Exercises. 1.1 The power delivered to a photodetector which collects 8.0 10 7 photons in 3.8 ms from monochromatic light is 0.72 microwatt. What is the frequency of the light? 1.2 The speed of a proton is 4.5 10 5 m s 1. If the indeterminacy of the momentum of the proton is to be reduced to 0.0100%, what indeterminacy in the location of the proton must be tolerated? 1.3 A particle with mass 6.65 10 27 kg is confined to an infinite square well of width L. The energy of the third level is 2.00 10 24 J. Calculate the value of L. 1.4 Calculate the spacing between the fourth and fifth energy levels for a mass of 3.3 10 27 kg in a one-dimensional box with length 5.0 nm. 1.5 Calculate the energy per photon, and the energy per mole of photons, when their wavelength is (a) 600 nm (red), (b) 550 nm (yellow), (c) 400 nm (blue), (d) 200 nm (ultraviolet), (e) 150 pm (X-ray), (f) 1 cm (microwave). 1.6 What are the momenta of the photons in the last Problem? What speed would a stationary hydrogen atom attain if the photon collided with it and was absorbed? 1.7 The peak in the sun s emitted energy occurs at about 480 nm. Assuming it to behave as a black-body emitter, what is the temperature of the surface? 1.8 The work function for caesium is 2.14 ev. What is the kinetic energy and the speed of the electrons emitted when the metal is irradiated with light of wavelength (a) 700 nm, (b) 300 nm? 1.9 The photoelectric effect is the basis of the spectrosopic technique known as photoelectron spectroscopy. An X-ray photon of wavelength 150 pm ejects an electron from the inner shell of an atom. The speed of the latter was measured as 2.14 10 7 m s 1. Calculate the electron s binding energy. 1.10 Calculate the size of the quantum involved in the excitation of (a) an electronic motion of period 10 15 s, (b) a molecular vibration of period 10 14 s, (c) a pendulum of period 1 s. Express your results in kj mol 1. 1.11 What is the de Broglie wavelength of (a) a mass of 1 g travelling at 1 cm s 1, (b) the same, travelling at 100 km s 1, (c) a helium atom travelling at its r.m.s. speed at 25 C, (d) an electron accelerated from rest through a potential difference of 100 V, 1 kv, 100 kv? 1.12 An electron is confined to a linear region with a length of the order of the diameter of an atom ( 0.1 nm). What are the minimum uncertainties in its linear momentum and speed? 1.13 In order to use the Born interpretation directly it is necessary that the wavefunction is normalized to unity. Normalize to unity the following wavefunctions:
2 (a) sin(nπx/l) for the range 0 x L, (b) c, a constant in the range L x L, (c) e r/a in these dimensions, (d) xe r/2a in three-dimensional space. In order to integrate over the three dimensions you need to know that the volume element is dτ = r 2 dr sin θdθdφ, with 0 r, 0 θ π, 0 φ 2π. Use xn e ax dx = n!/a n+1. 1.14 A wavefunction for a particle confined to a one-dimensional box of length L is ψ = 2/L sin(πx/l). Let the box be 10 nm long. What is the probability of finding the particle (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 and 10.00 nm, (d) in the right half of the box, (e) in the central third of the box? 1.15 The wavefunction for the electron in the ground state of the hydrogen atom is ψ(r, θ, φ) = 1/πa 3 e r/a, where a = 53 pm. What is the probability of finding the electron somewhere inside a small sphere of radius 1.0 pm centred on the nucleus? Now suppose the same tiny sphere is moved to surround a point at a distance 53 pm from the nucleus: what is the probability that the electron is inside it? 1.16 Which of the following functions are eigenfunctions ofd the operator d/dx (a) exp(ikx), (b) cos kx, (c) k, (d) kx, (e) exp( αx 2 )? Give the eigenvalue where appropriate. 1.17 Which of the functions in the last Problem are also eigenfunctions of d 2 /dx 2, and which are eigenfunctions only of d 2 /dx 2? Give the eigenvalues where appropriate. 1.18 A particle is in a state defined by the wavefunction ψ = cos χ e ikx +sin χe ikx where χ is a parameter. What is the probability that the particle will be found with a linear momentum (a) +k, (b) k. What form would the wavefunction take if it were 90% certain that the particle had a linear momentum +k? 1.19 Evaluate the kinetic energy of the particle described by the wavefunction in Problem 1.18. 