15.1 Preliminaries: Wave Motion and Light

15.2 Experimental Basis of Energy Quantization: Blackbody Radiation and the Photoelectric Effect

• The first rumblings of the quantum revolution occurred around the end of the 19th century when experimental results began to appear that could not be explained by classical physics. Most of these involved either the absorption of light by matter or the emission of light by matter. The quantum revolution was launched when Plank and Einstein introduced the radical concept of energy quantization to explain two of these results.

Blackbody Radiation• Every object emits energy through thermal radiation from its surface.• This energy is carried by electromagnetic waves, whose wavelengths

depend on the temperature of the object.• Classical mechanics attributes this radiation to the motion of charged

particles near the surface, accelerated by ordinary thermal movement.• Stefan’s law: IT = σeT4

• IT is the total energy emitted over the entire range of frequencies, per second and per m2, from the object at temperature T (K). The Stefan-Boltzmann constant measures the efficiency of converting thermal energy of the particle motions into thermal radiation. The parameter e, called the emissivity, is determined empirically for each surface and ranges from 0 to 1.

• Absorption of thermal radiation: absorptivity a = e.• Black bodies: a = e = 1, objects that absorb all radiation incident upon

them.

Blackbody radiation

Wien’s displacement law:

The Photoelectric Effect

Emax = eVmax

• Example 15.2Light with a wavelength of 400 nm strikes the surface of cesium in a

photocell, and the maximum kinetic energy of the electrons ejected is 1.54 x 10-19 J. Calculate the work function of cesium and the longest wavelength of light that is capable of ejecting electrons from the metal.

15.3 Experimental Demonstrations of Energy Quantization in Atoms

• In this section, we describe three sets of experiments that demonstrated quantization of energy in free atoms in the gas phase.

Atomic Spectra and Transitions between Energy States

neon argon mercury

Bohr proposed a model of the H atom that allowed energy only in discrete amounts corresponding to specific energy states. Bohr proposed that emission of light would correspond to a transition of the atoms between two such states.

Frank-Hertz Experiment and Energy Levels of Atoms

• Example 15.3The first two excitation voltage thresholds in the Frank-Hertz study of

mercury vapor were found at 4.9 V and 6.7 V. Calculate the wavelength of light that should be emitted by Hg atoms after excitation past each of these thresholds.

Photoelectron Spectroscopy and Binding Energies

BE = hνphoton – (1/2)meν2electron

• Example 15.4Construct the energy diagram for Ne from the data in the following figure.

The wavelength of x-ray is 9.890 x 10-11 m.

15.4 The Bohr Model: Predicting Discrete Energy Levels

• Bohr’s Model of the Atom

• Example 15.5Consider the n = 2 state of the Li2+ ion. Using the Bohr model, calculate

the radius of the electron orbit, the electron velocity, and the energy of the ion relative to the nucleus and electron separated by an infinite distance.

Atomic Spectra: Interpretation by Bohr’s Model

15.5 Waves, Particles, and the Schrodinger Equation

• Bohr’s theory gave a prescription for obtaining the discrete energy levels of a one-electron atom or ion, but did not explain the origin of energy quantization.

• A fundamental explanation was developed in 1926 by Schrodinger through an analogy with the theory of vibrations, where a form of quantization was already understood.

• A key step was the recognition by de Broglie that wave-particle duality, introduced by Einstein to describe photons, was equally a property of material particles such as electrons.

De Broglie Waves

Traveling waveStanding wave

node

Fundamentalor first harmonic

Second harmonic

• Example 15.6Calculate the de Broglie wavelengths of (a) an electron moving with

velocity 1.0 x 106 m s-1 and (b) a baseball of mass 0.145 kg, thrown with a velocity of 30 m s-1.

The Heisenberg Uncertainty Principle

• Example 15.7Suppose photons of green light (wavelength 5.3 x 10-7 m) are used to

locate the position of the baseball from Example 15.6 to an accuracy of one wavelength. Calculate the minimum uncertainty in the speed of the baseball.

The Schorodinger Equation

• De Broglie’s work attributes wave-like properties to electrons in atoms, and the uncertainty principle shows that detailed trajectories of electrons cannot be defined.

• Consequently, we must deal in terms of the probability of electrons having certain positions and momenta.

• These ideas are combined in the fundamental equation of quantum mechanics, the Schrodinger equation.

• He reasoned that an electron (or any other particle) with wave-like properties should be described by a wave function that has a value at each position in space.

• This wave function (ψ(x,y,z)) is the “height” of the wave at the point in space defined by the set of Cartesian coordinates (x, y, z).

• Schrodinger wrote down the equation satisfied by ψ for a given set of interactions between particles.

