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Thallium, a neurotoxin, is the heaviest member of Group 13 of the periodic table and is found most usually in the +1 oxidation state. Aluminum, which causes anaemia and dementia, is also a member of the group but its chemical properties are dominated by the +3 oxidation state. Examine this issue by plotting the first, second, and third ionization energies for the Group 13 elements against atomic number. Explain the trends you observe Hints. The third ionization energy, 13, is the minimum energy needed to remove an electron from the doubly charged cation: E2+ (g) → 1 E3+ (g) + e- (g), 13= E (E3+) – E (E2+). For data, see the links to databases of atomic properties provided in the text's web site.
Compare the approximations built into valence-bond theory and molecular-orbital theory.
Distinguish between the Pauling and Mulliken electro negativity scales.
Discuss the approximations built into the Huckel method.
Use concepts of molecular orbital theory to describe the biochemical reactivity of O2, N2, and NO.
Give the ground-state electron configurations of
(a) H-2
(b) N2, and
(c) Oz.
Give the ground-state electron configurations of
(a) CIF,
(b) CS, and
(c) 0-2.
Which of the molecules N2, NO, 02, C2, F2, and CN would you expect to be stabilized by
(a) The addition of an electron to form AB-,
(b) The removal of an electron to form AB+?
Sketch the molecular orbital energy level diagrams for BrCI and deduce its ground-state electron configurations. Is BrCl likely to have a shorter bond length than BrCl-?
Arrange the species 0+2, 02, 0-2, O2-2 in order of increasing bond length.
Normalize the molecular orbital ΨA + λ ΨB in terms of the parameter λ and the overlap integral S.
Can the function Ψ = x2 (L - 2x) be used as a trial wave function for the n = 1 state of a particle with mass m in a one-dimensional box of length L? If the answer is yes, then express the energy of this trial wave function in terms of h, m, and L and compare it with the exact result (eqn 9.4). If the answer is no, explain why this is not a suitable trial wave function.
Suppose that the function Ψ= Ae—ar2 with A being the normalization constant and a being an adjustable parameter, is used as a trial wave function for the is orbital of the hydrogen atom. The energy of this trial wave function is

Suppose that the function Ψ= Ae—ar2
Where e is the electron charge and j1 is the effective mass of the H atom. What is the minimum energy associated with this trial wave function?
What is the energy of an electron that has been ejected from an orbital of ionization energy 4.69 eV by a photon of radiation of wavelength 584 nm?
Construct the molecular orbital energy level diagrams of ethene (acetylene) on the basis that the molecule is formed from the appropriately hybridized CH2 or CH fragments.
Predict the electronic configurations of
(a) The benzene anion,
(b) The benzene cation. Estimate the π-electron binding energy in each case.
Use mathematical software to estimate the π-electron binding energy of
(a) Anthracene (8),
(b) Phenanthrene (9) within the Huckel approximation.
Suppose that a molecular orbital has the form N (O.l45A + 0.844B). Find a linear combination of the orbitals A and B that is orthogonal to this combination.
Show that, if a wave cos kx centred on A (so that x is measured from A) interferes with a similar wave cos k'» centred on B (with x measured from B) a distance R away, then constructive interference occurs in the intermediate region when k = k' = rr/2R and destructive interference if kR = ½ π and k' R = 3/2π.
Before doing the calculation below, sketch how the overlap between a is orbital and a 2p orbital can be expected to depend on their separation. The overlap integral between an H Is orbital and an H2p orbital on nuclei separated by a distance R and forming a σ orbital is S = (R/ao) 1 + (R/ao) + ½ (R/ao)2|e-Rlao. Plot this function, and find the separation for which the overlap is a maximum.
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|

Imagine a small electron-sensitive probe of volume 1.00 pm3 inserted into an H+2 molecule-ion in its ground state. Calculate the probability that it will register the presence of an electron at the following positions:

(a) At nucleus A,

(b) At nucleus B,

(c) half-way between A and B, (c) at a point 20 pm along the bond from A and 10 pm perpendicularly. Do the same for the molecule-ion the instant after the electron has been excited into the antibonding LCAO-MO.