1.20 The expectation value of momentum is evaluated by using eqn (24) in chapter 1. What is the average momentum of a particle described by the following wavefunctions: (a) exp(ikx), (b) cos kx (c) exp( αx 2 ) each one in the range x 1.21 The commutator of two operators  and ˆB is written [Â, ˆB] and is defined as the difference  ˆB ˆBÂ. It can be evaluated by taking some convenient function ψ (which can be left unspecified) and evaluating both  ˆB and ˆBÂ, and finding the difference in the form Ĉψ. Then Ĉ is identified as the commutator [Â, ˆB] Quite often Ĉ turns out to be a simple numerical factor. An extremely important commutator is that of the operator for the components of momentum and the components of position. Find [ˆx, ŷ], [ˆx, ˆx], [ˆp x, ˆp y ], [ˆx, ˆp x ], [ˆx, ˆp y ]. (N.B. The hat -notation emphasises that we are dealing with operators.) Are x and y complementary observables? Are x and p x? Are x and p y? 1.22 One of the reasons why the commutator is so important is that it lets us identify at a glance the observables that are restricted by the uncertainty relation. Thus, if  and ˆB have a non-zero commutator, the observables A and
0.0 Exercises. 3 B cannot in general be determined simultaneously. Can p x and x be determined simultaneously? Can p x and y? Can the three components of position be specified simultaneously? 1.23 Another important commutator is that for the components of angular momentum. From classical theory L x = yp z zp y, L y = zp x xp z, L z = xp y yp x ; hence write the corresponding operators. Show that [ˆL x, ˆL y ] = i L z. Can L x and L y be determined in general simultaneously? 2.1 Consider a particle of mass m in a cubic box with edge L and find the degeneracy of the level with energy which is three times that of the lowest level. 2.2 Calculate the zero point energy of a harmonic oscillator consisting of a particle with mass 2.33 10 26 kg and a force constant 155 N m 1. 2.3 For a harmonic oscillator consisting of a particle with mass 1.33 10 25 kg, the difference in the energies of levels with vibrational quantum numbers 5 and 4 is 4.82 10 21 J. Calculate the force constant k. 2.4 Calculate the wavelength of light one quantum of which has the energy corresponding to the spacing between levels of a particle of mass 1.67 10 27 kg oscillating harmonically with a force constant of 855 N m 1. 2.5 A point mass of 6.35 10 26 kg rotates about a centre. Its angular momentum is described by l = 2. The energy of rotation of the mass is 2.47 10 23 J. Calculate the distance of the mass from the centre of rotation. 2.6 Consider an electron in a box of length L. Suppose t that the box represents a long conjugated molecule. What are the energy separations in J, kj mol 1, ev, and cm 1 between the levels (a) n = 2 and n = 1, (b) n = 6 and n = 5, in both cases taking L = 1 nm (10 Å). 2.7 A gas molecule in a flask has quantized translational energy levels, but how important are the effects of quantization? Calculate the separation between the lowest two energy levels for an oxygen molecule in a one-dimensional container of length 5 cm. At what value of the quantum number n does the energy of the molecule equal 1 kt, when T = 300 K? What is the separation of 2 this level from the one below? 2.8 Set up the Schrödinger equation for a particle of mass m in a three-dimensional square well with sides L x, L y, and L z (and volume V = L x L y L z ). Show that the wavefunction requires three quantum numbers for its specification, and that ψ(x, y, z) can be written as the product of three wavefunctions for one-dimensional square wells. Deduce an expression for the energy levels, and specialize it to the case of a cubic box of side L. 2.9 The wavefunction for the lowest state of a harmonic oscillator has the form of a Gaussian function, e gx2, where x is the displacement from equilibrium. Show that this function satisfies the Schrödinger equation for a harmonic oscillator, and find g in terms of the mass m and the force-constant k. What is the (zero-point) energy of the oscillator with this wavefunction? What is its minimum excitation energy?