The meaning of wave function: probability density• The square of the wave function ψ2 for a particle as a probability

density for that particle.• In other words, ψ2(x, y, z)ΔxΔyΔz is the probability that the particle

will be found in a small volume ΔxΔyΔz about the point (x, y, z).

A Deeper Look….15.6 The Particle in a Box

• The particle in a box

• The particle in a box

• The particle in a box

• The particle in a box

• Example 15.8Consider the following two systems: (a) an electron in a one-dimensional

box of length 1.0 A and (b) a helium atom in a cube 30 cm on an edge. Calculate the energy difference between ground state and first excited state, expressing your answer in kJ mol-1. (The atomic mass unit u is defined in Section 14.1)

15.7 Hydrogen Atom• Energy Levels

n: principle quantum number

The energy of a one-electron atom depends only one the principal quantum number n, because the potential energy depends only on the radial distance.

• Schrodinger equation also quantizes L2, the square magnitude of the angular momentum as well as Lz, the projection of the angular momentum along the z-axis.

l: angular momentum quantum numberm: magnetic quantum number

• For n = 1 (ground state), the only allowed quantum numbers are (l = 0, m = 0)

• For n = 2, there are n2 = 4 allowed sets of quantum numbers, which are (l = 0, m = 0), (l = 1, m =1), (l = 1, m = 0), (l = 1, m = -1)

• The restrictions on l and m give rise to n2 sets of quantum numbers for every value of n.

• Each set (n, l m) identifies a specific quantum state of the atom in which the electron has

energy equal to En, angular momentum equal to [l(l + 1)]1/2 h/2π, andz-projection of angular momentum equal to mh/2π.

When n > 1, a total of n2 specific quantum states correspond to the single energy level En; consequently, this set of states is said to be degenerate.

Wave FunctionsFor each quantum state (n, l, m), solution of Schrodinger equation provides a wave function written in the form

ψnlm(r, θ, φ) = Rnl(r)Ylm(θ, φ)

The wave function itself is not measured directly. It is to be viewed as an intermediate step toward calculating the physically significant quantity ψ2, which gives the probability density for locating the electron at a particular point in the atom.

More precisely,[ψnlm(r, θ, φ)]2dτ = [Rnl(r)]2[Ylm(θ, φ)]2 dτ

gives the probability of locating the electron within a small three-dimensional volume dτ located at the position (r, θ, φ) when it is known that the atom is in the state (n, l, m).A wave function ψnlm(r, θ, φ) for a one-electron atom in the state (n, l, m) is called an orbital.

• Example 15.9Give the names of all the orbitals with n = 4, and state how many m values

correspond to each type of orbital.

Sizes and Shapes of Orbitals

• The sizes and shapes of the hydrogen atom orbitals are important in chemistry because they provide the foundations for the quantaldescription of chemical bonding and the resulting shapes of molecules.

• s orbitals

(a) Three-dimensional picture in which the shading is heaviest where ψ2 is largest and lighter where ψ2 is smaller. The probability of finding the electron and a measure of the electron density

(b) ψn00(r) ∝ Rn0(r): this gives the amplitude for finding the electron at a certain distance from the nucleus.

(c) the radial probability distribution r2Rn0(r): this gives the probability of finding the electron anywhere within a thin spherical shell of thickness dr, located at distance r from the nucleus.

(d) The spheres that enclose 90% of the electron probability.

* An ns orbital has n-1 radial nodes.

p orbitals

d orbitals

• Example 15.10Compare the 3p and 4d orbitals of a hydrogen atom with respect to (a)

number of radial and angular nodes and (b) energy of the corresponding atom.

Electron spin

spin quantum number

15.8 Many-Electron Atoms and the Periodic Table

• Schrodinger equation and its solutions become increasingly complicated as we move from one-electron to many-electron atoms.

• An explicit solution for even helium is not possible, but moderncomputers have enabled us to solve this equation numerically to very high accuracy.

• Although these numerical calculations demonstrate conclusively the usefulness of the Schrodinger equation for predicting atomic properties, they suffer from two defects: (1) difficult to interpret physically, and (2) more and more difficult with more electrons.

• As a result, approximate approaches to the many-electron Schrodingerequation has been developed: self-consistent field (SCF) orbital approximation method.

Hartree Orbitals: Shell Model of the Atom

• For any atom, Hartree’s method begins with the exact Schrodingerequation in which each electron is attracted to the nucleus and repelled by all the other electrons in accordance with the Coulomb potential.

• Two simplifying assumptions are made: (1) Each electron moves in an effective field due to all the other electrons, to be obtained by averaging over all the positions of the other electrons, and (2) the effective field is spherically symmetric; that is, it has no angular dependence.