The same data as in Problem 11.8 may be used to calculate the molecular potential energy curve for the antibonding orbital, which is given by

The same data as in Problem 11.8 may
Rydberg molecules are molecules with an electron in an atomic orbital with principal quantum number 11 one higher than the valence shells of the constituent atoms. Speculate about the existence of 'hyper Rydberg' H2 formed from two H atoms with 1005 electrons. Make reasonable guesses about the binding energy, the equilibrium intern clear separation, the vibrational force constant, and the rotational constant. Is such a molecule likely to exist under any circumstances?
Set up and solve the Huckel secular equations for the n electrons of NO-3. Express the energies in terms of the Coulomb integrals ao and aN and the resonance integral 13. Determine the delocalization energy of the ion.
The FEMO theory (Problem 11.14) of conjugated molecules is rather crude and better results are obtained with simple Huckel theory.
(a) For a linear conjugated polyene with each of N carbon atoms contributing an electron in a 2p orbital, the energies Ek of the resulting π molecular orbitals are given by (see also Section 20.9): Use this expression to determine a reasonable empirical estimate of the resonance integral 13 for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene given that π* ← π ultraviolet absorptions from the HOMO to the LUMO occur at 61 500, 46 080, 39 750, and 32900 cm-1, respectively.
(b) Calculate the π -electron delocalization energy, Ede1o, = En-n (a+ /3), of octatetraene, where E π is the total π –electron binding energy and n is the total number of π -electrons.
(c) In the context of this Huckel model, the π molecular orbitals are written as linear combinations of the carbon 2p orbitals. The coefficient of the jth atomic orbital in the kth molecular orbital is given by Determine the values of the coefficients of each of the six 2p orbitals in each of the six n molecular orbitals of hexatriene. Match each set of coefficients (that is, each molecular orbital) with a value of the energy calculated with the expression given in part (a) of the molecular orbital. Comment on trends that relate the energy of a molecular orbital with its 'shape', which can be inferred from the magnitudes and signs of the coefficients in the linear combination that describes the molecular orbital.

The FEMO theory (Problem 11.14)
If you have access to mathematical software that can perform matrix diagonalization, use it to solve Problems 11.15 and 11.16, disregarding the expressions for the energies and coefficients given there.
Electronic excitation of a molecule may weaken or strengthen some bonds because bonding and antibonding characteristics differ between the HOMO and the LUMO. For example, a carbon-carbon bond in a linear polyene may have bonding character in the HOMO and antibonding character in the LUMO.
Therefore, promotion of an electron from the HOMO to the LUMO weakens this carbon-carbon bond in the excited electronic state, relative to the ground electronic state. Display the HOMO and LUMO of each molecule in Problem 11.15 and discuss in detail any changes in bond order that accompany the π*← π ultraviolet absorptions in these molecules.
An sp2 hybrid orbital that lies in the xy-plane and makes an angle of 1200 to the x-axis has the form

An sp2 hybrid orbital that lies in the xy-plane
Use hydrogenic atomic orbitals to write the explicit form of the hybrid orbital. Show that it has its maximum amplitude in the direction specified.
Derive eqns 11.11 and 11.14 by working with the normalized LCAO-MOs for the H+2 molecule-ion (Section 11.3a). Proceed by evaluating the expectation value of the Hamiltonian for the ion. Make use of the fact that A and B each individually satisfy the Schrödinger equation for an isolated H atom.
We saw in Section 11.5 that, to find the energies of the bonding and antibonding orbitals of a heteronuclear diatomic molecule, we need to solve the secular determinant where aA et aB and we have taken S = O. Equations 11.34a and 11.34b give the general solution to this problem. Here, we shall develop the result for the case (aB - aA) 2>>{32.
(a) Begin by showing that where E+ and E_ are the energies of the bonding and antibonding molecular orbitals, respectively.
(b) Now use the expansion to show that which is the limiting result used in Justification 11.4.