4 2.10 Calculate the mean kinetic energy of a harmonic oscillator in the state v using the relations in Box 1, chapter 2. 2.11 The rotation of an HI molecule can be visualized as the orbiting of the hydrogen atom at a distance of 160 pm from a stationary iodine atom (this is quite a good approximation, but to be precise we would have to take into account the motion of both atoms around their joint centre of mass). Suppose that the molecule rotates only in a plane. How much energy (in kj mol 1 and cm 1 ) is needed to excite the stationary molecule into rotation? What, apart from zero, is the minimum angular momentum of the molecule? 2.12 Evaluate the z-component of angular momentum of a particle on a ring with (non-normalized) wavefunctions (a) e +iφ, (b) e 2iφ, (c) cos φ, and (d) cos χ e iφ + sin χ e iφ (where χ is a parameter). What is the particle s kinetic energy in each case? 2.13 What are the magnitudes of the angular momentum in the lowest four energy levels of a particle rotating on the surface of a sphere? How many states (as distinguished by the z-component of the angular momentum) are there in each case? 2.14 Calculate the energies of the first four rotational levels of HI, R = 160 pm, allowing it to rotate in three dimensions about its centre of mass. Express your answer in kj mol 1 and cm 1. Use I = µr 2, where µ is the reduced mass. 2.15 In the vector model of angular momentum a state with quantum numbers l, m l (or s, m s ) is represented by a vector length l(l + 1) units and of z-component m l units. Draw scale diagrams of the state of an electron with (a) s = 2, m s = 2, (b) l = 1, m l = +1, (c) l = 2, m l = 0. 2.16 Derive an expression for the half-angle of the apex of the cone of precession in terms of the quantum numbers l, m l (or s, m s ). What is its value for the α-state of an electron spin? Show that the minimum possible angle approaches zero as l approaches infinity. 3.1 The frequency of one of the lines in the Paschen series for H is 2.7415 10 14 Hz. Calculate the quantum number n 2 for the transition which produces this line. 3.2 One of the terms in the H atom has a value 27 414 cm 1. What is the value of the term with which it combines to produce light of wavelength 486.1 nm? 3.3 By differentiation, show that the radial part of the 2s hydrogen atomic orbital has a minimum and determine the value of r for this minimum. 3.4 At what values of r does the radial part of the 3s hydrogen orbital vanish? 3.5 An electron is known to have only the following values of total angular momentum quantum number: 3, 1, 1. What is the orbital angular momentum 2 2 of the electron? 3.6 Consider an electron in the ground state in the H atom, and calculate the value of r for which the probability density is 50% of its maximum value. 3.7 Consider an electron in the ground state in the H atom, and find the numerical values of r for which the radial distribution function is (a) 50%, (b) 75% of its maximum value.
0.0 Exercises. 5 3.8 A series of lines in the spectrum of atomic hydrogen lies at the wavelengths 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 energy is required to ionize the hydrogen atom when it is in the lower state involved in these transitions? 3.9 Calculate the mass of the deuteron on the basis that the first line of the Lyman series lies at 82 259.098 cm 1 for H, and at 82 281.476 cm 1 for D. 3.10 Positronium consists of an electron and a positron (same mass, opposite charge) orbiting around their common centre of mass. The broad features of the spectrum are therefore expected to resemble those of hydrogen, the differences arising largely from the mass relations. Where will the first three lines of the Balmer series of positronium lie? What is the binding energy in the ground state? 3.11 One of the most famous of the obsolete theories of the hydrogen atom was proposed by Bohr. It has been displaced by quantum mechanics, but by a remarkable coincidence (not the only one where the Coulomb potential is concerned) the energies it predicts agree exactly with those obtained from the Schrödinger equation. The Bohr atom is imagined as an electron circulating about a central nucleus. The Coulombic force of attraction, Ze 2 /4πε r 2, is balanced by the centrifugal effect of the circular orbiting motion of the electron. Bohr proposed that the angular momentum was limited to some integral multiple of = h/2π. When the two forces are balanced, the atom remains in a stationary state until it makes a spectral transition. Find the energies of the hydrogen-like atom on the basis of this model. 