In which you will learn about:•The quantum mechanical model•Heisenberg uncertainty principle

•Orbitals and their shapes•Quantum numbers

Unit 3: Atomic Theory & Quantum Mechanics

Section A.6 – A.7

A.6 The Quantum Mechanical Model of the AtomScientists in the mid-1920s, by then

convinced that the Bohr atomic model was incorrect, formulated new and innovative explanations of how electrons are arranged in atoms.

In 1924, a French graduate student in physics named Louis de Broglie (1892-1987) proposed an idea that eventually accounted for the fixed energy levels of Bohr’s model

Electrons as WavesDe Broglie had been thinking that Bohr’s quantized

electron orbits had characteristics similar to those of waves.Imagine the path of an electron around a circle of

fixed radius

Notice that in figure (a) there is an odd number of waves, and in (b) there is an even number of waves(a) works out perfectly and (b) does not

Electrons as Waves Cont’dOnly multiples of half-wavelengths are possible

on a plucked guitar string because the string is fixed at both ends

These finite waves for musical instruments and circles led de Broglie to ask an interesting questionIf light can act as both a wave and particle, can a

particle (like an electron) act like a wave?

De Broglie EquationThe de Broglie equation predicts that all moving

particles have wave characteristicsIt also explains why it is impossible to notice the

wavelength of a fast-moving car – a car moving at 25 m/s and weighing 910 kg would have a wavelength of 2.9 x 10-38 m (too small to be seen or detected!)

For comparison, an electron moving at the same speed has a wavelength of 2.9 x 10-5 m (which can be easily measured).

De Broglie knew that if an electron has wavelike motion and is restricted to circular orbits of fixed radius, only certain wavelengths, frequencies and energies are possible. This is summed up in the equation λ = h/mv, where λ

is still wavelength, h is still Planck’s constant, m is mass and v is velocity (speed).No, we will NOT be doing calculations with this equation.

The Heisenberg Uncertainty PrincipleWerner Heisenberg (1901-1976) showed that it

is impossible to take any measurement of an object without disturbing the objectScientists try to locate electrons by bombarding

them with photons of light, but once the electron is hit, it moves to a new location with a new speed!

In other words, the act of observing the electron produces a significant, unavoidable uncertainty in the position and motion of the electron

The Heisenberg uncertainty principle states that it is fundamentally impossible to know precisely both the velocity and position of a particle at the same time.

The Real Deal with HeisenbergSo we can’t know exactly where or how fast

an electron is moving

Which means that it is impossible to assign fixed paths for electrons like the circular orbits in Bohr’s model.

The only quantity that can be known is the probability for an electron to occupy a certain region around the nucleus.

The Schrödinger Wave EquationIn 1926, Austrian physicist Erwin Shrödinger (1887-

1961) furthered the wave-particle theory proposed by de Broglie.

Shrödinger derived an equation that treated the hydrogen atom’s electron as a wave

Shrödinger’s new model for the hydrogen atom seems to apply equally well to atoms of other elementsThis is where Bohr’s model failed

The atomic model in which electrons are treated as waves is called the wave mechanical model of the atom, or the quantum mechanical model of the atom.This model limits an electron’s energy to certain values

(like Bohr)Makes no attempt to describe the electron’s path

around the nucleus

Shrödinger is way too complicatedThe Shrödinger wave equation is too

complex to be considered hereMrs. Pford didn’t deal with it until junior year

of college and she had to use Calculus III-level math to solve it!

Solutions to the equation are called wave functionsWave functions are related to the probability

of finding the electron within a certain volume of space around the nucleus

Electron’s probable locationThe wave function predicts a three-

dimensional region around the nucleus, called an atomic orbital, which describes the electron’s probable location.An atomic orbital is like a fuzzy cloud in

which the density at a given point is proportional to the probability of finding the electron at that point

Density MapsThe density map can be thought of as a time-

exposure photograph of the electron moving around the nucleusThe electron cloud = all the probably places ONE

electron COULD beThe electron cloud ≠ all of the electrons in an atom

surrounding the nucleusTo overcome the inherent uncertainty about the

electron’s location, chemistry arbitrarily draw an orbital’s surface to contain 90% of the electron’s total probability distribution.Simply, there is a 90% chance you will find an

electron somewhere within the electron cloud

A.7 Quantum NumbersThere are four quantum numbers that are

used to describe the probable position of an electron.

