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Physical Chemistry 3rd edition Thomas Engel, Philip Reid - Solutions

Derive the ground-state term symbols for the following atoms or ions:a. Hb. F−c. Na+d. Sc
Use the transition frequencies shown in Example Problem 22.7 to calculate the energy (in joules and electron-volts) of the six levels relative to the 3s 2S1/2 level. State your answers with the correct number of significant figures.
Calculate the transition dipole moment,whereÎ¼z = ˆ’er cosÎ¸ for a transition from the 1s level to the 2s level in H. Show that this transition is forbidden. The integration is over r, Î¸, and ˆ…. = [v„(t)H=W»(t) dr
For a closed-shell atom, an anti-symmetric wave function can be represented by a single Slater determinant. For an open-shell atom, more than one determinant is needed. Show that the wave function for the MS = 0 triplet state of He 1s12s1 is a linear combination of two of the Slater determinants of
Classify the following functions as symmetric, antisymmetric, or neither in the exchange of electrons 1 and 2:a. [1s (1) 2s (2) + 2s (1) 1s (2)] [α (1) β (2) − β (1) α (2)]b. [1s (1) 2s (2) + 2s (1) 1s (2)] α (1) α (2)c. [1s (1) 2s (2) + 2s (1) 1s (2)] [α (1) β (2) + β (1) α (2)]d. [1s
You have commissioned a measurement of the second ionization energy from two independent research teams. You find that they do not agree and decide to plot the data together with known values of the first ionization energy. The results are shown here:The lowest curve is for the first ionization
In this problem, you will use the variational method to find the optimal 1s wave function for the hydrogen atom starting from the trial function Î¦(r) = eˆ’Î±rwith Î± as the variational parameter. You will minimizewith respect to Î±.a.
Show that the functions [α (1) β(2) + β (1) α (2)] / √2 and [α (1)β(2) − β (1)α (2)] 2 are eigenfunctions of Ŝ2total What is the eigenvalue in each case?
The operator for the square of the total spin of two electrons isGiven thatshow that Î±(1)Î± (2) and Î²(1)Î²(2) are eigenfunctions of the operator SÌ‚2total . What is the eigenvalue in each case? Sotal = ($ + $;}² = $} + $3 +
In this problem you will show that the charge density of the filled n = 2, l = 1 subshell is spherically symmetrical and that therefore L = 0. The angular distribution of the electron charge is simply the sum of the squares of the magnitude of the angular part of the wave functions for l = 1 and ml
In this problem you will prove that the ground-state energy for a system obtained using the variational method is greater than the true energy.a. The approximate wave function Î¦ can be expanded in the true (but unknown) eigenfunctions Ïˆ n of the total energy operator in the
In this problem we represent the spin eigenfunctions and operators as vectors and matrices.a. The spin eigenfunctions are often represented as the column vectorsShow that Î± and Î² are orthogonal using this representation.b. If the spin angular momentum operators are
Calculate the angles that a spin angular momentum vector for an individual electron can make with the z axis.
Is ψ (1, 2) = 1s(1)α (1)1s(2)β (2) + 1s(2)α (2)1s(1)β (1) an eigenfunction of the operator Ŝz? If so, what is its eigenvalue MS?
Would the trial wave functionhave been a suitable choice for the calculations carried out in Section 21.4? Justify your answer. но в3) 48) х Ф(х) %3D 0< x< a
Why is the s, p, d, … nomenclature derived for the H atom also valid for many-electron atoms?
The angular functions, Θ(θ )Φ(φ ), for the one-electron Hartree–Fock orbitals are the same as for the hydrogen atom, and the radial functions and radial probability functions are similar to those for the hydrogen atom. The contour coloring is explained in the caption to Figure 20.7. The
How can you tell if one basis set is better than another in calculating the total energy of an atom?
Is there a physical reality associated with the individual entries of a Slater determinant?
Show that calculating Eu in the manner described by Equation (23.21) gives the result Eu = (Haa − Hab) (1 − Sab).
Why is the total energy of a many-electron atom not equal to the sum of the orbital energies for each electron?
What is the functional dependence of the 1s orbital energy on Z in Figure 21.7? Check your answer against a few data points.Figure 21.7 0.5 4s Зd -0.5 зр 3s 2p -1.5 2s 15 -2.5 35 30 25 20 10 Atomic number -15
The total energy eigenvalues for the hydrogen atom are given by En= ˆ’e2 / (8Ï€Îµ0a0n), n = 1, 2, 3, 4,€¦, and the three quantum numbers associated with the total energy eigenfunctions are related by n = 1, 2, 3, 4,.....; l = 0,1 , 2, 3, ..., n
Calculate the mean value of the radius 〈r〉 at which you would find the electron if the H atom wave function is ψ210(r, θ, ∅).
