What atomic terms are possible for the electron configuration ns1nd1 Which term is

Describe the separation of variables procedure as it is applied to simplify the description of a hydrogenic atom free to move through space.

List and describe the significance of the quantum numbers needed to specify the internal state of a hydrogenic atom.

Specify and account for the selection rules for transitions in hydrogenic atoms.

Explain the significance of (a) a boundary surface and (b) the radial distribution function for hydrogenic orbitals.

Outline the electron configurations of many-electron atoms in terms of their location in the periodic table.

Describe and account for the variation of first ionization energies along Period 2 of the periodic table.

Describe the orbital approximation for the wavefunction of a manyelectron atom. What are the limitations of the approximation?

Explain the origin of spinorbit coupling and how it affects the appearance of a spectrum.

When ultraviolet radiation of wavelength 58.4 nm from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of 1.59 Mm s1. Calculate the ionization energy of krypton.

When ultraviolet radiation of wavelength \(58.4 \mathrm{~nm}\) from a helium lamp is directed on to a sample of xenon, electrons are ejected with a speed of \(1.79 \mathrm{Mm} \mathrm{s}^{-1}\). Calculate the ionization energy of xenon.

By differentiation of the 2s radial wavefunction, show that it has two extrema in its amplitude, and locate them.

By differentiation of the 3s radial wavefunction, show that it has three extrema in its amplitude, and locate them.

Locate the radial nodes in the 3s orbital of an H atom.

Locate the radial nodes in the 4p orbital of an H atom where, in the notation of Table 10.1, the radial wavefunction is proportional to 20 10 +2.

The wavefunction for the ground state of a hydrogen atom is Ner/a0. Determine the normalization constant N.

The wavefunction for the 2s orbital of a hydrogen atom is N(2 r/a0)er/2a0. Determine the normalization constant N.

Calculate the average kinetic and potential energies of an electron in the ground state of a hydrogen atom.

Calculate the average kinetic and potential energies of a 2s electron in a hydrogenic atom of atomic number Z.

Write down the expression for the radial distribution function of a 2s electron in a hydrogenic atom and determine the radius at which the electron is most likely to be found.

Write down the expression for the radial distribution function of a 3s electron in a hydrogenic atom and determine the radius at which the electron is most likely to be found.

Write down the expression for the radial distribution function of a 2p electron in a hydrogenic atom and determine the radius at which the electron is most likely to be found.

Write down the expression for the radial distribution function of a 3p electron in a hydrogenic atom and determine the radius at which the electron is most likely to be found.

What is the orbital angular momentum of an electron in the orbitals (a) 1s, (b) 3s, (c) 3d? Give the numbers of angular and radial nodes in each case.

What is the orbital angular momentum of an electron in the orbitals (a) 4d, (b) 2p, (c) 3p? Give the numbers of angular and radial nodes in each case.

Calculate the permitted values of j for (a) a d electron, (b) an f electron.

Calculate the permitted values of j for (a) a p electron, (b) an h electron.

An electron in two different states of an atom is known to have j = 32 and 12 . What is its orbital angular momentum quantum number in each case?

What are the allowed total angular momentum quantum numbers of a composite system in which j1 = 5 and j2 = 3?

State the orbital degeneracy of the levels in a hydrogen atom that have energy (a) hcRH; (b) 19 hcRH; (c) 1 25hcRH.

State the orbital degeneracy of the levels in a hydrogenic atom (Z in parentheses) that have energy (a) 4hcRatom (2); (b) 14 hcRatom (4), and (c) hcRatom (5).

What information does the term symbol 1D2 provide about the angular momentum of an atom?

What information does the term symbol 3F4 provide about the angular momentum of an atom?

At what radius does the probability of finding an electron at a point in the H atom fall to 50 per cent of its maximum value?

At what radius in the H atom does the radial distribution function of the ground state have (a) 50 per cent, (b) 75 per cent of its maximum value?

Which of the following transitions are allowed in the normal electronic emission spectrum of an atom: (a) 2s1s, (b) 2p1s, (c) 3d2p?

Which of the following transitions are allowed in the normal electronic emission spectrum of an atom: (a) 5d2s, (b) 5p3s, (c) 6p4f ?

(a) Write the electronic configuration of the Ni2+ ion. (b) What are the possible values of the total spin quantum numbers S andMS for this ion?

(a) Write the electronic configuration of the \(\mathrm{V}^{2+}\) ion. (b) What are the possible values of the total spin quantum numbers S and \(M_S\) for this ion?

Suppose that an atom has (a) 2, (b) 3 electrons in different orbitals. What are the possible values of the total spin quantum number S? What is the multiplicity in each case?

Suppose that an atom has (a) 4, (b) 5 electrons in different orbitals. What are the possible values of the total spin quantum number S? What is the multiplicity in each case?