Each quantum number is usually only referenced by name or variable, but there are also actual numbers, too

No two electrons can have the same exact set of four quantum numbers (more on this next time)

Principal Quantum Number (1st)The principal quantum number (n) indicates

the relative size and energy of atomic orbitals

In other words, n = energy level

n can have whole-number values ranging from 1-7.

If quantum numbers were an address, this is like telling you what state the electron lives in (not very specific if I want to find it)

Angular Momentum Quantum Number (2nd)The angular momentum quantum number (l) specifies

the shape of the orbital that the electron is in.This is sometimes referred to as the sublevel, but I find

this term to be confusing. I try to explain below (somewhat unsuccessfully?)Sublevel = shape of orbital or orbital type (can be s, p, d, or f)Orbital = specific orientation of sublevel (can be px, py, or pz

depending on the axis the density map is on)l can have whole number values ranging from 0 to n-1.

If l = 0 it’s an s orbital, if l = 1 p, if l = 2 d, if l = 3 fSee next slide for pictures of orbitals

In the address analogy, using this number helps specify which city the electron is in.

Before We Move On…Shapes of orbitals include:

Number of OrbitalsThere is only ONE type of s orbital

There are THREE types of p orbital

There are FIVE types of d orbital

There are SEVEN types of f orbital (not shown in previous slide)

Hydrogen’s First Four Principal Energy LevelsPrincipal Quantum Number (n)

Sublevels (Types of Orbitals) Present

Number of Orbitals Related to Sublevel

Total Number of Orbitals in Energy Level (n2)

1 s 1 1

2sp

13 4

3spd

135

9

4spdf

1357

16

Magnetic Quantum Number (3rd)The magnetic quantum number (m)specifies

which orientation of the orbitals an electron is in For example, if we know the electron is in

energy level 2 and it is in the p-type orbital, we need to know exactly which p-orbital it is in (there are three possibilities)

m can have integer values going from –l to +l.

In the address analogy, this is like giving the street the electron is on.

Spin Magnetic Quantum Number (4th)The spin magnetic quantum number (ms) is an

inherent property of electrons that separate them into individual positions. Up until this point, two electrons can share the first

3 quantum numbers, but since no two electrons can share ALL four, we use spin to indentify which is which

The electrons aren’t actually spinning! We refer to them as spin up or spin down but these are just arbitrary terms.

ms can only be +1/2 or -1/2

In the address analogy, this is like finally giving the house number where the electron is at.

Using Quantum NumbersIf you’ve been following along with the rules…

n = 1, 2, 3, 4, 5, 6, or 7l = 0 up to n-1 in integers (0 =s, 1 = p, 2 = d, 3 = f)m = - l up to +l in integersms = ±1/2

Example Problem: Write the quantum numbers associated with the first electron added to the 4f sublevel.

ANSWER: n = 4 (given), l = 3 (known because it’s an f orbital), m = 3 (-l up to +l in integers and in this case l = 3 – I’m choosing 3, it could be -3, -2, -1, 0, 1, 2 or 3), and ms = +1/2 (again, I’m choosing + because it can only be for ONE electron).

And now that you’re completely confused…Homework!1) Differentiate between the wavelength of visible

light and the wavelength of a moving soccer ball.2) List the number and types of orbitals contained

in the hydrogen atom’s first four energy levels. 3) Explain why the location of an electron in an

atom is uncertain using the Heisenberg uncertainty principle and de Broglie’s wave-particle duality. How is the location of electrons in the atom defined?

4) Compare and contrast Bohr’s model and the quantum mechanical model of the atom.

5) Write the numbers associated with each of the following:A) the fifth energy levelB) the 6s sublevelC) an orbital on the 3d level