The transitionhas two lines given by vÌ‚ = 25354.8 cmˆ’1 and vÌ‚ = 25242.7 cmˆ’1. The transitionhas three lines given by vÌ‚ = 32444.8 cmˆ’1, vÌ‚ = 32334.0 cmˆ’1, and vÌ‚ =
List the quantum numbers L and S that are consistent with the following terms:a. 4Sb. 4Gc. 3Pd. 2D
The Grotrian diagram in Figure 22.7 shows a number of allowed electronic transitions for He. Which of the following transitions shows multiple spectral peaks due to a splitting of terms into levels? How many peaks are observed in each case? Are any of the following transitions between energy levels
In the Na absorption spectrum, the following transitions are observed:4p2P → 3s2Sλ = 330.26 nm3p2P → 3s2Sλ = 589.593 nm, 588.996 nm5s2S → 3p2Pλ = 616.073 nm, 615.421 nmCalculate the energies of the 4p2P and 5s2S states with respect to the 3s2S ground state.
A general way to calculate the number of states that arise from a given configuration is as follows. Calculate the combinations of ml and ms for the first electron, and call that number n. The number of combinations used is the number of electrons, which we call m. The number of unused combinations
Given that the levels in the 3P term for carbon have the relative energies (expressed in wave numbers) of 3P1 – 3P0 = 16.4 cm-1 and 3P2 – 3P1 = 27.1 cm-1, calculate the ratio of the number of C atoms in the 3P2 and 3P0 levels at 200 and 1000. K.
Derive the ground-state term symbols for the following atoms or ions:a. Fb. Nac. P
The ground-state level for the phosphorous atom is 4S3/2. List the possible values of L, ML, S, MS, J, and MJ consistent with this level.
The following images show contours of constant electron density for H2calculated using the methods described in Chapter 26. The values of electron density are (a) 0.10, (b) 0.15, (c) 0.20, (d) 0.25, (e) 0.30 electron/ a30.a.b.c.d.e.a. Explain why the apparent size of the H2 molecule as approximated
Consider the molecular electrostatic potential map for the NH3molecule shown here. Is the hydrogen atom (shown as a white sphere) an electron acceptor or an electron donor in this molecule?
Write the Slater determinant for the ground-state configuration of Be.
The exact energy of a ground-state He atom is €“79.01 eV. Calculate the correlation energy and the ratio of the correlation energy to the total energy for He using the results in Table 21.1. TABLE 21.1 Total Energy and 1s Orbital Energy for He for Three Different Basis Sets Used to
The ground-state wave function of Li2+ is π-1/2 (Z/a0)3/2 e-Zr/a0, where Z is the nuclear charge. Calculate the expectation value of the potential energy for Li2+.
Calculate the position of the maximum in the radial distribution function for Li 2+ in its ground state using the wave function in P21.13.
Why is the magnitude of the electron affinity for a given element smaller than the magnitude of the first ionization energy? в с Na Ne Не Li Be Element First Ionization н N 13.6 24.6 5.4 13.6 9.3 8.3 14.5 17.4 21.6 Energy Electron 11.3 5.1 0.3
The electron affinities of He, Be, and Ne are negative, meaning that the negative ion is less stable than the neutral atom. Explain why this is so for these three elements.
Are the effective nuclear charges listed in Figure 21.13 helpful in explaining the trend in the first ionization energy with increasing atomic number? Explain your answer. Н Не 1s 1 1s 1.69 Li 2s 1.28 2s 1.91 Be Ne 2s 2.58 2s 3.22 2s 3.85 2s 4.49 2s 5.13 2s 5.76 2p 2.42 2p 3.14 2p 3.83 2p 4.45
Are the effective nuclear charges listed in Figure 21.13 helpful in explaining the trend in the electron affinity with increasing atomic number? Explain your answer. Н Не 1s 1 1s 1.69 Li 2s 1.28 2s 1.91 Be Ne 2s 2.58 2s 3.22 2s 3.85 2s 4.49 2s 5.13 2s 5.76 2p 2.42 2p 3.14 2p 3.83 2p 4.45 2p 5.10
Explain why the electron affinity of N is negative.
Explain why the first ionization energy and electron affinity for F are larger than for O.
Justify the statement that the Coulomb integral J defined in Equation (22.20) is positive by explicitly formulating the integral that describes the interaction between two negative classical charge clouds.
Without invoking equations, explain why the energy of the triplet state is lower than that of the singlet state for He in the 1s12s1 configuration.
How can the width of a laser line be less than that determined by Doppler broadening?
Why is an electronically excited atom more reactive than the same ground-state atom?
Why is atomic absorption spectroscopy more sensitive in many applications than atomic emission spectroscopy?
