Download Exercises. 1.1 The power delivered to a photodetector which collects

Exercises.
1.1 The power delivered to a photodetector which collects 8.0 × 107 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 × 105 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 × 107 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 volumeRelement is dτ = r2 dr sin θdθdφ, with 0 ≤ r ≤ ∞, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π.
∞
Use −∞ xn e−ax dx = n!/an+1 .
1.14 A wavefunction
for a particle confined to a one-dimensional box of length
p
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
electron in the ground state of the hydrogen
p for the
−r/a◦
3
, where a◦ = 53 pm. What is the probability
atom is ψ(r, θ, φ) = 1/πa◦ e
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(−αx2 )? Give the eigenvalue where
appropriate .
1.17 Which of the functions in the last Problem are also eigenfunctions of d2 /dx2 ,
and which are eigenfunctions only of d2 /dx2 ? Give the eigenvalues where appropriate.
1.18 A particle is in a state defined by the wavefunction ψ = cos χ eikx +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(−αx2 ) each one in the
range −∞ ≤ x ≤ ∞
1.21 The commutator of two operators  and B̂ is written [Â, B̂] and is defined
as the difference ÂB̂ − B̂ Â. It can be evaluated by taking some convenient
function ψ (which can be left unspecified) and evaluating both ÂB̂ and B̂ Â,
and finding the difference in the form Ĉψ. Then Ĉ is identified as the commutator [Â, B̂] 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 [x̂, ŷ], [x̂, x̂], [p̂x , p̂y ], [x̂, p̂x ],
[x̂, p̂y ]. (N.B. The ’hat’-notation emphasises that we are dealing with operators.)
Are x and y complementary observables? Are x and px ? Are x and py ?
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 B̂ have a non-zero commutator, the observables A and
0.0 Exercises.
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B cannot in general be determined simultaneously. Can px and x be determined simultaneously? Can px 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 Lx = ypz − zpy , Ly = zpx − xpz , Lz = xpy − ypx ;
hence write the corresponding operators. Show that [L̂x , L̂y ] = i~Lz . Can Lx
and Ly 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 ` = 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 12 kT , when T = 300 K? What is the separation of
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 Lx , Ly , and Lz (and volume V = Lx Ly Lz ). 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
2
of a Gaussian function, e−gx , 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?
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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 eiφ +
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 = µR2 , where µ is the reduced mass.
2.15 In the vector model of angular momentum a state
pwith quantum numbers `, m` (or s, ms ) is represented by a vector length `(` + 1) units and of
z-component m` units. Draw scale diagrams of the state of an electron with (a)
s = 2, ms = 2, (b) ` = 1, m` = +1, (c) ` = 2, m` = 0.
2.16 Derive an expression for the half-angle of the apex of the cone of precession in terms of the quantum numbers `, m` (or s, ms ). What is its value for the
α-state of an electron spin? Show that the minimum possible angle approaches
zero as ` approaches infinity.
3.1 The frequency of one of the lines in the Paschen series for H is 2.7415 × 1014
Hz. Calculate the quantum number n2 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: 23 , 1, 21 . What is the orbital angular momentum
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.
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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, Ze2 /4πε◦ r2 , 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 −RH . Now modify the nuclear charge from e to Ze. What is the binding
energy of the electron in the ion F8+ ?
3.17 What is the orbital degeneracy of the level of the hydrogen atom that has
the energy (a) −RH , (b) −RH /9, (c) −RH /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 × 106 m s−1 . The same radiation releases electrons from
rubidium vapour with a speed of 2.45 × 106 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: Li2 , Be2 , C2 .
Give the bond order in each case.
4.2 Use the ground state configurations of B2 and C2 to predict which should
have the greater bond dissociation energy. Explain.
4.3 One of the upper states of the C2 molecule has the electron configuration
(1sσg )2 (1sσu∗ )2 (2sσg )2 (2sσu∗ )2 (2pπu )3 (2pπg )1 . Give the multiplicity and the parity of this state.
4.4 Use the electron configurations of NO and O2 to predict which should have
the shorter internuclear distance.
4.5 One of the sp2 -hybrid orbitals has the form
√ 1 √ s+p 2 .
3
Show that it is normalized.
4.6 The sp2 -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
r !
1
px
3
√
.
s − √ + py
2
3
2
Using hydrogen 2s, 2px , and 2py 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 H2 S is 92.2◦ . Estimate the per cent s-character in the H2 S
bonding hybrid orbital constructed from s, px , and py 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 2px and 2py 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 H2 molecule. Take 1sA = exp (−r/a◦ ) with r measured
from A (and questions of normalization ignored at this stage), and 1sB = 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 1sA − 1sB , and
notice how the node appears in the internuclear region.
4.11 Normalize the orbital (1sA ) + λ(1sB ) in terms of the parameter λ and the
overlap integral.
R
4.12 One orbital is orthogonal to another if the integral ψ 0∗ ψdτ = 0. Confirm
that the 1sσ- and 1sσ ∗ -bonding and antibonding orbitals are mutually orthogonal, and find the orbital orthogonal to (1sA ) + λ(1sB ), with λ arbitrary.
4.13 Imagine a small electron-sensitive probe of volume 1 pm3 being inserted
into a H2 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
H2 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: H2 ,
N2 , O2 , CO, NO
4.16 Which of the species N2 , NO, O2 , C2 , F2 , 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 1s2 2p3 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: CO2 , NO2 ,
−
2+
NO+
2 , NO2 , SO2 , H2 O, H2 O ? Give reasons.
4.21 Which of the following species do you expect to be planar: NH3 , NH2+
3 ,
−
2−
CH3 , NO3 , CO+3 ? 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 CH2 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
CH2 =CH−CH=CH−CH=CH−CH=CH2 and regard the electrons as being in a
box of length 8RCC (as in this case, an extra 2RCC 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 RCC = 140 pm.
4.26 As an example of the application of the variation principle, take a trial
2
function for the hydrogen atom of the form (a) e−kr , (b) e−kr 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
2
that of the optimum form of e−kr . In the calculation, use the following form
for the kinetic energy operator:
~2 1 d2
T =−
r,
2µ r dr2
which is the radial part of the full operator.
4.27 Write down the secular determinant for (a) linear H3 , (b) cyclic H3 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.