# Chapter 1

```Chapter 1
What is a wavefunction?
Wavefunctions are the permitted solutions of the Schrödinger equation describing
electrons on atoms as waves.
They give the energy and probability of location of the
electron in any region around the nucleus.
The wavefunctions for hydrogen-like atoms,
with only one electron can be found analytically.
There is only one wavefunction for the
lowest energy state but the states of higher energy each have a number of different
wavefunctions, all of which have the same energy.
are said to be degenerate.
Wavefunctions with the same energy
The probability of encountering the electron in a certain small
volume of space surrounding a point with co-ordinates x, y and z is proportional to the
square of the wavefunction at that point.
Wavefunctions of many electron atoms are derived by approximate methods.
The
simplest model for these wavefunctions suggests that they can be divided into two parts, a
radial part multiplied by an angular part.
The maximum probability of finding the electron
depends on both the radial and angular parts of the wave function and the resulting
boundary surfaces have complex shapes.
For many purposes, however, it is sufficient to
describe only the angular part of the wavefunction and these are the familiar orbital shapes
of chemistry.
What is an atomic term?
An atomic term is a symbol that represents a set of energy levels in energy level
diagrams for many electron atoms obtained experimentally from atomic spectra.
Theoretically the energy levels are explained using Russell-Saunders coupling, which is
used to derive the total spin angular momentum quantum number S, the spin multiplicity,
(2S+1) and the total angular momentum quantum number, L.
combinations of S and L are written in the form
The compatible
2S+1
L, called a term symbol, which
represents a set of energy levels; a term. Some examples are: 3P, 1D, 1S.
Although Russell-Saunders coupling allows the terms for an atom to be obtained, the
energies of the terms must be calculated using quantum mechanical procedures.
How are the energy levels of atoms labelled?
The term symbol does not account for the true energy level complexity found in atoms.
This is because the interaction between the spin and the orbital momentum, (spin-orbit
coupling), is ignored in Russell-Saunders coupling.
To make up for this a new quantum
number, J, is needed which is incorporated, as a subscript to the term and written
Each value of J represents a different energy level and is called a level.
labels attached to the energy levels of atoms.
composed of three levels 3P0, 3P1 and 3P2.
2S+1
LJ.
These are the
For example, a ground state term 3P is
The magnitude of the energy difference
between these levels depends upon the strength of the interaction between L and S.
Quick quiz
1. c; 2. b; 3. b; 4. c; 5. a; 6. c; 7. b; 8. b; 9. b; 10. b; 11. c; 12. a; 13. c; 14. a;
15. b; 16. b; 17. c.
Calculations and questions
1. 1.51 eV, 2.42 x 10-19 J.
2. 3.02 eV, 4.84 x 10-19 J.
3. He+, 54.4 eV, 8.72 x 10-18 J; Li2+, 122.4 eV, 1.96 x 10-17 J.
4. to n = 1, , 97.3 nm, , 3.08 x 1015 s-1; to n = 2, , 486 nm, , 6.17 x 1014 s-1; to n = 3, ,
1.88 m, , 1.60 x 1014 s-1.
5. to n = 1, , 95.0 nm, , 3.16 x 1015 s-1; to n = 2, , 434 nm, , 6.91 x 1014 s-1; to n = 3, ,
1.28 m, , 2.34 x 1014 s-1.
6. to n = 1, , 11.4 nm, , 2.63 x 1016 s-1; to n = 2, , 73.0 nm, , 4.12 x 1015 s-1.
7. to n = 1, , 24.3 nm, , 1.23 x 1016 s-1; to n = 2, , 122 nm, , 2.47 x 1015 s-1; to n = 2, ,
489 nm, , 6.40 x 1014 s-1,
8. 3.37 x 10-19 J.
9. 4.56 x 10-19 J.
10. n, 2; l, 1; ml, 1, 0, -1; ms, +½, -½.
11. J, 4, 3, 2; 3F2.
12. J, 3/2; 4S3/2.
13. J, 5/2, 3/2; 2D3/2.
14. J, 3/2, ½; 2P1/2.
15. 18/12.
16. 60/40.
17. Diagram required.
Solutions
1 What energy is required to liberate an electron in the n = 3 orbital of a hydrogen atom.
E = E - E3 = 13.6 [(1 / 32) - (1 / 2)] = 13.6 [ 1/9 ] = 1.51 eV
or
2.18 x 10-18 [1/9] = 2.42 x 10-19 J
2
What is the energy change when an electron moves from the n = 2 orbit to the n = 6
orbit in a hydrogen atom?
E = E2 - E36 = 13.6 [(1 / 22) - (1 / 62)] = 13.6 [0.2222] = 3.02 eV
2.18 x 10-18 [0.2222] = 4.84 x 10-19 J
or
3
Calculate the energy of the lowest orbital, (the ground state), of the single electron
hydrogen-like atoms with Z = 2, (He+) and 3, (Li2+).