Why does the Doppler effect lead to a shift in the wavelength of a star, but to a broadening of a transition in a gas?
Why are n, l, ml, and ms not good quantum numbers for many-electron atoms?
Write an equation giving the relationship between the Rydberg constant for H and for Li2+.
Can the individual states in Table 22.1 be distinguished experimentally? States and Terms for the np? Configuration TABLE 22.1 M, =m,1 + m,2 Term ML=mn + M2 m;1 m,2 'D -2 1/2 -1/2 -1 3p -1/2 -1/2 1/2 -1/2 1/2 -1 -1/2 1/2 1/2 'D, ³P -1 -1 3p 1/2 -1/2 -1/2 1/2 'D, ³P, 's 1/2 -1/2 1/2 -1/2 1 -1 -1/2
How is it possible to determine the L and S values of a term knowing only the ML and MS values of the states?
What is the origin of the chemical shift in XPS?
Why are two medium-energy photons rather than one high-energy photon used in laser isotope separation?
Why does one need to put a sample in a vacuum chamber to study it with XPS or AES?
Why is XPS a surface-sensitive technique?
Explain the direction of the chemical shifts for Fe(0), Fe(II), and Fe(III) in Figure 22.20. Fe 2p at normal emission Fe 2p32 Fe 2p12 Fe (0) Fe/MgO(001) Fe(0) Fe;OM9O(001) Fe(ll & II) Fe(ll) Fe(lI) y-Fe,O3/MgO(001) Fe(llI) "shake up" "shake up" a-Fe,Og/a-Al2Og (0001) Fe(llI) 740 730 720 710 700
The principal line in the emission spectrum of sodium is yellow. On close examination, the line is seen to be a doublet with wavelengths of 589.0 and 589.6 nm. Explain the source of this doublet.
The absorption spectrum of the hydrogen atom shows lines at 5334, 7804, 9145, 9953, and 10,478 cm−1. There are no lower-frequency lines in the spectrum. Use the graphical methods discussed in Example Problem 22.6 to determine ninitial and the ionization energy of the hydrogen atom in this state.
Using Table 22.3, which lists the possible terms that arise from a given configuration, and Hund€™s rules, write the term symbols for the ground state of the atoms H through F in the form(2s+1)LJ. Possible Terms for Indicated Configurations TABLE 22.3 Electron Configuration Term Symbol 25
In this problem you will supply the missing steps in the derivation of the formula Esinglet= E1s+ E2s+ J + K for the singlet level of the 1s12s1configuration of He.a. Expand Equation (22.17) to obtainb. Starting from the equations 1 2 HË† i1s(i) = E s1s(i) and HË† i 2s(i) = E s
What J values are possible for a 6H term? Calculate the number of states associated with each level and show that the total number of states is the same as that calculated from the term symbol.
Using Table 22.3, which lists the possible terms that arise from a given configuration, and Hund€™s rules, write the configurations and term symbols for the ground state of the ions Fˆ’and Ca2+in the form(2s+1)LJ. Possible Terms for Indicated Configurations TABLE 22.3 Term
The Doppler broadening in a gas can be expressed aswhere M is the molar mass. For the sodiumtransition, Î½0 5.0933 Ã— 1014 s-1. Calculate Î”Î½ and Î”Î½ /Î½0 at 500.K. Av = (2vo/c)/2 In2(RT/M)) Bp P3/2
Calculate the transition dipole moment, for a transition from the 1s level to the 2pz level in H. Show that this transition is allowed. The integration is over r, Î¸, and ˆ…. Usefor the 2pz wave function. = [wm(7)H,V„(7) dr, where u. = -er cose 3/2 V2о(г.0,0) -
Consider the 1s np 3P → 1s nd 3D transition in He. Draw an energy-level diagram, taking the spin-orbit coupling that splits terms into levels into account. Into how many levels does each term split? The selection rule for transitions in this case is ΔJ = 0, ± 1. How many transitions
Atomic emission experiments of a mixture show a calcium line at 422.673 nm corresponding to a 1P1 → 1S0 transition and a doublet due to potassium 2P3/2 → 2S1/2 and 2P1/2 → 2S1/2 transitions at 764.494 and 769.901nm, respectively.a. Calculate the ratio gupper /glower for
How many ways are there to place three electrons into an f sub-shell? What is the ground-state term for the f3 configuration, and how many states are associated with this term? See Problem P 22.36.
Calculate the wavelengths of the first three lines of the Lyman, Balmer, and Paschen series, and the series limit (the shortest wavelength) for each series.