What atomic terms are possible for the electron configuration ns1nd1? Which term is likely to lie lowest in energy?

What atomic terms are possible for the electron configuration np1nd1? Which term is likely to lie lowest in energy?

What values of J may occur in the terms (a) 1S, (b) 2P, (c) 3P? How many states (distinguished by the quantum number MJ) belong to each level?

What values of J may occur in the terms (a) 3D, (b) 4D, (c) 2G? How many states (distinguished by the quantum number MJ) belong to each level?

Give the possible term symbols for (a) Li [He]2s1, (b) Na [Ne]3p1.

Give the possible term symbols for (a) \(\mathrm{Sc}[\mathrm{Ar}] 3 d^1 4 s^2\), (b) \(\operatorname{Br}[\mathrm{Ar}] 3 d^{10} 4 s^2 4 p^5\).

The Zeeman effect is the modification of an atomic spectrum by the application of a strong magnetic field. It arises from the interaction between applied magnetic fields and the magnetic moments due to orbital and spin angular momenta (recall the evidence provided for electron spin by the SternGerlach experiment, Section 9.8). To gain some appreciation for the socalled normal Zeeman effect, which is observed in transitions involving singlet states, consider a p electron, with l = 1 and ml = 0, }1. In the absence of a magnetic field, these three states are degenerate. When a field of magnitude B is present, the degeneracy is removed and it is observed that the state with ml = +1 moves up in energy by BB, the state with ml = 0 is unchanged, and the state with ml = 1 moves down in energy by BB, where B = e$/2me = 9.274 1024 J T 1 is the Bohr magneton (see Section 15.1). Therefore, a transition between a 1S0 term and a 1P1 term consists of three spectral lines in the presence of a magnetic field where, in the absence of the magnetic field, there is only one. (a) Calculate the splitting in reciprocal centimetres between the three spectral lines of a transition between a 1S0 term and a 1P1 term in the presence of a magnetic field of 2 T (where 1 T = 1 kg s2 A1). (b) Compare the value you calculated in (a) with typical optical transition wavenumbers, such as those for the Balmer series of the H atom. Is the line splitting caused by the normal Zeeman effect relatively small or relatively large?

In 1976 it was mistakenly believed that the first of the superheavy elements had been discovered in a sample of mica. Its atomic number was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)

What is the most probable point (not radius) at which a 2p electron will be found in the hydrogen atom?

Show by explicit integration that (a) hydrogenic 1s and 2s orbitals, (b) 2px and 2py orbitals are mutually orthogonal.

Explicit expressions for hydrogenic orbitals are given in Tables 10.1 and 9.3. (a) Verify both that the 3px orbital is normalized (to 1) and that 3px and 3dxy are mutually orthogonal. (b) Determine the positions of both the radial nodes and nodal planes of the 3s, 3px, and 3dxy orbitals. (c) Determine the mean radius of the 3s orbital. (d) Draw a graph of the radial distribution function for the three orbitals (of part (b)) and discuss the significance of the graphs for interpreting the properties of many-electron atoms. (e) Create both xy-plane polar plots and boundary surface plots for these orbitals. Construct the boundary plots so that the distance from the origin to the surface is the absolute value of the angular part of the wavefunction. Compare the s, p, and d boundary surface plots with that of an f-orbital; e.g. f x(5z2 r2) sin (5 cos2 1)cos .

Determine whether the px and py orbitals are eigenfunctions of lz. If not, does a linear combination exist that is an eigenfunction of lz?

Show that lz and l2 both commute with the hamiltonian for a hydrogen atom. What is the significance of this result?

The size of an atom is sometimes considered to be measured by the radius of a sphere that contains 90 per cent of the charge density of the electrons in the outermost occupied orbital. Calculate the size of a hydrogen atom in its ground state according to this definition. Go on to explore how the size varies as the definition is changed to other percentages, and plot your conclusion.

Some atomic properties depend on the average value of 1/r rather than the average value of r itself. Evaluate the expectation value of 1/r for (a) a hydrogen 1s orbital, (b) a hydrogenic 2s orbital, (c) a hydrogenic 2p orbital.

One of the most famous of the obsolete theories of the hydrogen atom was proposed by Bohr. It has been replaced 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 Schrdinger equation. In the Bohr atom, an electron travels in a circle around the nucleus. The Coulombic force of attraction (Ze2/40r2) is balanced by the centrifugal effect of the orbital motion. Bohr proposed that the angular momentum is limited to integral values of $. When the two forces are balanced, the atom remains in a stationary state until it makes a spectral transition. Calculate the energies of a hydrogenic atom using the Bohr model.

The Bohr model of the atom is specified in Problem 10.18. What features of it are untenable according to quantum mechanics? How does the Bohr ground state differ from the actual ground state? Is there an experimental distinction between the Bohr and quantum mechanical models of the ground state?