E = 13.6 Z2 / [(1 / 12)] = 2.18 x 10-18 / [(1 / 12)]
For He, Z = 2, E = 13.6 x 4 = 54.4 eV
=
2.18 x 10-18 x 4 = 8.72 x 10-18 J
For Li, Z = 3, E = 13.6 x 9 = 122.4 eV
=
2.18 x 10-18 x 9 = 1.96 x 10-17 J
4 What are the frequencies and wavelengths of the photons emitted from a hydrogen atom
when an electron makes a transition from n = 4 to the lower levels n = 1, 2 and 3?
E = h  = h c /  = 2.18 x 10-18 / [(1 / n12) – (1 / n22)]
For n2 = 4 to n1 = 1
E = 2.18 x 10-18 [(1 / 1) – (1 / 16)] = 2.18 x 10-18 x (15 / 16)

= h c / E = (6.626 x 10-34 x 2.999 x 108 x 16) / (2.18 x 10-18 x 15)
= 9.73 x 10-8 m = 97.3 nm
 = E / h = 2.18 x 10-18 x 16 / (15 x 6.626 x 10-34) = 3.08 x 1015 s-1
For n2 = 4 to n1 = 2
E = 2.18 x 10-18 x [(1 / 4) – (1 / 16)] = 2.18 x 10-18 x (0.1875)

= h c / E = (6.626 x 10-34 x 2.999 x 108 ) / (2.18 x 10-18 x 0.1875)
= 4.86 x 10-7 m
 = E / h = (2.18 x 10-18 x 0.1875) / (6.626 x 10-34) = 6.17 x 1014 s-1
For n2 = 4 to n1 = 3
E = 2.18 x 10-18 x [(1 / 9) – (1 / 16)] = 2.18 x 10-18 x (0.0486)

= h c / E = (6.626 x 10-34 x 2.999 x 108 ) / (2.18 x 10-18 x 0.0486)
= 1.88 x 10-6 m
 = E / h = (2.18 x 10-18 x 0.0486) / (6.626 x 10-34) = 1.60 x 1014 s-1
5 What are the frequencies and wavelengths of the photons emitted from a hydrogen atom
when an electron makes a transition from n = 5 to the lower levels n = 1, 2 and 3?
E = h  = h c /  = 2.18 x 10-18 / [(1 / n12) – (1 / n22)]
For n2 = 5 to n1 = 1
E = 2.18 x 10-18 [(1 / 1) – (1 / 25)] = 2.18 x 10-18 x (0.9600)

= h c / E = (6.626 x 10-34 x 2.999 x 108 x 16) / (2.18 x 10-18 x 0.9600)
= 9.50 x 10-8 m = 95.0 nm
 = E / h = 2.18 x 10-18 x 0.9600) / 6.626 x 10-34) = 3.16 x 1015 s-1
For n2 = 5 to n1 = 2
E = 2.18 x 10-18 x [(1 / 4) – (1 / 25)] = 2.18 x 10-18 x (0.2100)

= h c / E = (6.626 x 10-34 x 2.999 x 10-8 ) / (2.18 x 10-18 x 0.2100)
= 4.34 x 10-7 m
 = E / h = (2.18 x 10-18 x 0.2100) / (6.626 x 10-34) = 6.91 x 1014 s-1
For n2 = 5 to n1 = 3
E = 2.18 x 10-18 x [(1 / 9) – (1 / 25)] = 2.18 x 10-18 x (0.0711)

= h c / E = (6.626 x 10-34 x 2.999 x 108 ) / (2.18 x 10-18 x 0.0711)
= 1.28 x 10-6 m
 = E / h = (2.18 x 10-18 x 0.0711) / (6.626 x 10-34) = 2.34 x 1014 s-1
6
What are the frequencies and wavelengths of photons emitted when an electron on a
Li2+ ion makes a transition from n = 3 to the lower levels n = 1 and 2?
E = h  = h c /  = 2.18 x 10-18 x 9 / [(1 / n12) – (1 / n22)]
= 1.96 x 10-17 / [(1 / n12) – (1 / n22)]
For n2 = 3 to n1 = 1
E = 1.96 x 10-17 [(1 / 1) – (1 / 9)] = 1.96 x 10-17 x (0.8889)

= h c / E = (6.626 x 10-34 x 2.999 x 108 x 16) / (1.96 x 10-17 x (0.8889)
= 1.14 x 10-8 m = 11.4 nm
 = E / h = 1.96 x 10-17 x (0.8889) / 6.626 x 10-34) = 2.63 x 1016 s-1
For n2 = 3 to n1 = 2
E = 1.96 x 10-17 [(1 / 4) – (1 / 9)] = 1.96 x 10-17 (0.1389)

= h c / E = (6.626 x 10-34 x 2.999 x 108 ) / (1.96 x 10-17 x 0.1389)
= 7.30 x 10-8 m
 = E / h = (1.96 x 10-17 x 0.1389) / (6.626 x 10-34) = 4.12 x 1015 s-1
7
What are the frequencies and wavelengths of photons emitted when an electron on a
He+ ion makes a transition from n = 4 to the lower levels n = 1, 2 and 3?