The Lyman series in the hydrogen atom corresponds to transitions that originate from the n = 1 level in absorption or that terminate in the n = 1 level for emission. Calculate the energy, frequency (in inverse seconds and inverse centimeters), and wavelength of the least and most energetic
The inelastic mean free path of electrons in a solid, λ, governs the surface sensitivity of techniques such as AES and XPS. The electrons generated below the surface must make their way to the surface without losing energy in order to give elemental and chemical shift information. An empirical
The effective path length that an electron travels before being ejected into the vacuum is related to the depth below the surface at which it is generated and the exit angle by d = λcosθ , where λ is the inelastic mean free path and θ is the angle between the surface normal and the exit
List the allowed quantum numbers ml and ms for the following subshells and determine the maximum occupancy of the subshells:a. 2pb. 3dc. 4 fd. 5g
What are the levels that arise from a 4F term? How many states are there in each level?
As discussed in Chapter 20, in a more exact solution of the SchrÃ¶dinger equation for the hydrogen atom, the coordinate system is placed at the center of mass of the atom rather than at the nucleus. In that case, the energy levels for a one-electron atom or ion of nuclear charge Z are
Derive the ground-state term symbols for the following configurations:a. d5b. f3c. p4
Calculate the terms that can arise from the configuration np1n′p1, n ≠ n′. Compare your results with those derived in the text for np2. Which configuration has more terms and why?
Calculate the terms that can arise from the configuration np1n′p1, n ≠ n′. Compare your results with those derived in the text for np2. Which configuration has more terms and why?
Show using an example that the following two formulations of the Pauli exclusion principle are equivalent:a. Wave functions describing a many-electron system must change sign under the exchange of any two electrons.b. No two electrons may have the same values for all four quantum numbers.
Are the elements of a basis set observable in an experiment? Explain your reasoning.
Explain why shielding is more effective by electrons in a shell of lower principal quantum number than by electrons having the same principal quantum number.
The angular functions, Î˜(Î¸ )Î¦(Ï† ), for the one-electron Hartree€“Fock orbitals are the same as for the hydrogen atom, and the radial functions and radial probability functions are similar to those for the hydrogen atom. The contour
Using the result of Problem P20.12, calculate the probability of finding the electron in the 1s state outside a sphere of radius 0.75a0, 2.5a0, and 4.5a0.
How is the effective nuclear charge related to the size of the basis set in a Hartree Fock calculation?
There are more electrons in the n = 4 shell than for the n = 3 shell in krypton. However, the peak in the radial distribution in Figure 21.6 is smaller for the n = 4 shell than for the n = 3 shell. Explain this fact.
Why does the effective nuclear charge for the 1s orbital increase by 0.99 in going from oxygen to fluorine but only increases by 0.65 for the 2p orbital?
Show that ψ2px (r, θ, ∅) and ψ2py (r, θ, ∅) can be written in the form Nxe−r/2a0 and N′ye−r / 2a0, where N and N′ are normalization constants.
How many radial and angular nodes are there in the following H orbitals?a. b.c.d. 42p,(r. 0, 4) b28(r)
Calculate r and the most probable value of r for the H atom in its ground state. Explain why they differ with a drawing.
Show that the total energy eigenfunctions ψ210 (r, θ, ∅) and ψ211 (r, θ, ∅) are orthogonal. Do you have to integrate over all three variables to show that the functions are orthogonal?
Is the total energy wave functionAn eigenfunction of any other operators? If so, which ones? What are the eigenvalues? 2. 572 W310(r. 0, ø) cose -r/3a, .2 81 af do ao
As will be discussed in Chapter 21, core electrons shield valence electrons so that they experience an effective nuclear charge Zeff rather than the full nuclear charge. Given that the first ionization energy of Li is 5.39 eV, use the formula in Problem P20.20 to estimate the effective nuclear
Show that the total energy eigenfunctions ψ100 (r) and ψ200 (r) are orthogonal.
Show by substitution that Î¨100(r, Î¸, ˆ…) = 1/ˆšÏ€(1/a0)3/2e-r/a0is a solution ofWhat is the eigenvalue for total energy? Use the relationa0 = Îµ0h2 / (Ï€mee2). aự (r, 0, Ø) ,2 ay (r, 0. Ø) 1 sıne p sine de ae ar
Calculate the average value of the kinetic and potential energies for the H atom in its ground state.
Locate the radial and angular nodes in the H orbitals ψ3px (r, θ, ∅) and ψ3pz (r, θ, ∅).
The energy levels for ions with a single electron such as He+, Li2+, and Be3+ are given by En = −Z2e2 / (8πε0a0n2), n = 1, 2, 3, 4, . . . Calculate the ionization energies of H, He+, Li2+, and Be3+ in their ground states in units of electron-volts (eV).
Calculate the expectation value of the moment of inertia of the H atom in the 2s and 2pz states in terms of μ and a0.

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