The Bohr Model and the Quantum Mechanical Model of the Atom

Physics 12

Clip of the day:•Minutesphysics on origins of Quantum

model•http://www.youtube.com/watch?v=i1TVZI

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The history lesson continues…

Dalton’s Atomic Model•Atom = solid, indivisible sphere

Plum Pudding Model (Thomson)•Proton and electrons spread through the

atom

Rutherford Model:•Nuclear model

▫Positive charge and most of the mass concentrated in centre of atom

▫Electrons circling nucleus

The Bohr Model:•The Bohr Model built upon

earlier models of the atom▫Dalton – Billiard Ball▫Thompson – Raisin Bread▫Rutherford – Nuclear Model

•Bohr began investigating the line spectra of hydrogen in order to determine the behaviour of electrons

Bohr Model:

•Electrons orbit the nucleus in circular paths of fixed energy (energy levels).

Energy levels:•Electrons can jump from energy level to

energy level.•Electrons absorb or emit light energy when

they jump from one energy level to another.▫Energy is emitted by the electron as it leaps

from the higher to the lower energy level ▫Energy is absorbed by the electron as it moves

from the lower to the higher energy level •The energy is proportional to the frequency of

the light wave.•Frequency defines the color of visible light

emitted or absorbed• http://higheredbcs.wiley.com/legacy/college/halliday/

0471320005/simulations6e/index.htm?newwindow=true

Hydrogen Line Spectra:

• Bohr studied gas discharge tubes filled with individual gases▫ Particularly hydrogen

Hydrogen Line Spectra:• When hydrogen is bombarded with cathode rays

(beam of electrons), it will absorb specific wavelengths of light

• Similarly, if a large amount of energy is passed through hydrogen gas, it will emit specific wavelengths of light

• The line represent the specific levels of energy that are possible

Balmer and Rydberg:

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•Balmer showed that the visible lines could be predicted using:

•Rydberg went on to show that all hydrogen lines could be predicted using:

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Bohr Postulates:

1. Electrons exist in circular orbits

2. Electrons exist only in allowed orbits

3. Electrons do not radiate energy within an orbit

4. Electrons can jump between orbits

So….• The Bohr model explained the emission

spectrum of the hydrogen atom but did NOT always explain those of other elements.

• Since the Bohr Model does well with hydrogen, it is likely that the theory needs to be expanded, not discarded!

Principal Quantum Number: The Bohr model

actually used a single quantum number (n) to describe an orbit (energy level/ring)

The Quantum model uses four quantum numbers to describe an orbital

Lead to the Quantum Mechanical Model:•1920’s

•Credit to..▫Werner Heisenberg (Uncertainty Principle)▫Louis de Broglie (electron has wave

properties)▫Erwin Schrodinger (mathematical equations

using probability, quantum numbers)

de Broglie Wavelength and the Electron:• de Broglie realized that

as a result of his matter wave equation, the wavelength of an electron would play a role in how it orbits the nucleus

• The orbital circumference would have to be an integral number of wavelengths and “pilot waves”

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Schrödinger Wave Equation:

•Erwin Schrödinger developed the Schrödinger wave equation that forms the foundation of quantum mechanics

•This equation leads to the ability to plot an electron’s orbital

•The Schrödinger Wave Equation leads to the addition of two additional quantum numbers in addition to the principal quantum number (n) from Bohr

Paul Dirac:•Paul Dirac modified the

Schrödinger Wave Equation using a relativistic correction

•Once this was applied, Bohr’s Model was able to predict the behaviour of the hydrogen atom even more accurately

•Further, this correction allows the Schrödinger Wave Equation to work with other atoms and also predicts behaviour that had not even been discovered when Dirac did his original work

Quantum Model:• is based on mathematics and quantum theory,

which says matter also has properties associated with waves.

• It’s impossible to know the exact position and momentum of an electron at the same time (known as the Heisenberg Uncertainty Principle).

• Uses complex shapes of orbitals (electron clouds), volumes of space in which there is likely to be an electron.

• Based on probability not certainty

Orbitals:•A region in space in which there is high

probability of finding an electron.•Electrons, instead of traveling in defined

orbits or hard, spherical “shells,” as Bohr proposed, travel in diffuse clouds around the nucleus.

Quantum Numbers:

•Specify the properties of atomic orbitals and their electrons.

•There are four Quantum Numbers1. Principal Quantum Number2. Orbital Quantum Number3. Magnetic Quantum Number4. Spin Quantum Number

The principle quantum number (n):

•Has integral values ▫n = 1, 2, 3, 4…

•The maximum number of electrons in a principal energy level is given by:

2n2

•As n increases the electron has a higher energy and is less tightly bound to the nucleus

The orbital (second) quantum number(l ):

•The value of ℓ ranges from 0 to n − 1•describes the shape of the orbital

▫l = 0, 1, 2, 3, …▫l = s, p, d, f, …

•Example: if n =2

than l = 1, 0

Magnetic Quantum Number, ml

:

•Indicates the orientation of the orbital in space

•ml = - l , …,0,…, l

•Example: for l= 2

ml = -2, -1, 0, +1, +2

Spin Quantum Number:

•Finally, the Spin Quantum Number (ms) comes about due to the relativistic wave equation

•This described the magnetic field that a spinning electron creates and explains why electrons pair up in an orbital

•ms = -½ , ½

Pauli Exclusion Principle:•The Pauli Exclusion Principle states that

no two electrons in the same atom can occupy the same state

•This means that of the four quantum numbers, no two electrons in the same atom can have the same four quantum numbers

Recap:

•Thus, it takes three quantum numbers to define an orbital but four quantum numbers to identify one of the electrons that can occupy the orbital.

n l Subshell ml Number of orbitals in subshell

3 0 3s 0 1

1 3p -1,0,+1 3

2 3d -2,1,0,+1,+2 5

Ex: the electron orbitals with a principle quantum number (n) of 3

Practice #1:•What are the possible values of l and

ml for an electron with the principle quantum number n=4?

•If l=0, ml=0

•If l=1, ml= -1, 0, +1

•If l=2, ml= -2,-1,0,+1, +2

•If l=3, ml= -3, -2, -1, 0, +1, +2, +3

Practice #2:•Can an electron have the quantum

numbers n=2, l=2 and ml=2?•No, because l cannot be greater than n-1,

so l may only be 0 or 1. •ml cannot be 2 either because it can never

be greater than l

Practice #3:•List the values of the four quantum

numbers for orbitals in the 3d (n=3, l=2) sublevel.

•Answer: n=3l = 2ml = -2,-1, 0, +1, +2

ms = +1/2, -1/2 for each pair of electrons

Orbital Energy Levels:• In any give atom, the electrons will fill the

orbitals starting from the lowest energy state▫Remember the number of electrons is equal to

the atomic number of an atom•The energy of each orbital can be calculated

in order to determine the filling order•However, there is also a diagram that

provides this information without calculations

Energy Level Diagrams:• This model can be used to

create an energy level diagram

• Also, this model predicts the structure of the periodic table:▫Groups 1A and 2A – s▫Groups 3B – 8B – p▫Transition Metals – d▫Rare Earth/Synthetics -

f

Let’s try some:•Draw energy level diagrams for:

a. sodium b. siliconc. berylliumd. strontiume. chlorinef. carbong. copperh. bromine