We saw in Section 11.5 that
There is some indication that other hydrogen ring compounds and ions in addition to H3 and D3 species may play a role in interstellar chemistry.
According to J.S. Wright and G.A. DiLabio (J. Phys. Chem. 96, 10793 (1992)), H-5 H6, and H+7 are particularly stable whereas H4 and H+3 are not. Confirm these statements by Huckel calculations.
Molecular orbital calculations may be used to predict trends in the standard potentials of conjugated molecules, such as the quinones and flavins that are involved in biological electron transfer reactions (Impact I7.2). It is commonly assumed that decreasing the energy of the LUMO enhances the ability of a molecule to accept an electron into the LUMO, with an attendant increase in the value of the molecule's standard potential. Furthermore, a number of studies indicate that there is a linear correlation between the LUMO energy and the reduction potential of aromatic hydrocarbons (see, for example, J.P. Lowe, Quantum chemistry, Chapter 8, Academic Press (1993)).
(a) The standard potentials at pH = 7 for the one-electron reduction of methyl-substituted lA-benzoquinone (13) to their respective SEM quinone radical anions are
Using molecular modeling software and the computational method of your choice (semi-empirical, ab initio, or density functional theory methods), calculate EWMO' the energy of the LUMO of each substituted lA-benzoquinone, and plot EWMO against E~. Do your calculations support a linear relation between EWMO and E?
(b) The lA-benzoquinone for which Rz = R3 = CH3 and Rs = R6 = OCH3 is a suitable model of ubiquinone, a component of the respiratory electron transport chain (Impact I7.2). Determine EWMO of this quinone and then use your results from part (a) to estimate its standard potential.
(c) The l, 4-benzoquinone for which R2 = R3 = CH3 and R6 = H is a suitable model of plastoquinone, a component of the photosynthetic electron transport chain (Impact I7.2). Determine EWMO of this quinone and then use your results from part (a) to estimate its standard potential. Is plastoquinone expected to be a better or worse oxidizing agent than ubiquinone?
(d) Based on your predictions and on basic concepts of biological electron transport (Impact I7.2 and I23.2), suggest a reason why ubiquinone is used in respiration and plastoquinone is used in photosynthesis.

Molecular orbital calculations
Explain how a molecule is assigned to a point group.
Explain the symmetry criteria that allow a molecule to be polar?

Explain what is meant by

(a) A representative and

(b) A representation in the context of group theory.

Explain how spectroscopic selection rules arise and how they are formulated by using group theory.
The CCl4 molecule belongs to the point group Td. List the symmetry elements of the group and locate them in the molecule.
Which of the following molecules may be polar?
(a) CH3Cl (C3J,
(b) HW2 (CO)10 (D4h),
(c) SnCl4 (Td).
Use symmetry properties to determine whether or not the integral f Px zPz dτ is necessarily zero in a molecule with symmetry D6
Is the transition A1g → E2u forbidden for electric dipole transitions in a D6h molecule?
Show that the function xyz has symmetry species AI in the group D2.
Molecules belonging to the point groups Th or Td cannot be chiral. Which elements of these groups rule out chirality’s?
The group C4v consists of the elements E, 2C4, C2, and 2σv, 2σd construct the group multiplication table.
Identify the point groups to which the following objects belong:
(a) A sharpened cylindrical pencil,
(b) A three-bladed propeller,
(c) A four-legged table,
(d) Yourself (approximately).
List the symmetry elements of the following molecules and name the point groups to which they belong:
(a) Naphthalene,
(b) Anthracene,
(c) The three dichlorobenzenes.
Assign the following molecules to point groups:
(a) HF,
(b) IF7 (pentagonal bipyramid),
(c) XeO2F2, (see-saw),
(d) Fe,(CO)9 (22),
(e) Cubane, C8H8,
(f) Tetrafluorocubane, C8H4F4 (23)

Assign the following molecules
Which of the molecules in Exercises 12.9b and 12.10b can be?
(a) Polar,
(b) Chiral?
Consider the C3Yion NO;. Is there any orbital of the central N atom that can have a nonzero overlap with the combination 2pz (A) – pz (B) – pz (C) of the three O atoms (with z perpendicular to the plane). What would be the case in 503' where 3d orbitals might be available?
The CIO2 molecule (which belongs to the group C, 2v) was trapped in a solid. Its ground state is known to be BJ• Light polarized parallel to the y-axis (parallel to the 00 separation) excited the molecule to an upper state. What is the symmetry of that state?
What states of
(a) Anthracene,
(b) Coronene (24) may be reached by electric dipole transitions from their (totally symmetrical) ground states?

