The Bohr Model and the Quantum Mechanical Model of the Atom Physics 12

Clip of the day: Minutesphysics on origins of Quantum model

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: Balmer showed that the visible lines could be predicted using: Rydberg went on to show that all hydrogen lines could be predicted using:

Bohr Postulates: Electrons exist in circular orbits Electrons exist only in allowed orbits Electrons do not radiate energy within an orbit 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”

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 Numbers Principal Quantum Number Orbital Quantum Number Magnetic Quantum Number 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.

Ex: the electron orbitals with a principle quantum number (n) of 3 Subshell ml Number of orbitals in subshell 3 3s 1 3p -1,0,+1 2 3d -2,1,0,+1,+2 5

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=3 l = 2 ml = -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. silicon c. beryllium d. strontium e. chlorine f. carbon g. copper h. bromine