1. Chapter 7 The Quantum-Mechanical Model of the Atom

2. The Quantum Mechanical Model of the Atom: Orbitals and Quantum Numbers
electron energy and position are complimentary
because KE = ½mv2
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 – called an orbital
a probability distribution map of a region where the electron is likely to be found
distance vs. y2 (wave function)
many of the properties of atoms are related to the energies of the electrons
Probability of finding
electron in a region
of space (Y 2)
Solve
Wave
equation
Wave function
or orbital (Y)
calculations show that the size, shape and orientation in space of an orbital are determined be three integer terms in the wave function
added to quantize the energy of the electron
these integers are called quantum numbers
principal quantum number, n
angular momentum quantum number, l
magnetic quantum number, ml
Since we can’t ever be certain of the electron’s position, we work with probabilities.

3. Electron Energy electron energy and position are complimentary
because KE = ½mv2
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 – called an orbital
a probability distribution map of a region where the electron is likely to be found
distance vs. y2 (wave function)
many of the properties of atoms are related to the energies of the electrons

4. Wave Function, y
calculations show that the size, shape and orientation in space of an orbital are determined be three integer terms in the wave function
added to quantize the energy of the electron
these integers are called quantum numbers
principal quantum number, n
angular momentum quantum number, l
magnetic quantum number, ml

5. The Quantum Mechanical Model of the Atom: Orbitals and Quantum Numbers
Principal Quantum Number (n)
Describes the size and energy level of the orbital
Commonly called a shell
Positive integer (n = 1, 2, 3, 4, …)
As the value of n increases,
the energy increases.
the average distance of the e– from the nucleus increases.

6. The Quantum Mechanical Model of the Atom: Orbitals and Quantum Numbers
Angular-Momentum Quantum Number (l)
Defines the three-dimensional shape of the orbital
Commonly called a subshell
There are n different shapes for orbitals.
If n = 1, then l = 0.
If n = 2, then l = 0 or 1.
If n = 3, then l = 0, 1, or 2.
Commonly referred to by letter (subshell notation)
l = 0 s (sharp)
l = 1 p (principal)
l = 2 d (diffuse)
l = 3 f (fundamental)
Scientists working in spectroscopy first used sharp, principal, diffuse, and fundamental with respect to atomic spectra.
After f, the series goes alphabetically (g, h, etc.).
the energy of the subshell increases with l (s < p < d < f).

7. The angular Momentum Quantum number (l)
Is an integer that determines the shape of the orbital.
Quantum number n
(shell)
Value of l
Letter designation
(subshell)
n = 1
l = 0 s
n = 2
l = 1 p
n = 3
l = 2 d
n = 4
l = 3 f
the energy of the subshell increases with l (s < p < d < f).

8. The Quantum Mechanical Model of the Atom: Orbitals and Quantum Numbers
Magnetic Quantum Number (ml )
Defines the spatial orientation of the orbital
There are 2l + 1 values of ml, and they can have any integral value from -l to +l.
If l = 0, then ml = 0.
If l = 1, then ml = –1, 0, or 1.
If l = 2, then ml = –2, –1, 0, 1, or 2. n l
# Orbitals ml 1 2
0, 1
-1, 0, 1 3
0, 1, 2
-2, -1, 0, 1, 2 4
0, 1, 2, 3
-3, -2, -1, 0, 1, 2, 3, 4

9. Examples
Give the possible combination of quantum numbers for the following orbitals
3s orbital 2 p orbitals
Give orbital notations for electrons in orbitals with the following quantum numbers:
n = 2, l = 1 ml = 1 n = 3, l = 2, ml = -1

10. Principal Quantum Number, n
characterizes the energy of the electron in a particular orbital
corresponds to Bohr’s energy level
n can be any integer ³ 1
the larger the value of n, the more energy the orbital has
energies are defined as being negative
an electron would have E = 0 when it just escapes the atom
the larger the value of n, the larger the orbital
as n gets larger, the amount of energy between orbitals gets smaller
–The negative sign means that the energy of the electron bound to the nucleus is lower than it would be if the electron were at an infinite distance (n = ∞) from the nucleus, where there is no interaction.
for an electron in H

11. Principal Energy Levels in Hydrogen

12. Electron Transitions
in order to transition to a higher energy state, the electron must gain the correct amount of energy corresponding to the difference in energy between the final and initial states
electrons in high energy states are unstable and tend to lose energy and transition to lower energy states
energy released as a photon of light
each line in the emission spectrum corresponds to the difference in energy between two energy states

13. Hydrogen Energy Transitions

14. Predicting the Spectrum of Hydrogen
the wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states
for an electron in energy state n, there are (n – 1) energy states it can transition to, therefore (n – 1) lines it can generate
both the Bohr and Quantum Mechanical Models can predict these lines very accurately
Since the energy must be conserved, the exact amount of energy emitted by the atom is carried away by the photon
ΔEatom = - ΔEphoton

15. Example
Determine the wavelength of light emitted when an electron in a hydrogen atom makes a transition from an orbital in n= 6 to an orbital in n=5
As electron in the n=6 level of the hydrogen atom relaxes to a lower energy level, emitting light of λ= 93.8 nm. Find the principle level to which the electron relaxed

16. Probability Density Function

17. The Shapes of Atomic Orbitals
the l quantum number primarily determines the shape of the orbital
l can have integer values from 0 to (n – 1)
each value of l is called by a particular letter that designates the shape of the orbital
s orbitals are spherical
p orbitals are like two balloons tied at the knots
d orbitals are mainly like 4 balloons tied at the knot
f orbitals are mainly like 8 balloons tied at the knot

18. l = 0, the s orbital each principal energy state has 1 s orbital
lowest energy orbital in a principal energy state
spherical
number of nodes = (n – 1)

19. l = 1, p orbitals
each principal energy state above n = 1 has 3 p orbitals
ml = -1, 0, +1
each of the 3 orbitals point along a different axis
px, py, pz
2nd lowest energy orbitals in a principal energy state
two-lobed
node at the nucleus, total of n nodes
Tro, Chemistry: A Molecular Approach

20. p orbitals

21. l = 2, d orbitals
each principal energy state above n = 2 has 5 d orbitals
ml = -2, -1, 0, +1, +2
4 of the 5 orbitals are aligned in a different plane
the fifth is aligned with the z axis, dz squared
dxy, dyz, dxz, dx squared – y squared
3rd lowest energy orbitals in a principal energy state
mainly 4-lobed
one is two-lobed with a toroid
planar nodes
higher principal levels also have spherical nodes

22. d orbitals

23. l = 3, f orbitals
each principal energy state above n = 3 has 7 d orbitals
ml = -3, -2, -1, 0, +1, +2, +3
4th lowest energy orbitals in a principal energy state
mainly 8-lobed
some 2-lobed with a toroid
planar nodes
higher principal levels also have spherical nodes

24. f orbitals

25. Now we know why atoms are spherical