Atomic units of length and energy may be based on the properties of a particular atom. The usual choice is that of a hydrogen atom, with the unit of length being the Bohr radius, a0, and the unit of energy being the (negative of the) energy of the 1s orbital. If the positronium atom (e+,e) were used instead, with analogous definitions of units of length and energy, what would be the relation between these two sets of atomic units?

Some of the selection rules for hydrogenic atoms were derived in Justification 10.4. Complete the derivation by considering the x- and ycomponents of the electric dipole moment operator.

SternGerlach splittings of atomic beams are small and require either large magnetic field gradients or long magnets for their observation. For a beam of atoms with zero orbital angular momentum, such as H or Ag, the deflection is given by x = (BL2/4EK)dB/dz, where B is the Bohr magneton (Problem 10.9), L is the length of the magnet, EK is the average kinetic energy of the atoms in the beam, and dB/dz is the magnetic field gradient across the beam. (a) Use the MaxwellBoltzmann velocity distribution to show that the average translational kinetic energy of the atoms emerging as a beam from a pinhole in an oven at temperature T is 2kT. (b) Calculate the magnetic field gradient required to produce a splitting of 1.00 mm in a beam of Ag atoms from an oven at 1000 K with a magnet of length 50 cm.

The wavefunction of a many-electron closed-shell atom can expressed as a Slater determinant (Section 10.4b). A useful property of determinants is that interchanging any two rows or columns changes their sign and therefore, if any two rows or columns are identical, then the determinant vanishes. Use this property to show that (a) the wavefunction is antisymmetric under particle exchange, (b) no two electrons can occupy the same orbital with the same spin.

Hydrogen is the most abundant element in all stars. However, neither absorption nor emission lines due to neutral hydrogen are found in the spectra of stars with effective temperatures higher than \(25000 \mathrm{~K}\). Account for this observation.

The distribution of isotopes of an element may yield clues about the nuclear reactions that occur in the interior of a star. Show that it is possible to use spectroscopy to confirm the presence of both 4He+ and 3He+ in a star by calculating the wavenumbers of the n = 3n = 2 and of the n = 2n = 1 transitions for each isotope.

Highly excited atoms have electrons with large principal quantum numbers. Such Rydberg atoms have unique properties and are of interest to astrophysicists . For hydrogen atoms with large n, derive a relation for the separation of energy levels. Calculate this separation for n = 100; also calculate the average radius, the geometric cross-section, and the ionization energy. Could a thermal collision with another hydrogen atom ionize this Rydberg atom? What minimum velocity of the second atom is required? Could a normal-sized neutral H atom simply pass through the Rydberg atom leaving it undisturbed? What might the radial wavefunction for a 100s orbital be like?

The spectrum of a star is used to measure its radial velocity with respect to the Sun, the component of the stars velocity vector that is parallel to a vector connecting the stars centre to the centre of the Sun. The measurement relies on the Doppler effect, in which radiation is shifted in frequency when the source is moving towards or away from the observer. When a star emitting electromagnetic radiation of frequency moves with a speed s relative to an observer, the observer detects radiation of frequency receding = f or approaching = /f, where f = {(1 s/c)/(1 + s/c)}1/2 and c is the speed of light. It is easy to see that receding < and a receding star is characterized by a red shift of its spectrum with respect to the spectrum of an identical, but stationary source. Furthermore, approaching > and an approaching star is characterized by a blue shift of its spectrum with respect to the spectrum of an identical, but stationary source. In a typical experiment, is the frequency of a spectral line of an element measured in a stationary Earth-bound laboratory from a calibration source, such as an arc lamp. Measurement of the same spectral line in a star gives star and the speed of recession or approach may be calculated from the value of and the equations above. (a) Three Fe I lines of the star HDE 271 182, which belongs to the Large Magellanic Cloud, occur at 438.882 nm, 441.000 nm, and 442.020 nm. The same lines occur at 438.392 nm, 440.510 nm, and 441.510 nm in the spectrum of an Earth-bound iron arc. Determine whether HDE 271 182 is receding from or approaching the Earth and estimate the stars radial speed with respect to the Earth. (b) What additional information would you need to calculate the radial velocity of HDE 271 182 with respect to the Sun?

The d-metals iron, copper, and manganese form cations with different oxidation states. For this reason, they are found in many oxidoreductases and in several proteins of oxidative phosphorylation and photosynthesis (Impact I7.2 and I23.2). Explain why many d-metals form cations with different oxidation states.

Thallium, a neurotoxin, is the heaviest member of Group 13 of the periodic table and is found most usually in the +1 oxidation state. Aluminium, 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, I3, is the minimum energy needed to remove an electron from the doubly charged cation: E2+(g)E3+(g) + e(g), I3 = E(E3+ ) E(E2+). For data, see the links to databases of atomic properties provided in the texts web site.