Line Spectra & Quantization Last day we stated that, in the absorption and emission spectra of atoms, only a small number of frequencies of light (or wavelengths)

Line Spectra & Quantization• Last day we stated that, in the absorption and

emission spectra of atoms, only a small number of frequencies of light (or wavelengths) are absorbed or emitted. A reminder slide (emission spectrum) follows. This slide is consistent only with energy quantization – an atom possesses, at a given time, only one energy from a possible set of allowed energies.

The Balmer series for hydrogen atoms – a line spectrumFIGURE 8-10

Copyright © 2011 Pearson Canada Inc. General Chemistry: Chapter 8 Slide 2 of 50

H Atom Electronic Emission Spectrum

Bohr Theory for the Hydrogen Atom• Bohr used the existing physics of the early 20th

century and a set of quantum numbers to model the observed absorption and emission spectra of the H atom.

• Tenets of the Bohr Theory:• (1) Energies of the H atom/electron are

quantized. Line spectra prove this!!!• (2) The electron in a H atom moves around the

nucleus in a circular orbit. (Not true!)

Bohr Theory for the H Atom

• (3) The angular momentum of the electron in a H atom is quantized.

• (4) Energy and angular momentum values for the electron in a H atom are calculated using a quantum number, n. Further, n has integer values n = 1, 2, 3, 4, 5….. infinity.

• Angular momentum (electron) = nh/2π • Radius of orbit = n2ao = rn (Not true!)

• Electron energy = En = -RH/n2 (Rydberg constant)

H Atom Energies - Bohr

• Electron energy = En = -RH/n2

• Notes: En is always negative. Why? (Coulombic rationalization?) The quantity ao is called the Bohr radius and specifies the radius of the electron orbit for the lowest energy state of the H atom. (Again, actual H atoms do not have an e- circling the nucleus at a fixed distance/radius ao).

A Stamp and Bohr’s Model

• The next slide shows a stamp with Bohr’s picture and the very important and generally valid equation ∆E = Efinal state – Einitial state = hν.

• The H atom model of Bohr (Newtonian and deterministic!) is not valid. Electrons do not “circle” nuclei in regular orbits. However, energy is required to move an electron away from the nucleus (and vice versa).

The Bohr Atom


FIGURE 8-13

•Bohr model of the hydrogen atom

E = -RH

n2

RH = 2.179 10-18 J

Energy Ladders• We could draw a simple energy ladder to

represent the gravitational potential energy of a can of paint on various steps of a ladder. The possible potential energy values would be equally spaced if the ladder steps are equally spaced. For atoms we can construct similar “energy ladders”. For one electron atoms the energy ladders are simple but the energy levels are not equally spaced. The H atom “energy ladder” is shown on the next slide.

Energy-level diagram for the hydrogen atomFIGURE 8-14


ΔE = Ef – Ei = -RH

nf2

-RH

ni2

–

= RH ( ni2

1

nf2

–1) = h = hc/λ

Absorption & Emission Spectra• Very hot H atoms can exist in electronically

excited states (the single electron is in a high energy state with n > 1). Such atoms can emit light as they move to a lower energy state. A small # of light frequencies are emitted. “Cold” H atoms can absorb light as they move to states with larger n values. What is the next slide telling us?

Emission and absorption spectroscopyFIGURE 8-15


Emission Absorption

Coulombic Interactions in the H Atom• The Coulombic force of attraction/repulsion

between stationary point charges is described by the familiar Coulomb’s Law

• F = keQ1Q2

r2

• where Q1 and Q2 specify the magnitude and sign of the two point charges and r is the distance between them. Since the H atom nucleus (a proton) and the lone electron have opposite charges there is a strong attractive force between these two subatomic particles.

• Coulomb’s Law → to pull the proton and electron apart we must do work/supply energy. Conversely, energy must be released if the proton and electron come closer to each other. The closer the e- comes to the nucleus the greater the amount of energy released. The application of Coulomb’s Law to atomic structure is not straightforward since electrons in an atom are not stationary and, in fact, have wavelike properties! In many electron atoms the rapidly moving electrons also interact with each other as well as the nucleus.

Calculations with the Bohr Expression• Again, by experiment• En = -RH/n2 for the single e- H atom.• We can use this energy expression to

calculate:• (1) Energies for levels with different values of

the quantum number n).• (2) Ionization energies (energy required for

removal of an electron from an atom!).• (3) Any ΔE for a transition nFinal ← nInitial.• (4) the frequency of light absorbed (or

emitted) for a transition nFinal ← nInitial.

H Atom Ionization – Example:• Example: How much energy is needed to

ionize (a) one H atom and (b) one mole of H atoms initially in their ground (lowest energy) state.

• Hint: The problem is easily solved if we use the Bohr energy expression for H and choose appropriate initial and final values for n.

