F atoms). Using the reduced point group D4 rather than the full symmetry point group of the
What is the maximum possible degree of degeneracy of the orbitals in benzene?
Discuss the strengths and limitations of the parabolic and Morse functions as descriptors of the potential energy curve of a diatomic molecule.
Give the ground-state electron configurations of (i) F2−, (ii) N2, (iii) O22−.
Describe the Pauling and Mulliken electronegativity scales. Why should they be approximately in step?
Equation 10D.9 follows from eqn 10D.8a by making the approximation |αB – αA|≫2|β| and setting S=0. Explore the consequences of not setting S=0.Data in Equation 10D.9 E+ = 0 + B2 OA-OB E_=0B- B2 XA-OB (10D.9)
Give the ground-state electron configurations of (i) CO, (ii) NO, and (iii) CN−.
Discuss the scope, consequences, and limitations of the approximations on which the Hückel method is based.
Write down the secular determinants for (i) linear H4, (ii) cyclic H4 within the Hückel approximation.
Set up and solve the Hückel secular equations for the π electrons of CO32−. Express the energies in terms of the Coulomb integrals αO and αC and the resonance integral β. Determine the delocalization energy of the ion.
The languages of valence-bond theory and molecular orbital theory are commonly combined when discussing unsaturated organic compounds. Construct the molecular orbital energy level diagrams of ethene on the basis that the molecule is formed from the appropriately hybridized CH2 or CH fragments.
Use the 2px and 2pz hydrogenic atomic orbitals to construct simple LCAO descriptions of 2pσ and 2pπ molecular orbitals.(a) Make a probability density plot, and both surface and contour plots of the xz-plane amplitudes of the 2pzσ and 2pzσ* molecular orbitals.(b) Make surface and contour plots
Why do both ionization energy and electron affinity play a role in estimating the energy of an atomic orbital to use in a molecular structure calculation?
Sketch the molecular orbital energy level diagram for IF and deduce its ground-state electron configuration. Is IF likely to have a shorter bond length than IF− or IF+?
Suppose that a molecular orbital of a heteronuclear diatomic molecule is built from the orbital basis A, B, and C, where B and C are both on one atom (they can be envisaged as F2s and F2p in HF, for instance). Set up the secular equations for the optimum values of the coefficients and the
Predict the electron configurations of (i) the allyl radical, (ii) the cyclobutadiene cation. Estimate the π-electron binding energy in each case.
For monocyclic conjugated polyenes (such as cyclobutadiene and benzene) with each of N carbon atoms contributing an electron in a 2p orbital, simple Hückel theory gives the following expression for the energies Ek of the resulting π molecular orbitals: (a) Calculate the energies of the π
Which has the higher dissociation energy, F2 or F2+ ?
Show, if overlap is ignored,(a) That any molecular orbital expressed as a linear combination of two atomic orbitals may be written in the form ψ=ψA cos θ +ψB sin θ, where θ is a parameter that varies between 0 and π, (b) That if ψA and ψB are orthogonal and normalized to 1, then ψ is
Discuss the steps involved in the calculation of the energy of a system by using the variation principle. Are any assumptions involved?
Use the electron configurations of SO− and SO+ to predict which is likely to have the shorter bond length.
Continue the preceding problem by setting αA =−7.2 eV, αB =−10.4 eV, αB =−8.4 eV, βAB =−1.0 eV, βAC =−0.8 eV, and calculate the orbital energies and coefficients with (i) both S=0, (ii) both S=0.2.
Outline the computational steps used in the self-consistent field approach to electronic structure calculations.
Compute the delocalization energy and π-bond formation energy of (i) the benzene anion, (ii) the benzene cation.
Set up the secular determinants for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene and diagonalize them by using mathematical software. Use your results to show that the π molecular orbitals of linear polyenes obey the following rules:• The π molecular
Evaluate the bond order of each Period 2 homonuclear diatomic molecule.
In a particular photoelectron spectrum using 21.21 eV photons, electrons were ejected with kinetic energies of 11.01 eV, 8.23 eV, and 5.22 eV. Sketch the molecular orbital energy level diagram for the species, showing the ionization energies of the three identifiable orbitals.
What is the physical significance of the Coulomb and resonance integrals?
