Quantum MechanicalQuantum MechanicalModel of the AtomModel of the Atom

Mathematical laws can identify the regions outside of the nucleus where electrons are most likely to be found.

These laws are beyond the scope of this class…

Schrodinger Wave EquationSchrodinger Wave Equation

22

2 2

8dh EV

m dx

Equation for probabilityprobability of a single electron being found along a single axis (x-axis)

Erwin Schrodinger

Heisenberg Uncertainty PrincipleHeisenberg Uncertainty Principle

You can find out where the electron is, but not where it is going.

OR…

You can find out where the electron is going, but not where it is!

“One cannot simultaneously determine both the position and momentum of an electron.”

WernerHeisenberg

QUANTUM NUMBERSQUANTUM NUMBERSQUANTUM NUMBERSQUANTUM NUMBERSWe will focus now on We will focus now on polyelectronicpolyelectronic atoms – atoms atoms – atoms

with with moremore than one electron. than one electron.The The shape, size, shape, size, andand energy energy of each orbital is a function of each orbital is a function

of 3 quantum numbers which describe the location of of 3 quantum numbers which describe the location of an electron within an atom or ionan electron within an atom or ion

nn ((principalprincipal)) ---> energy level---> energy levell (l (orbitalorbital) ) ---> shape of orbital---> shape of orbitalmmll ((magneticmagnetic) ---> designates a particular ) ---> designates a particular

suborbitalsuborbitalThe fourth quantum number is not derived from the The fourth quantum number is not derived from the

wave functionwave functionss ((spinspin) ---> spin of the electron ) ---> spin of the electron

(clockwise or counterclockwise: ½ or – ½)(clockwise or counterclockwise: ½ or – ½)

Principal Quantum NumberThe principal quantum number, n, denotes the probable

distance of an electron from the nucleus.

n = energy levels (shells) = 1, 2, 3, 4, ….

distance of e- from the nucleus

The angular momentum quantum number, l, denotes the orbital (subshell or sublevel) in which an electron is located.

for a given value of n, l = 0, 1, 2, 3, … n-1

n = 1, l = 0n = 2, l = 0 or 1

n = 3, l = 0, 1, or 2

Shape of the “volume” of space that the e- occupies

l = 0 s orbitall = 1 p orbitall = 2 d orbitall = 3 f orbital

Angular Momentum Quantum NumberAn orbital is a region within an energy level where there is a probability of finding an electron.

The s orbital (l = 0) has a spherical shape centered aroundthe origin of the three axes in space.

ss Orbital shape Orbital shape

There are three dumbbell-shaped p orbitals (l = 1) in each energy level above n = 1, each assigned to its own axis (x, y and z) in space.Note: there is a planar node through the nucleus, which is a area of zero probability of finding an electron.

pp orbital shape orbital shape

Planar node

Things get a bit more complicated with the five d orbitals (l = 2) that are found in the d sublevels beginning with n = 3. To remember the shapes, think of “double dumbells”

…and a “dumbell with a donut”!

dd orbital shapes orbital shapes

Shape of Shape of ff ( (ll = 3) orbitals = 3) orbitals

The magnetic quantum number m, denotes the orientation of an electron’s orbital with respect to the three axes in space.

for a given value of lml = -l, …., 0, …. +l

orientation of the orbital in space

if l = 1 (p orbital), ml = -1, 0, or 1if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2

Magnetic Quantum Number

Assigning the NumbersAssigning the Numbers The three quantum numbers (n, l, and m) are integers.

The principal quantum number (n) cannot be zero.

n must be 1, 2, 3, etc.

The angular momentum quantum number (l ) can be any integer between 0 and n - 1.

For n = 3, l can be either 0, 1, or 2.

The magnetic quantum number (ml) can be any integer between -l and +l.

For l = 2, m can be either -2, -1, 0, +1, +2.

Quantum numbers for the first four levels of orbitals in the hydrogen atom

n l Orbital designation

ml # of orbitals

1 0 1s 0 1

2 0 2s 0 1

1 2p -1, 0, 1 3

3 0 3s 0 1

1 3p -1, 0, 1 3

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

4 0 4s 0 1

1 4p -1, 0, 1 3

2 4d -2, -1, 0, 1, 2 5

3 4f -3, -2, -1, 0, 1, 2, 3 7

spin quantum number ms

ms = +½ or -½

Spin Quantum Number

ms = -½ms = +½

The spin quantum number, ms, describes the behavior of an electron in a magnetic field.

Pauli exclusion principle - no two electrons in an atom can have the same four quantum numbers.

Each seat is uniquely identified (E, R12, S8)Each seat can hold only one individual at a time

Shell – electrons with the same value of n

Subshell or sublevel – electrons with the same values of n and lOrbital – electrons with the same values of n, l, and ml

How many electrons can an orbital hold?

If n, l, and ml are fixed, then ms = ½ or - ½

An orbital can hold 2 electrons

s orbitalss orbitals d orbitalsd orbitals

Number ofNumber oforbitalsorbitals

Number of Number of electronselectrons

11 33 55

22 66 1010

p orbitalsp orbitals f orbitalsf orbitals

77

1414

How many How many electrons electrons can be in a can be in a sublevel sublevel subshell?subshell?

Remember: A maximum of two electrons can be Remember: A maximum of two electrons can be placed in an orbital.placed in an orbital.