3.12 What features of the Bohr model are untenable in the light of quantum mechanics? How does the Bohr ground state differ from the actual ground state? If numerical agreement is exact, is there no experimental way of eliminating the Bohr model in favour of the quantum mechanical model? 3.13 A hydrogen-like 1s-orbital in an atom of atomic number Z is the exponential function ψ = (Z 3 /πa ) 1/2 exp( Zr/a. Form the radial distribution function and derive an expression for the most probable distance of the electron from the nucleus. What is its value in the case of (a) helium, (b) fluorine? 3.14 What is the magnitude of the angular momentum of an electron that occupies the following orbitals: (a) 1s, (b) 3s, (c) 3d, (d) 2p, (e) 3p? Give the number of radial and angular nodes in each case, and infer a rule. 3.15 Is an electron on average further away from the nucleus when it occupies a 2p-orbital in hydrogen than when it occupies a 2s-orbital? What is the most probable distance of an electron in a 3s-orbital from the nucleus? 3.16 Take the exponential 1s-orbital for the ground state of hydrogen and confirm that it satisfies the Schrödinger equation for the atom and that its energy is R H. Now modify the nuclear charge from e to Ze. What is the binding energy of the electron in the ion F 8+? 3.17 What is the orbital degeneracy of the level of the hydrogen atom that has the energy (a) R H, (b) R H /9, (c) R H /25? 3.18 How many electrons can enter the following sets of atomic orbitals: (a) 1s, (b) 3p, (c) 3d, (d) 6g?
6 3.17 Write the configurations of the first 18 elements of the periodic table (Z = 1 to 18). 3.18 An alternative method for measuring the ionization energy is to expose the atom to high energy monochromatic radiation, and to measure the kinetic energy or the speed of the electrons it ejects. When 58.4 nm light from a helium discharge lamp is directed into a sample of krypton, electrons are ejected with a velocity of 1.59 10 6 m s 1. The same radiation releases electrons from rubidium vapour with a speed of 2.45 10 6 m s 1. What are the ionization energies of the two species? 3.19 By how much does the ionization energy of deuterium differ from that of ordinary hydrogen atoms? 3.20 What values of the total angular momentum quantum number L, and magnitude of the total angular momentum, may a single electron with l = 3 possess? 3.21 Suppose an atom has (a) two, (b) three, (c) four electrons in different orbitals. What values of the total spin quantum number S may the atom possess? What would be the multiplicity in each case? 4.1 Give the ground state configuration of the following species: Li 2, Be 2, C 2. Give the bond order in each case. 4.2 Use the ground state configurations of B 2 and C 2 to predict which should have the greater bond dissociation energy. Explain. 4.3 One of the upper states of the C 2 molecule has the electron configuration (1sσ g ) 2 (1sσ u) 2 (2sσ g ) 2 (2sσ u) 2 (2pπ u ) 3 (2pπ g ) 1. Give the multiplicity and the parity of this state. 4.4 Use the electron configurations of NO and O 2 to predict which should have the shorter internuclear distance. 4.5 One of the sp 2 -hybrid orbitals has the form 1 (s + p ) 2. 3 Show that it is normalized. 4.6 The sp 2 -hybrid orbital that extends in a direction lying in the xy-plane and making an angle of 120 with the x-axis has the form ( 1 s p ) x 3 + p y 3 2 2. Using hydrogen 2s, 2p x, and 2p y orbitals, write the wave function explicitly. Show it is symmetric about the above direction by identifying the angular dependence of the hybrid orbital with that of the projection of the vector r on a line in the above direction. 4.7 The bond angle in H 2 S is 92.2. Estimate the per cent s-character in the H 2 S bonding hybrid orbital constructed from s, p x, and p y orbitals. Estimate the per cent s-character in the non-bonding hybrid orbital obtained in the above hybridization.