• Under these two assumptions, each electron is then described by a one-electron orbital similar to those of the H atom.

• These two assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and unknown one-electron orbitals.

• These equations must be solved by iteration until a self-consistent solution is obtained.

• Hartree’s method produces two principal results: energy levels and orbitals.

1s

2p

2s

3p

3s

These Hartree orbitals resemble the atomic orbitals of hydrogen in many ways.Their angular dependence is identical, so it is straightforward to associate quantum numbers l and m with each atomic orbital.The radial dependence of the orbitals in many-electron atoms differs from that of one-electron orbitals because the effective field differs from the Coulomb potential, but a principal quantum number n can still be defined.

shell

subshell

Shielding Effects: Energy Sequence of Hartree Orbitals

• The energy-level diagrams calculated for many-electron atoms by Hartree’s method resemble the diagram for the hydrogen atom, but differ in two important respects.

• First, the degeneracy of the p, d, f orbitals is removed. Because the effective field in Hartree’s method is different from the Coulomb field in the hydrogen atom, the energy levels of Hartree orbitals depend on both n and l.

• Second, the energy values, especially at smaller values of n, are distinctly shifted from the values of corresponding hydrogen orbitalsbecause of the stronger attractive force exerted by nuclei with Z > 1.

• Vneff(r) = -(Zeff(n)e2)/r where Zeff(n) is the effective nuclear charge in

that shell.

• Detailed Hartree calculations for argon show that Zeff(1) ≈ 16, Zeff(2)

≈ 8, Zeff(3) ≈ 2.5.

• The effect of shielding on the energy and radius of a Hartree orbital is easily estimated in this simplified picture by using the hydrogen atom equations with Z replaced by Zeff(n).

• En≈ -[Zeff(n)]2/n2 (rydbergs)

• Thus, electrons in inner shells (small n) are tightly bound to the nucleus, and their average position is quite near the nucleus because they are only slightly shielded from the full nuclear charge Z. Electrons in outer shells are only weakly attracted to the nucleus, and their average position is far from the nucleus because they are almost fully shielded from the nuclear charge Z.

• Example 15.11Estimate the energy and average of r in the 1s orbital of argon. Compare

the results with the corresponding values for hydrogen.

• The dependence of the energy on l in addition to n can be demonstrated by considering the effective charge within a subshell.

• Only the s orbitals penetrated to the nucleus; both p and d orbitals have nodes at the nucleus. Consequently, the shielding will be smallest and the effective charge greatest in s orbitals:

• Zeff(ns) > Zeff(np) > Zeff(nd)• Ens < Enp < End

The Aufbau (“Building Up”) Principle

• The ground-state electronic structures of atoms are built up by arranging the Hartree atomic orbitals in order of increasing energy and filling the one electron at a time.

• Two additional restrictions: (1) the Pauli exclusion principle and (2) Hund’s rule

• (1) The Pauli exclusion principle: no two electrons in an atom can have the same set of four quantum numbers (n, l, m, ms). Each Hartreeatomic orbital (characterized by a set of three quantum numbers, n, l, m) holds at most two electrons, one with spin up and the other with spin down.

• (2) Hund’s rule: when electrons are added to Hartree orbitals of equal energy, a single electron enters each orbital before a second one enters any orbital. In addition, the spins remain parallel if possible.

Building Up from Helium to Argon

Paramagnetic: attracted into a magnetic field – unpaired electronDiamagnetic: pushed out of a magnetic field – no unpaired electron

s-block

p-block

1s22s22p63s23p2

[Ne]3s23p2

[Ar]3d14s2 Sc+: [Ar]3d14s1

Transition-Metal Elements and Beyond

For a series of elements or ions in the same group of the periodic table, the radius usually increases with increasing atomic number – Pauliexclusion principle, increase in shell

On the other hand, Coulomb forces cause the radii of atoms to decreasewith increasing atomic number across a period – incomplete shielding

Lanthanide contraction

Example 15.13(a) Kr vs Rb(b) Y vs Cd(c) F- vs Br-

In oxygen, two electrons in the same 2p orbital, leading to greater electron-electron repulsion and diminished binding

In Be, the fifth electron is I a higher energy 2p orbital

(1) A 2s electron is much farther from the nucleus than a 1s electron (2) screening effect in Li: in Li, 1s electrons screen the nucleus so effectively that the 2s electron sees a net positive charge close to +1 than the large charge seen by the electrons in He

The ionization energy tends to decrease down a group in the periodic table (for example, from Li to Na to K) – as the principle quantum number increases, so does the distance of the outer electrons from the nucleus.

Periodic Trends in Ionization Energies

Electron Affinity

Example 15.14Consider Se and Br. Which has the higher first ionization energy, and which the higher electron affinity?