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Chapter 7 : The Quantum-Mechanical Model of the Atom

- Outline
- Intro to Quantum Mechanics
- The Nature of Light
- Atomic Spectroscopy and the Bohr Model
- The Wave of Nature of Matter
- Quantum Mechanics and the Atom
- The Shapes of Atomic Orbitals

The Behavior of the Very Small

- electrons are incredibly small

- electron behavior determines much of the behavior of atoms

- directly observing electrons in the atom is impossible

Intro to QM

A Theory that Explains Electron Behavior

- the quantum-mechanical model explains the manner electrons exist and behave in atoms

- helps us understand and predict the properties of atoms that are directly related to the behavior of the electrons

- why some elements are metals while others are nonmetals
- why some elements are very reactive while others are practically inert

Intro to QM

Before we introduce quantum mechanics we must first understand a few things about light.

- light is a form of electromagnetic radiation
- composed of perpendicular oscillating waves, one for the electric field and one for the magnetic field

- all electromagnetic waves move through space at the same constant speed

Light

EM radiation can be described as a wave composed of oscillating electric and magnetic fields.

Light

time

Characterizing Waves

Maximum height above centre line (or the maximum depth below the centre line) is the amplitude

Distance between successive peaks is called the wavelength (λ, “lambda”)

Number of peaks (or troughs) that pass through a given point in a unit of time is the frequency, (υ, “nu”).

The difference in time between successive occurrences of the same displacement is the period. (τ)

Light

Relating Wavelength and Frequency

- for waves traveling at the same speed, the shorter the wavelength, the more frequently they pass

- this means that the wavelength and frequency of electromagnetic waves are inversely proportional

- since the speed of light is constant, if we know wavelength we can find the frequency, and vice versa

Light

Wavelength x Frequency = “speed of light”

Speed of light is known and equal to 3.00 x 108 m s-1

This equation applies to all the forms of electromagnetic radiation (not just visible light).

Light

Types of Electromagnetic Radiation

There are many forms of EM radiation that you may already be familiar with :

Light

Which colour has the higher frequency ?

Green ?

or Orange ?

A Closer Look at Visible Light

The colour of visible light depends on its wavelength.

Visible light wavelengths are on the order of 100s of nm.

Light

1. Many cordless phones operate on signals at 600 MHz. What is the equivalent wavelength ?

2. Helium–Neon lasers (the light used to scan your groceries at the checkout) produce light at 633 nm. What is the frequency of the laser’s light ?

Light

Interference

When two sets of waves (for example water waves) intersect, there are places where the waves disappear and other places where the waves persist.

Light

When the waves are in-step, (called being in-phase) the waves add together to give the highest crests and the deepest troughs.

When the waves are out-of-step, (called being out-of-phase) the waves cancel each other out.

Light

Diffraction

When a wave encounters an obstacle or a slit that is comparable in size to its wavelength, it bends around. This phenomenon is called diffraction.

Light

The Photoelectric Effect

When light strikes the surface of certain metals, electrons are detected. This was first observed by Heinrich Hertz in 1888 (12 years before Planck’s quantum theory).

- Electron emission only occurs when the _________ of the light exceeds a threshold value.
- The number of electrons emitted depends on the ________ of the light but…
- The kinetic energy of the emitted electrons depends on the __________of the light.

Light

- Einstein proposed that the light energy was delivered to the atoms in packets, called quanta or photons

- the energy of a photon of light was directly proportional to its frequency
- __________ proportional to it wavelength
- the proportionality constant is called Planck’s Constant, (h)and has the value 6.626 x 10-34 J∙s

Light

1 photon at the threshold frequency has just enough energy for an electron to escape the atom

- for higher frequencies, the electron absorbs more energy than is necessary to escape

- this excess energy becomes kinetic energy of the ejected electron

Light

The atoms of group 1 give a characteristic colour when placed in a flame.

Atomic Spectroscopy

Exciting Gas Atoms to Emit Light with Electrical Energy

Atomic Spectroscopy

Emission Spectra

Atomic Spectroscopy

Visible Lines (aka - The Balmer Series)

410.1 nm

434.0 nm

486.1 nm

656.3 nm

Rydberg’s Equation

The Atomic Spectrum of Hydrogen

Atomic Spectroscopy

- Use the Rydberg equation for n = 3. Does it agree with the experimental atomic spectra for hydrogen ?
- Repeat the exercise for n = 7. Can you explain why this line is not observed by the human eye ?

