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.