Physical Chemistry Third Edition

21.3 The CH 4 ,NH 3 , and H 2 O Molecules and thesp^3 Hybrid Orbitals 873

PROBLEMS

Section 21.2: The BH 3 Molecule and thesp^2 Hybrid
Orbitals

21.5 a.Using EXCEL
or some other graphing program,

construct an accurate graph of the value of the boron

ψ 2 sp (^2) (1)hybrid orbital along thexaxis, usingx/aas
the independent variable. Use a value ofZeffequal to
3.776.
b.Sketch the orbital region of theψ 3 sp (^2) (1)hybrid orbital.
21.6 a.Construct an accurate graph of the value of the boron
ψ 3 sp (^2) (1)hybrid orbital along thexaxis, usingx/aas
the independent variable. Use a value ofZeffequal
to 3.776.
b.Sketch the orbital region of theψ 3 sp (^2) (1)hybrid orbital.
21.7 Show that the 2sp^2 hybrid orbitals of Eq. (21.2-1) are
normalized and orthogonal to each other.
21.8In Chapter 20, the double bond in the C 2 molecule was
described as two pi bonds using theπu 2 pxandπu 2 py
bonding orbitals. Describe the bonding using a sigma and
pi bond. What hybrid orbitals are used for the sigma
bond? What orbitals are occupied by the lone pairs? Is the
Lewis octet rule satisfied? Do you think that the molecule
would have any unpaired electrons?
21.9Attempt a description of the O 2 molecule using a sigma
bond and a pi bond for the double bond instead of the
bonding described in Table 20.2. Does your description
include the two unpaired electrons?
21.10Using LCAOMOs containing no more than two atomic
orbitals, describe the bonding in the CO 2 molecule. How
do the two pi bonding LCAOMOs differ from each other
in their orientation?

21.3 The CH 4 ,NH 3 , and H 2 O Molecules

and thesp^3 Hybrid Orbitals

The Methane Molecule

Methane, CH 4 , has an equilibrium nuclear conformation with four bonds oriented

symmetrically, with all bond angles equal to 109.5◦. In order to create LCAOMOs that

contain only two atomic orbitals and point in these directions, we create a third set of

hybrid orbitals, the 2 sp^3 hybrid orbitals:

ψ 2 sp (^3) (1)

1

2

[−ψ 2 s+ψ 2 px+ψ 2 py+ψ 2 pz] (21.3-1a)

ψ 2 sp (^3) (2)

1

2

[−ψ 2 s+ψ 2 px−ψ 2 py−ψ 2 pz] (21.3-1b)

ψ 2 sp (^3) (3)

1

2

[−ψ 2 s−ψ 2 px+ψ 2 py−ψ 2 pz] (21.3-1c)

ψ 2 sp (^3) (4)

1

2

[−ψ 2 s−ψ 2 px−ψ 2 py+ψ 2 pz] (21.3-1d)

Exercise 21.3

Using the fact that the 2s,2px,2py, and 2pzatomic orbitals are normalized and are orthogonal

to each other, choose one of the hybrid orbitals in Eq. (21.3-1) and show that it is normalized.

Choose a pair of orbitals in Eq. (21.3-1) and show that they are orthogonal to each other.