What states of  (a) Anthracene,  (b) Coronene
Determine whether the integral over /J and /, in Exercise 12.15a is zero over a symmetrical range about e = 0 in the group C3v
List the symmetry elements of the following molecules and name the point groups to which they belong:
(a) Staggered CH3CH3,
(b) Chair and boat cyclohexane,
(c) B, H6,
(d) [Co (en) 3J3+, where en is ethylenediamine (ignore its detailed structure),
(e) crown-shaped Ss' which of these molecules can be (i) polar, (ii) chiral?
The group D 1h has a C, axis perpendicular to the principal axis and a horizontal mirror plane. Show that the group must therefore have a centre of inversion.
Confirm that the z-component of orbital angular momentum is a basis for an irreducible representation of A, symmetry in C3v'
Construct the multiplication table of the Pauli spin matrices, CY, and the 2 x 2 unit matrix:
Do the four matrices form a group under multiplication?

Construct the multiplication table
Suppose that a methane molecule became distorted to
(a) C3v symmetry by the lengthening of one bond,
(b) C2v symmetry, by a kind of scissors action in which one bond angle opened and another closed slightly. Would more d orbitals become available for bonding?
R. Eujen, B. Hoge, and D.J. Brauer (Inorg. Chem. 36, 1464 (1997» prepared and characterized several square-planar Ag (III) complex anions. In the complex anion [trans-Ag (CF3)2 (CN), 2|-, the Ag-CN groups are collinear.
(a) Assuming free rotation of the CF3 groups (that is, disregarding the AgCF and AgCH angles), name the point group of this complex anion.
(b) Now suppose the CF3 groups cannot rotate freely (because the ion was in a solid, for example). Structure (26) shows a plane that bisects the NC-Ag-CN axis and is perpendicular to it. Name the point group of the complex if each CF3 group has a CF bond in that plane (so the CF3 groups do not point to either CN group preferentially) and the CF3 groups are (i) staggered, (ii) eclipsed.

R. Eujen, B. Hoge, and D.J. Brauer
The algebraic forms of the [orbitals are a radial function multiplied by one of the factors (a) z (5z2 - 3r2),
(b) y (5y2 - 3r2),
(c) x (5x2 - 3r2),
(d) Z(x2 – Y2),
(e) Y(x2- z2), (f) x (z2 – y2), (g) Xyz. Identify the irreducible representations spanned by these orbitals in (a) C2v (b) C3v, (c) Td, (d) Oh Consider a lanthanide ion at the centre of (a) a tetrahedral complex, (b) an octahedral complex. What sets of orbitals do the seven [orbitals split into?
The NO2, molecule belongs to the group C2v' with the C2 axis bisecting the ONO angle. Taking as a basis the N2s, N2p, and 02p orbitals, identify the irreducible representations they span, and construct the symmetry-adapted linear combinations.
The phenanthrene molecule (29) belongs to the group C2v with the C, axis in the plane of the molecule.
(a) Classify the irreducible representations spanned by the carbon 2pz orbitals and find their symmetry adapted linear combinations.
(b) Use your results from part (a) to calculate the Huckel secular determinant. (c) What states of phenanthrene may be reached by electric dipole transitions from its (totally symmetrical) ground state?

The phenanthrene molecule (29) belongs
The H3 molecular ion, which plays an important role in chemical reactions occurring in interstellar clouds, is known to be equilateral triangular.
(a) Identify the symmetry elements and determine the point group of this molecule.
(b) Take as a basis for a representation of this molecule the three H1s orbitals and set up the matrices that group in this basis.
(c) Obtain the group multiplication table by explicit multiplication of the matrices.
(d) Determine if the representation is reducible and, if so, give the irreducible representations obtained.
Some linear polyenes, of which ,6-carotene is an example, are important biological co-factors that participate in processes as diverse as the absorption of solar energy in photosynthesis (Impact 123.2) and protection against harmful biological oxidations. Use as a model of, 6-carotene a linear polyene containing 22 conjugated C atoms.
(a) To what point group does this model of, 6-carotene belong?
(b) Classify the irreducible representations spanned by the carbon 2pz orbitals and find their symmetry-adapted linear combinations.
(c) Use your results from part (b) to calculate the Huckel secular determinant.
(d) What states of this model of, 6-carotene may be reached by electric dipole transitions from its (totally symmetrical) ground state?
Describe the physical origins of line widths in the absorption and emission spectra of gases, liquids, and solids.
Discuss the physical origins of the gross selection rules for rotational and vibrational Raman spectroscopy.
Suppose that you wish to characterize the normal modes of benzene in the gas phase. Why is it important to obtain both infrared absorption and Raman spectra of your sample?