Write down the secular determinants for (i) anthracene (1), (ii) phenanthrene (2) within the Hückel approximation and using the C2p orbitals as the basis set. 1 Anthracene 2 Phenanthrene
As a variation of the preceding problem explore the consequences of increasing the energy separation of the B and C orbitals (use S=0 for this stage of the calculation). Are you justified in ignoring orbital C at any stage?
Explain why the use of Gaussian-type orbitals is generally preferred over the use of hydrogenic (exponential) orbitals in basis sets.
Set up the secular determinants for cyclobutadiene, benzene, and cyclooctatetraene and diagonalize them by using mathematical software. Use your results to show that the π molecular orbitals of monocyclic polyenes with an even number of carbon atoms follow a pattern in which:• The π molecular
Use mathematical software to estimate the π-electron binding energy of (i) azulene (3), (ii) acenaphthalene (4) within the Hückel approximation. 3 Azulene 4 Acenaphthalene
For each of the species in Exercise 10C.4(b), specify which molecular orbital is the HOMO.Data in Exercise 10C.4(b),Evaluate the bond order of each Period 2 homonuclear diatomic cation, X2+ , and anion, X2−.
Discuss how the properties of carbon explain the bonding features that make it an ideal biological building block.
What is the speed of a photoelectron ejected from a molecule with radiation of energy 21 eV and known to come from an orbital of ionization energy 12 eV?
Write the electronic hamiltonian for HeH+.
Prove that for an open chain of N conjugated carbons the characteristic polynomial of the secular determinant (the polynomial obtained by expanding the determinant), PN(x), where x=(α−β)/β, obeys the recurrence relation PN =xPN−1−PN−2, with P1=x and P0=1.
Now repeat Exercise 10D.6(a), but with S=0.20.Data in Exercise 10D.6(a),Use the values derived in Exercise 10D.5(a) to estimate the molecular orbital energies in HCl; Use S=0.
Use appropriate electronic structure software and basis sets of your or your instructor’s choosing, perform self-consistent field calculations for the ground electronic states of H2 and F2. Determine ground-state energies and equilibrium geometries. Compare computed equilibrium bond lengths to
Use an appropriate semi-empirical method to compute the equilibrium bond lengths and standard enthalpies of formation of (a) ethanol, C2H5OH, (b) 1,4-dichlorobenzene, C6H4Cl2. Compare to experimental values and suggest reasons for any discrepancies.
Molecular orbital calculations based on semi-empirical, ab initio, and DFT methods describe the spectroscopic properties of conjugated molecules better than simple Hückel theory.(a) Use the computational method of your choice (semi-empirical, ab initio, or density functional methods) or your
Explain what is meant by a ‘group’.
Use as a basis the valence pz orbitals on each atom in BF3 to find the representative of the operation σh. Take z as perpendicular to the molecular plane.
Identify and list four applications of character tables.
Use symmetry properties to determine whether or not the integral ∫pxzpzdτ is necessarily zero in a molecule with symmetry D3h.
Use the matrix representatives of the operations σh and C3 in a basis of valence pz orbitals on each atom in BF3 to find the operation and its representative resulting from σhC3. Take z as perpendicular to the molecular plane.
In the square-planar complex anion [trans-Ag(CF3)2(CN)2]−, the AgeCN groups are collinear.(a) Assume free rotation of the CF3 groups (that is, disregarding the AgCF and AgCH angles) and name the point group of this complex ion.(b) Now suppose the CF3 groups cannot rotate freely (because the ion
Explain how symmetry arguments are used to construct molecular orbitals.
Is the transition A1→A2 forbidden for electric dipole transitions in a C3v molecule?
Identify the group to which the naphthalene molecule belongs and locate the symmetry elements in a drawing of the molecule.
Show that all three C2 operations in the group D3h belong to the same class.
State and explain the symmetry criteria that allow a molecule to be polar.
Explain what is meant by the reduction of a representation to a direct sum of representations.
What is the maximum degeneracy of a particle confined to the interior of an octahedral hole in a crystal?
State the symmetry criteria that allow a molecule to be optically active.
List the symmetry elements of the following molecules and name the point groups to which they belong: (i) NO2, (ii) N2O, (iii) CHCl3, (iv) CH2=CH2.