E = h  = h c /  = 2.18 x 10-18 x 4 / [(1 / n12) – (1 / n22)]
= 8.72 x 10-18 / [(1 / n12) – (1 / n22)]
For n2 = 4 to n1 = 1
E = 8.72 x 10-18 [(1 / 1) – (1 / 16)] = 8.72 x 10-18 x (0.9375)

= h c / E = (6.626 x 10-34 x 2.999 x 108 x 16) / (8.72 x 10-18 x 0.9375)
= 2.43 x 10-8 m = 24.3 nm
 = E / h = 8.72 x 10-18 x (0.9375) / 6.626 x 10-34) = 1.23 x 1016 s-1
For n2 = 4 to n1 = 2
E = 8.72 x 10-18 x [(1 / 4) – (1 / 16)] = 8.72 x 10-18 x (0.1875)

= h c / E = (6.626 x 10-34 x 2.999 x 108 ) / (8.72 x 10-18 x 0.1875)
= 1.22 x 10-7 m
 = E / h = (1.96 x 10-17 x 0.1875) / (6.626 x 10-34) = 2.47 x 1015 s-1
For n2 = 4 to n1 = 3
E = 8.72 x 10-18 x [(1 / 9) – (1 / 16)] = 8.72 x 10-18 x (0.0486)

= h c / E = (6.626 x 10-34 x 2.999 x 108 ) / (8.72 x 10-18 x 0.0486)
= 4.69 x 10-7 m
 = E / h = (8.72 x 10-18 x 0.0486) / (6.626 x 10-34) = 6.40 x 1014 s-1
8
Sodium lights emit yellow colour, with photons of wavelength 589 nm.
What is the
energy of these photons?
E = h  = h c /  = 6.626 x 10-34 x 2.999 x 108 / 589 x 10-9
= 3.37 x 10-19 J
9
Mercury lights emit photons with a wavelength 435.8 nm.
What is the energy of the
photons?
E = h  = h c /  = 6.626 x 10-34 x 2.999 x 108 / 435.8 x 10-9
= 4.56 x 10-19 J
10 What are the possible quantum numbers for an electron in a 2p orbital?
n = 2, I = 1, ml = -1, 0, +1, ms = -½, ½
11
Titanium has the term symbol 3F.
What are the possible values of J?
ground state level?
J = (L + S), (L + S -1), ….. |L – S|
F corresponds to L = 3,
multiplicity [(2S + 1) = 3] corresponds to S = 1
J = 4, 3, 2
What is the
Titanium has a less tan half-filled shell so the ground state is given by the lowest value of J,
i.e. 3F2.
12 Phosphorus has the term symbol 4S. What are the possible values of J? What is the
ground state level?
J = (L + S), (L + S -1), ….. |L – S|
S corresponds to L = 0,
multiplicity [(2S + 1) = 4] corresponds to S = 3/2
J = 3/2
the ground state is 4S3/2.
13
Scandium has a term symbol 2D.
What are the possible values of J?
What is the
ground state level?
J = (L + S), (L + S -1), ….. |L – S|
D corresponds to L = 2,
multiplicity [(2S + 1) = 2] corresponds to S = ½
J = 5/2 3/2
Scandium has a less tan half-filled shell so the ground state is given by the lowest value of
J, i.e. 2D3/2.
14 Boron has a term symbol 2P.
state level?
What are the possible values of J? What is the ground
J = (L + S), (L + S -1), ….. |L – S|
P corresponds to L = 1,
multiplicity [(2S + 1) = 2] corresponds to S = ½
J = 3/2, 1/2
Boron has a less tan half-filled shell so the ground state is given by the lowest value of J,
i.e. 2P3/2.
15 What is the splitting gJ, for sulphur, with ground state 3P2?
gJ = 1 + {[(J(J + 1) - L(L + 1) + S(S + 1)] / (2J (J + 1)}
For 3P2
multiplicity [(2S + 1) = 3] corresponds to S = 1
P corresponds to L = 1
J = 2
gJ = 1 + {[2 (3) - 1 (2) + 1 (2)] / (2 x 2(3)} = 1 + 6/12 = 18/12
16 What is the splitting gJ, for iron, with a ground state 5D4?
gJ = 1 + {[(J(J + 1) - L(L + 1) + S(S + 1)] / (2J (J + 1)}
For 3P2
multiplicity [(2S + 1) = 5] corresponds to S = 2
D corresponds to L = 2
J = 4
gJ = 1 + {[4 (5) - 2 (3) + 2 (3)] / (2 x 4(5)} = 1 + 20/40 = 60/40
17 Draw a diagram equivalent to Figure 1.9, for the ground state of a chlorine atom, with a
ground state 2P3/2.
The 2P state has two J values, ½ and 3/2 (Q14).
The valence shell is more than half-full,
so this corresponds to the largest value of J, 3/2, as stated. The first excited state is 2P1/2.
In a magnetic field these are split into 2J + 1 levels:
The other terms for a p5 electron configuration can be worked out as detailed above.
It
turns out hat these are like those for a p1 configuration, which is easier to calculate.
However, it is not possible to put hese into an order of increasing energy without complex
calculations, so extension of the diagram to higher levels is not possible without this
knowledge.
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