3.4 QUANTUM NUMBERS

WHAT IS A QUANTUM NUMBER?

From the solutions to Schrodinger’s wave equations, these are numbers that describe the properties of each orbital.

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Think of locating

an electron by

address...

THE PRINCIPAL QUANTUM NUMBER (n)

Energy levels in an atom are sometimes called shells. The principal quantum number (n) describes the size and energy of an atomic orbital. (n = 1, 2, 3, and so on)

SUBSHELLS

Albert Michelson - the distinct lines we see in Bohr’s line spectrum are actually made of many smaller lines.

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Arnold Sommerfeld - proposed the secondary quantum number to describe electron energy sublevels or subshells.

SUBSHELLS:

Staircase

Analogy

THE SECONDARY QUANTUM NUMBER (l)

- describes the shape and energy of an atomic orbital (represents subshells - think of small energy level steps within the main energy level)

- whole number values range from 0 to n-1 for each value of n

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ex. when n = 1, l = 0

when n = 3, l = 0, 1, and 2

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We use letters to represent l to avoid confusion

SECONDARY QUANTUM NUMBERS AND THEIR ENERGY LEVELS

How many orbitals exist at n=1?

How many orbitals exist at n=3? What are they?

MAGNETIC QUANTUM NUMBER (m_l)

- represents the orientations of subshells

(ie. the orientation of an atomic orbital in space relative to the other orbitals in the atom)

- represented by whole number values between +l and -l including 0

- the number of different values that m_l can have is equal to the number of orbitals that are possible

MAGNETIC QUANTUM NUMBER (m_l)

EXAMPLE:

When l = 1, how many different orbitals are there?

SHAPES AND ORIENTATION OF ORBITALS

Each orbital has a unique probability distribution, shape, and orientation.

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Let’s start with “s” orbitals. Think of

s = spherical!

p ORBITALS

- have two lobes separated by a node

at the nucleus (think dumbbell shaped)

- can exist in any of 3 dimensions and are labeled using

xyz coordinate

system

VIDEO : s and p orbitals

d ORBITALS

- the first d orbitals occur in n=3 energy level

- there are 5 of them…

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SPIN QUANTUM NUMBER (m_s)

This is the quantum number that relates to the spin of the electron. It can either be +½ or -½.

THE PAULI EXCLUSION PRINCIPLE

“In a given atom, no two electrons can have the same set of four quantum numbers (n, l, m_l, and m_s)”

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Since electrons in the same orbital have the same values for n, l, and m_l, they must have different m_s values.

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This means that an orbital can only hold 2 electrons which must have opposite spins.

EXAMPLE 1:

For the principal quantum number n=5, determine the value(s) of the secondary quantum number l and the types of orbitals in each case.

EXAMPLE 2:

How many possible values of m_l are there for

l = 0, 1, 2, and 3? What pattern do you notice in these numbers?

HOMEWORK:

p. 158 #1, 2

p. 159 #1, 3-9

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QUIZ THURSDAY!

3.1->3.4

Trying to learn about Quantum Theory from Einstein himself!