Calculate the ratio of the Einstein coefficients of spontaneous and stimulated emission, A and B, for transitions with the following characteristics:

(a) 500 MHz radiofrequency radiation,

(b) 3.0 cm microwave radiation.

At what speed of approach would a red (660 nm) traffic light appear green (520 mu)?
Estimate the lifetime of a state that gives rise to a line of width
(a) 100 MHZ
(b) 2.14 cm-1
A molecule in a gas undergoes about 1.0 X 109 collisions in each second. Suppose that
(a) Every collision is effective in deactivating the molecule rotationally and
(b) That one collision in 10 is effective. Calculate the width (in hertz) of rotational transitions in the molecule.
Calculate the frequency of the J = 3 f-- 2 transition in the pure rotational spectrum of 12CI60. The equilibrium bond length is 112.81 pm.
If the wave number of the J = 1 f-- 0 rotational transition of IH81Br considered as a rigid rotator is 16.93 cm-1, what is
(a) The moment of inertia of the molecule,
(b) The bond length?
Given that the spacing of lines in the microwave spectrum of 35Cjl9p is constant at 1.033 cm-1, calculate the moment of inertia and bond length of the molecule (me(35Cl) = 34.9688 u, m(19F) = 18.9984 u)
The rotational constant of 12/16OC is 0.39021 cm. Calculate the bond length of the molecule (m (l2C) = 12 u exactly, m (16O) = 15.9949 u).
Determine the CO and CS bond lengths in OCS from the rotational constants B(16Ol2C32S) = 6081.5 MHz, B(16012C34S) = 5932.8 MHz.
The wave number of the incident radiation in a Raman spectrometer is 20623 cm-1. What is the wave number of the scattered Stokes radiation for the J = 4 f- 2 transition of 160O2?
The rotational Raman spectrum of 19F2 (m (19F) = 18.9984 u) shows a series of Stokes lines separated by 3.5312 cm-1 and a similar series of anti-Stokes lines. Calculate the bond length of the molecule.
Which of the following molecules may show a pure rotational microwave absorption spectrum?
(a) H20,
(b) H202,
(c) NH4
(d) N20?
Which of the following molecules may show a pure rotational Raman spectrum?
(a) CH2Cl2
(b) CH3CH3,
(c) SP6,
(d) N2O?
An object of mass 2.0 g suspended from the end of a spring has a vibrational frequency of 3.0 Hz. Calculate the force constant of the spring.
Calculate the percentage difference in the fundamental vibration wave number of IH35Cl and 2H37CIon the assumption that their force constants are the same.
The wave number of the fundamental vibrational transition of 79BrB1Bris 323.2 cm-1, Calculate the force constant of the bond (mC9Br) = 78.9183 u, m(8IBr) = 80.9163 u).
Calculate the relative numbers of Br, molecules (v = 321 cm-1) in the second and first excited vibrational states at
(a) 298 K,
(b) 800 K.
Prom the data in Exercise 13.18a, predict the fundamental vibrational Wave numbers of the deuterium halides.
For 14N2 ∆G values for the transitions v = 1 ← 0, 2 ← 0, and 3 ← 0 are, respectively, 2345.15, 4661.40, and 6983.73 cm-1. Calculate v and xe- Assume ye to be zero.
The first five vibrational energy levels of HI are at 1144.83, 3374.90, 5525.51, 7596.66, and 9588.35 cm-1 Calculate the dissociation energy of the molecule in reciprocal centimeters and electron volts.
Infrared absorption by I H81 12 Br gives rise to an R branch from v = O. What is the wave number of the line originating from the rotational state with J = 2? Use the information in Table 13.2.
Which of the following molecules may show infrared absorption spectra?
(a) CH3CH3,
(b) CH4,
(c) CH3CI,
(d) N2
How many normal modes of vibration are there for the following molecules:
(a) C6H6,
(b) C6H6CH3,
(c) HC=C-C=CH.