- the electrons traveled in orbits that were a fixed distance from the nucleus
- therefore the _______of the electron was proportional to the distance the orbital was from the nucleus

- Niels Bohr proposed that the electrons could only have very specific amounts of energy

- electrons emitted radiation when they “jumped” from an orbit with higher energy down to an orbit with lower energy

Bohr’s Model of the Atom

if electrons behave like particles, there should only be two bright spots on the target

Wave Nature of Matter

de Broglie proposed that ____particles could have wave-like character

- Incredibly, electrons which we were thought of as negatively charged _______also exhibit ________ properties
- because it is so small, the wave character of electrons is significant
- de Broglie predicted that the wavelength of a particle was _________ proportional to its momentum

Wave Nature of Matter

1. Calculate the de Broglie length of an electron travelling at one-tenth the speed of light.

2. In last night’s ALCS, a fastball was clocked at 97 miles an hour (43 m/s). Given that a baseball weighs 145 g. Calculate the de Broglie length of his fastball and comment on whether that was a feasible reason why the batters couldn’t hit the pitches.

- Heisenberg stated that the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass
- x = position,
- v = velocity,
- m = mass
- the means that the more accurately you know the position of a small particle, like an electron, the less you know about its speed
- and vice-versa

Wave Nature of Matter

Quantum Mechanics

Standing waves are waves where the magnitude of the oscillation is different from point to point along the wave. Points that undergo no displacement are called nodes.

Consider a plucked guitar string of length, l.

x direction

Schrödinger suggested that if an electron in an atom has wave-like properties then it should be describable using a mathematical equation called a wavefunction (ψ).

The wavefunction must be a solution to Schrödinger’s equation

The wavefunction should correspond to a standing wave within the boundary of the system being described..

1. The energy of the particle in a 1D PIAB is quantized.

- The minimum energy of the particle in a 1D PIAB is never zero

3. n is called the __________ quantum number

4. The square of the wavefunction, ψ2, at a given point in space represents the __________ of finding the particle there.

Use de Broglie’s equation for matter waves, the fact that the kinetic energy of a particle is given by the following expression

and the equation for the wavelength of a standing wave

to derive the equation for the energy of a 1D PIAB.

- The energy of an electron dictates the properties of an element. For example, bonding.

- However, if we very accurately know the energy of an electron, Heisenberg says we can’t precisely know its position.

- for an electron with a given energy, the best we can do is describe a region in the atom of high probability of finding it

- To determine the energy of an electron the Schrödinger equation must be solved.

Quantum Mechanics

- A wavefunction, ψ, is just a mathematical function that is a solution to the Schrödinger equation.

- The square of the wavefunction, ψ2 gives a probability map of finding the electron in a region of space.

- calculations show that the size, shape and orientation in space of an orbital are determined be three integer terms in the wave function

- these integers are calledquantum numbers
- __________quantum number, n
- __________ momentum quantum number, l
- __________ quantum number, ml

Quantum Mechanics

- characterizes the energy of the electron in a particular orbital

- n can be any integer ³ 1
- the larger the value of n,

- energies are defined as being negative
- the larger the value of n, the larger the orbital

Quantum Mechanics

Angular Momentum Quantum Number, l

- The angular quantum number is an integer that determines the shape of the orbital (see later).

- Possible values for l are 0,1,2,…,(n-1).

Quantum Mechanics

- The magnetic quantum number is an integer that determines the orientation of the orbital (see later).

- Possible values for ml are +l, +(l-1), +(l-2)…-l.

- Each specific combination of n,l,mlspecifies one atomic orbital.

- Orbitals with the same principal quantum number are said to be in the same principal level (shell).

- Orbitals with the same value of n and m are said to be in the same sublevel (subshell).

Quantum Mechanics

Quantum Mechanics

Orbital energies for a hydrogen atom depend only on the principal quantum number n. This means that all the subshells within a principal shell have the same energy. Orbitals at the same energy level are said to be __________.

Electronic Orbitals of Hydrogen

Quantum Mechanics

Question 32 from Tro (Chapter 7 – End of Chapter Problems)

List all the orbitals in each of the following principal levels. Specify the three quantum numbers for each orbital.

- n =1
- n = 2
- n = 3
- n = 4

Quantum Mechanics

Principal Energy Levels in Hydrogen

Quantum Mechanics

The Hydrogen Spectrum Explained !

- both the Bohr and Quantum Mechanical Models can predict these lines very accurately

Quantum Mechanics

Recall that ψ2gives the probability density

The Shapes of Atomic Orbitals

The Radial Distribution Function

The Shapes of Atomic Orbitals

A node is a point where both ψand ψ2all equal zero.

The ns orbitals (n > 1) are spherically symmetric like the 1s orbital. They are just bigger and have nodes.

The Shapes of Atomic Orbitals

There are three types of p orbitals. Each corresponds to a different ml quantum number.

The Shapes of Atomic Orbitals

The Shapes of Atomic Orbitals

Write an orbital designation corresponding to the quantum numbers n = 4, l = 2, ml = 0.

Write an orbital designation corresponding to the quantum numbers n = 3, l = 1, ml = 1.

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