0.0 Exercises. 7 4.8 Verify the fact that the hydrogen 2p x and 2p y orbitals are orthogonal by showing that the φ portion of their overlap integral vanishes. 4.9 In molecules the atomic wavefunctions are not extended waves (like cos kx) but are localized around the nuclei. Their amplitudes spread into common regions of space. The expression in eqn. 134 (Ch. 4) gives the form of the superposition involved in the H 2 molecule. Take 1s A = exp ( r/a ) with r measured from A (and questions of normalization ignored at this stage), and 1s B = exp ( r/a ) with r measured from B. Plot the amplitude of the bonding molecular orbital along the internuclear axis for an internuclear distance of 106 pm. 4.10 Repeat the calculation for the antibonding combination 1s A 1s B, and notice how the node appears in the internuclear region. 4.11 Normalize the orbital (1s A ) + λ(1s B ) in terms of the parameter λ and the overlap integral. 4.12 One orbital is orthogonal to another if the integral ψ ψdτ = 0. Confirm that the 1sσ- and 1sσ -bonding and antibonding orbitals are mutually orthogonal, and find the orbital orthogonal to (1s A ) + λ(1s B ), with λ arbitrary. 4.13 Imagine a small electron-sensitive probe of volume 1 pm 3 being inserted into a H 2 molecule in its ground state. What is the probability that it will register the presence of an electron when it is inserted at the following positions: (a) at nucleus A, (b) at nucleus B, (c) half-way between A and B, (d) at a point arrived at by moving 20 pm from A along the axis towards B and then 10 pm perpendicularly? 4.14 What probabilities would occur if the same probe were inserted into a H 2 molecule in which the electron had just been excited into the antibonding orbital (and the molecule had not yet dissociated)? 4.15 Use Fig. 9 (Ch. 4) to give the configurations of the following species: H 2, N 2, O 2, CO, NO 4.16 Which of the species N 2, NO, O 2, C 2, F 2, CN would you expect to be stabilized (a) by the addition of an electron to form AB, (b) by ionization to AB +? 4.17 Draw the molecular orbital diagram for (a) CO, (b) XeF, and use the aufbau principle to put in the appropriate number of electrons. Is XeF + likely to be more stable than XeF? 4.18 What is the energy required to remove a potassium ion from its equilibrium distance of 294 pm in the K + Br ion pair? Assume negligible repulsive forces. 4.19 Derive an expression for the extent of promotion of a 1s 2 2p 3 atom needed in order to achieve three equi- valent and coplanar bonds making an angle 120 to each other. 4.20 Which of the following species do you expect to be linear: CO 2, NO 2, NO + 2, NO 2, SO 2, H 2 O, H 2 O 2+? Give reasons. 4.21 Which of the following species do you expect to be planar: NH 3, NH 2+ 3, CH 3, NO 3, CO+3 2? Give reasons.
8 4.22 Construct the molecular orbital diagram of (a) ethene, (b) ethyne (acetylene) on the basis that the molecules are formed from the appropriately hybridized CH 2 or CH fragments. 4.23 A simple model of conjugated polyenes allows their electrons to roam freely along the chain of atoms. The molecule is then regarded as a collection of independent particles confined to a box, and the molecular orbitals are taken to be the square-well wave functions. This is the free-electron molecular orbital (FEMO) approximation. As a first example of employing it, take the butadiene molecule with its four π-electrons and show that for its FEMO description we require the lowest two particle-in-a-box wavefunctions. Sketch the form of these two. 4.24 An advantage of the FEMO approach is that it lets us draw some quantitative conclusions with very little effort. What is the minimum excitation energy of butadiene? 4.25 Consider the FEMO description of the molecule CH 2 =CH CH=CH CH=CH CH=CH 2 and regard the electrons as being in a box of length 8R CC (as in this case, an extra 2R CC is often added at each end of the molecule). What is the minimum excitation energy of this molecule? What colour does the molecule absorb from white light? What colour does it then appear? Sketch the form of the uppermost filled orbital. Take R CC = 140 pm. 4.26 As an example of the application of the variation principle, take a trial function for the hydrogen atom of the form (a) e kr, (b) e kr2 and find the optimum value of k in each case. Observe that e kr has the form of the true ground state wavefunction, and that the energy it corresponds to is lower than that of the optimum form of e kr2. In the calculation, use the following form for the kinetic energy operator: ( ) T = 2 1 d 2 r, 2µ r dr 2 which is the radial part of the full operator. 4.27 Write down the secular determinant for (a) linear H 3, (b) cyclic H 3 using the Hückel approximation. Which of the two molecules is more stable? 4.28 Predict the electronic configuration of (a) the benzene negative ion, (b) the benzene cation. Estimate the π-bond energy in each case.