In a spectroscopic study of C60, Negri et al. (J. Phys. Chem. 100, 10849 (1996)) assigned peaks in the fluorescence spectrum. The molecule has icosahedral symmetry (Ih). The ground electronic state is A1g, and the lowestlying excited states are T1g and Gg.(a) Are photon-induced transitions allowed
Discuss the significance of the letters and subscripts used to denote the symmetry species of a representation.
Find the representatives of the operations of the group Td in a basis of four H1s orbitals, one at each apex of a regular tetrahedron (as in CH4).
Assign (i) cis-dichloroethene and (ii) trans-dichloroethene to point groups.
A set of basis functions is found to span a reducible representation of the group C4v with characters 4,1,1,3,1 (in the order of operations in the character table in the Resource section). What irreducible representations does it span?
Which of the following molecules may be polar? (i) pyridine, (ii) nitroethane, (iii) gas-phase HgBr2, (iv) B3N3H6.
What states of (i) benzene, (ii) naphthalene may be reached by electric dipole transitions from their (totally symmetrical) ground states?
Identify the point groups to which all isomers of dichloronaphthalene belong.
Can molecules belonging to the point groups D2h or C3h be chiral? Explain your answer.
What is the physical interpretation of a selection rule?
Describe the physical origins of the gross selection rules for infrared spectroscopy.
Calculate the ratio A/B for transitions with the following characteristics: (i) 70.8pm X-rays, (ii) 500nm visible light, (iii) 3000 cm−1 infrared radiation.
Account for the rotational degeneracy of the various types of rigid rotor. Would their lack of rigidity affect your conclusions?
Calculate the moment of inertia around the C2 axis (the bisector of the OOO angle) and the corresponding rotational constant of an 16O3 molecule (bond angle 117°; OO bond length 128 pm).
Which of the following molecules may show a pure rotational microwave absorption spectrum: (i) H2O, (ii) H2O2, (iii) NH3, (iv) N2O?
An object of mass 100 g suspended from the end of a rubber band has a vibrational frequency of 2.0Hz. Calculate the force constant of the rubber band.
Describe the physical origins of linewidths in absorption and emission spectra. Do you expect the same contributions for species in condensed and gas phases?
Describe the physical origins of the gross selection rules for vibrational Raman spectroscopy.
The molar absorption coefficient of a substance dissolved in hexane is known to be 723 dm3mol−1 cm−1 at 260nm. Calculate the percentage reduction in intensity when light of that wavelength passes through 2.50mm of a solution of concentration 4.25mmol dm−3.
Describe the differences between an oblate and a prolate symmetric rotor and give several examples of each.
A Dubosq colorimeter consists of a cell of fixed path length and a cell of variable path length. By adjusting the length of the latter until the transmission through the two cells is the same, the concentration of the second solution can be inferred from that of the former. Suppose that a plant dye
Calculate the frequency and wavenumber of the J=3←2 transition in the pure rotational spectrum of 14N16O. The equilibrium bond length is 115 pm. Does the frequency increase or decrease if centrifugal distortion is considered?
Describe the effect of vibrational excitation on the rotational constant of a diatomic molecule.
Describe the basic experimental arrangements commonly used for absorption, emission, and Raman spectroscopy.
How many vibrational modes are there for the molecule NC–(C≡C–C≡C–)10CN detected in an interstellar cloud?
A solution of an unknown component of a biological sample when placed in an absorption cell of path length 1.00 cm transmits 18.1 per cent of light of 320nm incident upon it. If the concentration of the component is 0.139mmol dm−3, what is the molar absorption coefficient?
The Beer–Lambert law is derived on the basis that the concentration of absorbing species is uniform. Suppose, instead, that the concentration falls exponentially as [J]=[J]0e−x/λ. Develop an expression for the variation of I with sample length; suppose that L≫λ.
Classify the following rotors: (i) O3, (ii) CH3CH3, (iii) XeO4, (iv) FeCp2 (Cp denotes the cyclopentadienyl group, C5H5).
The wavenumber of the J=3←2 rotational transition of 1H35Cl considered as a rigid rotor is 63.56 cm−1; what is the H–Cl bond length?
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