• Hint(?): Buzz Lightyear would not be happy!

Bohr Theory and the Ionization Energy of Hydrogen


ΔE = RH ( ni2

1

nf2

–1) = h

As nf goes to infinity for hydrogen starting in the ground state:

h = RH ( ni2

1) = RH

This also works for hydrogen-like species such as He+ and Li2+.

En = RH ( ni2

-Z2

)

One Electron Monatomic Species• The modified Bohr energy expression on the

previous slide can be used to calculate electronic energies for the H atom (atomic number = Z = 1), He+ (Atomic number = Z = 2), Li2+ (Z = 3), etc. The next slide shows a few energy levels calculated using the modified Bohr equation for both H and He+. Can we account for the differences by considering Coulombic forces and potential energies?

H Atom and He+ Energies (kJ mol∙ -

1)

Energy

-5000

0

-3000

n = 1

n = 2n = 1

n = 2

n = 3n = 4

(kJ mol∙ -1)

Experimental energy spacings are much larger for He+ then for the H atom. Why?

Observed energy gaps are much larger for He+ than for the H

atom. Why?

H Atom (1e-) He+ Ion (1e-)

n = n =

The Bohr Energy Expression• The Bohr Energy expression can be used to

calculate energy differences between any two “levels” in the H atom. The energy differences can be quoted on a J/atom or kJ/mol basis. Energy level differences can be calculated for monatomic one e- ions if we include the atomic number in the Bohr expression.

• Special case: Ionization• H(g) → H+(g) + e- or He+(g) → He2+(g) + e-

Copyright 2011 Pearson Canada Inc. 8 - 20

Copyright 2011 Pearson Canada Inc. 8 - 21

Wave Particle Duality – Light and Subatomic Particles

• In high school physics light was treated as having both wave like and particle like character. Diffraction and refraction of light both exemplify its wave like properties. Particle like properties of light can also be demonstrated readily. In one such experiment – Compton effect - light hitting black blades attached to a “wind mill” cause the wind mill to spin. This implies, surprisingly, that light photons have momentum.

Subatomic Particles – Wave Character

• A number of experiments show that small particles have observable wave like properties. Such wave like properties become increasingly important as one moves to particles of smaller mass. The electron is the most important of these particles. Interesting diffraction and refraction experiments have been conducted with electrons.

Mathematical Description of Electrons

• The fact that electrons exhibit wave like behavior suggested that equations used to describe waves, and light waves in particular, might be modified to describe electrons. We will see some familiar mathematical functions used to describe the electron (e.g. cos θ, sin θ, eiθ). We will use so-called wave functions (Ψ’s) to gain insight into the behavior of electrons in atoms.

De Broglie’s Contribution

• De Broglie used results/equations from classical physics to rationalize experimental results which proved that subatomic particles (and some atoms and molecules) had wave like properties. He proposed, in particular, that particles with a finite rest mass had a characteristic wave length – as do light waves.

8-5 Two Ideas Leading to a New Quantum Mechanics

• Wave-Particle Duality– Einstein suggested particle-like properties of light could explain the photoelectric effect.–Diffraction patterns suggest photons are wave-like.

• deBroglie, 1924– Small particles of matter may at times display wavelike properties.


Louis de BroglieNobel Prize 1918

Wave-Particle Duality


E = mc2

h = mc2

h/c = mc = p

p = h/λ

λ = h/p = h/mu

Wave properties of electrons demonstratedFIGURE 8-16


Probabalistic Description of Electrons

• Classical physics suggests that we should be able (given sufficient information) to describe the behaviour of any body – its velocity, kinetic energy, potential energy and so on at any point in time. Classical physics suggests that all energies are continuously variable – a result which very clearly is contradicted by experimental results for atoms and molecules (line spectra/quantized energies).

Uncertainty Principle

• The quantum mechanical description used for atoms and molecules suggests that for some properties only a probabalistic description is possible. Heisenberg suggested that there is a fundamental limitation on our ability to determine precise values for atomic or molecular properties simultaneously. The mathematical statement of Heisenberg’s so-called Uncertainty Principle is given on the next slide.

The Uncertainty Principle


FIGURE 8-17

•The uncertainty principle interpreted graphically

Δx Δp ≥ h

4π

Heisenberg and Bohr

Atomic “Diagrams”• Many simple diagrams of atoms/atomic

structure have limitations. In class we’ll consider some limitations of the C atom diagram shown on the next slide.

• Representations of molecules are even more challenging – at least if we want to consider the electrons!

The “Carbon Atom”

• Limitations?

electron

neutron

proton

A Molecular Model

Documents

Line Spectra & Quantization Last day we stated that, in the absorption and emission spectra of atoms, only a small number of frequencies of light (or wavelengths)