Which of the vibrations of an AB3 molecule are infrared or Raman active when it is
(a) Trigonal planar,
(b) Trigonal pyramidal?
Consider the vibrational mode that corresponds to the boat-like bending of a benzene ring. Is it
(a) Raman,
(b) Infrared active?
A carbon disulfide molecule belongs to the point group D_h. The nine displacements of the three atoms span A1g+ AIU+ A2g+ 2E1u+ E1g.What are the symmetries of the normal modes of vibration?
Use mathematical software to evaluate the Planck distribution at any temperature and wavelength or frequency, and evaluate integrals for the energy density of the radiation between any two wavelengths. Calculate the total energy density in the visible region (700 nm to 400 nm) for a black body at
(a) 1500 K, a typical operating temperature for globars,
(b) 2500 K, a typical operating temperature for tungsten filament lamps,
(c) 5800 K, the surface temperature of the Snn. What are the classical values at these temperatures?
The collision frequency z of a molecule of mass m in a gas at a pressure pis z = 4(J (kTlnm) 1/2p/kT, where o is the collision cross-section. Find an expression for the collision-limited lifetime of an excited state assuming that every collision is effective. Estimate the width of rotational transition in HCI ((J= 0.30 nm2) at 25°C and 1.0 atm. To what value must the pressure of the gas be reduced in order to ensure that collision broadening is less important than Doppler broadening?
The rotational constant for CO is 1.9314 cm-1 and 1.6116 cm-1 in the ground and first excited vibrational states, respectively. By how much does the intern clear distance change as a result of this transition?
Rotational absorption lines from 1H35CI gas were found at the following wave numbers (R.1, Hausler and R.A. Oetjen, f. Chem. Phys. 21, 1340 (1953)): 83.32, 104.13, 124.73, 145.37, 165.89, 186.23, 206.60, and 226.86 cm-1, calculate the moment of inertia and the bond length of the molecule. Predict the positions of the corresponding lines in 2H35Cl.
Thermodynamic considerations suggest that the copper monohalides CuX should exist mainly as polymers in the gas phase, and indeed it proved difficult to obtain the monomers in sufficient abundance to detect spectroscopically. This problem was overcome by flowing the halogen gas over copper heated to 1100 K (E.1, Manson, F.e. de Lucia, and W. Gordy, f. Chem. Phys. 63, 2724 (1975)).
For CuBr the J = 13-14, 14-15, and 15-16 transitions occurred at 84 421.34, 90 449.25, and 96 476.72 MHz, respectively. Calculate the rotational constant and bond length of CuBr.
In a study of the rotational spectrum of the linear Fe CO radical, K. Tanaka, M. Shirasaka, and T. Tanaka (J. Chem. Phys. 106,6820 (1997)) report the following J+ I (,- J transitions:
J 24 25 26 27
28 29 V/m-1 214777.7 223379.0 231981.2 240584.4
249188.5 257793.5 Evaluate the rotational constant of the molecule. Also, estimate the value of J for the most highly populated rotational energy level at 298 K and at 100 K.
Predict the shape of the nitronium ion, NO from its Lewis structure and the VSEPR model. It has one Raman active vibrational mode at 1400 cm-1, two strong IR active modes at 2360 and 540 cm-1, and one weak IR mode at 3735 cm-I Are these data consistent with the predicted shape of the molecule? Assign the vibrational wave numbers to the modes from which they arise.
The HCI molecule is quite well described by the Morse potential with De = 5.33 eV, V = 2989.7 cm-1, and XV = 52.05 cm-I. Assuming that the potential is unchanged on deuteration, predict the dissociation energies (Do) of
(a) HCI,
(b) DCI.
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