Quantum+Numbers+and+Atomic+Orbitals

```Lecture PowerPoint
Chemistry
The Molecular Nature of
Matter and Change
Seventh Edition
Martin S. Silberberg
and Patricia G. Amateis
7-1
Quantum Numbers and Atomic Orbitals
7-2
Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three quantum numbers.
The set of numbers used to describe the position and energy
of the electron in an atom are called quantum numbers.
The principal quantum number (n) is a positive integer.
The value of n indicates the relative size of the orbital and therefore its
relative distance from the nucleus.
The angular momentum quantum number (l) is an integer
from 0 to (n –1).
The value of l indicates the shape of the orbital.
The magnetic quantum number (ml) is an integer with
values from –l to +l.
The value of ml indicates the spatial orientation of the orbital.
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Angular Momentum Quantum Number
The angular momentum quantum number (l) describes the shape
of the orbital.
The values of l are integers that depend on the value of the
principal quantum number
The allowed values of l range from 0 to n – 1.
&Oslash; Example: If n = 2, l can be 0 or 1.
l
0
1
2
3
Orbital designation
s
p
d
f
A collection of orbitals with the same value of n and l is referred to
as a subshell.
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Magnetic Quantum Number
The magnetic quantum number (ml) describes the orientation of
the orbital in space.
The values of ml are integers that depend on the value of the
angular moment quantum number:
– l,…0,…+l
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Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals
Name, Symbol
(Property)
Allowed Values
Quantum Numbers
Principal, n
Positive integer
(size, energy)
(1, 2, 3, ...)
1
Angular
momentum, l
0 to n – 1
(shape)
0
0
0
0
Magnetic, ml
-l,…,0,…,+l
(orientation)
2
3
1
0
2
0
-1 0 +1
-1 0 +1
-2
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1
-1
0
+1 +2
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Sample Problem 7.6
PROBLEM:
Determining Quantum Numbers for an
Energy Level
What values of the angular momentum (l) and magnetic
(ml) quantum numbers are allowed for a principal quantum
number (n) of 3? How many orbitals are allowed for n = 3?
PLAN: Values of l are determined from the value for n, since l can take
values from 0 to (n – 1). The values of ml then follow from the
values of l.
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Sample Problem 7.6
PROBLEM:
Determining Quantum Numbers for an
Energy Level
What values of the angular momentum (l) and magnetic
(ml) quantum numbers are allowed for a principal quantum
number (n) of 3? How many orbitals are allowed for n = 3?
PLAN: Values of l are determined from the value for n, since l can take
values from 0 to (n – 1). The values of ml then follow from the
values of l.
SOLUTION:
For n = 3, allowed values of l are = 0, 1, and 2
For l = 0, ml = 0
For l = 1, ml = –1, 0, or +1
For l = 2, ml = –2, –1, 0, +1, or +2
There are 9 ml values and therefore 9 orbitals with n = 3.
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7-12
Sample Problem 7.7
Determining Sublevel Names and Orbital
Quantum Numbers
PROBLEM: Give the name, magnetic quantum numbers, and number
of orbitals for each sublevel with the following quantum
numbers:
(a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3
PLAN: Combine the n value and l designation to name the sublevel.
Knowing l, we can find ml and the number of orbitals.
SOLUTION:
n
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l
sublevel name possible ml values # of orbitals
Sample Problem 7.7
Determining Sublevel Names and Orbital
Quantum Numbers
PROBLEM: Give the name, magnetic quantum numbers, and number
of orbitals for each sublevel with the following quantum
numbers:
(a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3
PLAN: Combine the n value and l designation to name the sublevel.
Knowing l, we can find ml and the number of orbitals.
SOLUTION:
n
l
sublevel name possible ml values # of orbitals
(a)
3
2
3d
(b)
2
0
2s
0
1
(c)
5
1
5p
–1, 0, 1
3
(d)
4
3
4f
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–2, –1, 0, 1, 2
–3, –2, –1, 0, 1, 2, 3
5
7
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Sample Problem 7.8
Identifying Incorrect Quantum Numbers
PROBLEM: What is wrong with each of the following quantum numbers
designations and/or sublevel names?
l
n
ml
Name
SOLUTION:
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(a)
1
1
0
1p
(b)
4
3
+1
4d
(c)
3
1
–2
3p
Sample Problem 7.8
Identifying Incorrect Quantum Numbers
PROBLEM: What is wrong with each of the following quantum numbers
designations and/or sublevel names?
l
n
ml
Name
(a)
1
1
0
1p
(b)
4
3
+1
4d
(c)
3
1
–2
3p
SOLUTION:
(a) A sublevel with n = 1 can only have l = 0, not l = 1. The only possible
sublevel name is 1s.
(b) A sublevel with l = 3 is an f sublevel, not a d sublevel. The name
should be 4f.
(c) A sublevel with l = 1 can only have ml values of –1, 0, or +1, not –2.
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Figure 7.17
7-19
The 1s, 2s, and 3s orbitals.
Figure 7.18
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The 2p orbitals.
Figure 7.19
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The 3d orbitals.
Figure 7.19 continued
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Figure 7.20
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The 4fxyz orbital, one of the seven 4f orbitals.
Figure 7.21
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Energy levels of the H atom.
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