Chemistry - A Molecular Science

this second factor by considering how molecules move. The motion of molecules is a combination of three different types of movement (translation, rotation, and vibration) called

degrees of freedom

.

1.

Translation

is the straight line motion of the ent

ire molecule through space, which can be

described in terms of its x, y, and z component

s. Thus, all particles have three degrees of

translational freedom. A molecule with N atoms must have 3N degrees of freedom, three of which are translational, because each of

its atoms can move in three directions.

2.

Rotation

is the circular motion of a molecule about an axis fixed to the molecule. There

are three axes in a molecule (X, Y, and Z),

so there are three degrees of rotational

freedom. However, energy cannot be stored in

a rotation if all of the atoms lie on the

axis, so linear molecules have only two rotational degrees of freedom.

3.

Vibration

is the motion of the atoms in a molecu

le relative to one another that causes

small changes in bond lengths and/or bond angles. A molecule has 3N degrees of freedom, but six are due to translation and rotation, so non-linear molecules have 3N-6 vibrational degrees of freedom. Linear molecules have 3N-5 degrees because they have only two degrees of rotational freedom.

Figure 9.2 shows the six degrees of freedom

of CO: 3 translational, 2 rotational, and

3N - 5 = 3(2) - 5 = 1 vibrational.

Consider a system of ten books. There are 10! = 3,628,800

† ways in which they can be

distributed on a shelf, but only one is alphabetical. Thus, constraining (restricting or confining) the books to an alphabetical order

greatly reduces the number of ways in which

they can be distributed. Removing and returning books from the alphabetical arrangement without intervention (no librarian), would result in a random arrangement because a random arrangement is favored on strictly

statistical grounds (3,628,799 random to 1

alphabetical). Similarly, energy is more likely to be found in systems that can distribute the energy in more ways. For this purely sta

tistical reason, systems that have many ways

in which to distribute their energy are favored over those that have only a few.

The number of ways in which the energy of

a molecular system can be distributed

decreases as constraints on

molecular motion are added

. For example, all of the thermal

energy of an O atom is translational, while that of an O

molecule is a combination of 2

translational, rotational, and vibrational. Thus, the energy of an O atom, whose energy is disbursed over only three degrees of freed

om is more constrained than an O

molecule, 2

whose energy can be spread over six degrees of

freedom.* Freedom of movement not only

applies to the number of degrees of freedom; it

also applies to how freely a molecule can

move in each degree of freedom. For example, a molecule in the gas phase has a great deal

x

y

z

C

O

C

O

C

O C

O

C

O C

O

(a)

(b)

(c)

Rx

Ry

Tx

Ty

TZ n

Figure 9.2 Six Degrees of Freedom of a Diatomic Molecule a) Translations result when all atoms move in the same direction. T

(^) x
is a translation along the x-axis. b) Rotations occur when atoms move in opposite directions perpendicular to the bond. R
is a y
rotation about the y-axis. There is no rotation about the bonding axis in a linear molecule. c) A vibration is caused by atoms moving in opposite directions along the bond axis.
ν is called a C
≡O stretch.

• Increasing the number of atoms always increases the number of
  degrees of freedom and the number
  of ways in which the energy
  can be distributed. Thus, O
  has nine degrees of freedom and 3
  can distribute its energy in more ways than can O

. 2

† 10! (spoken ten factorial) = 10

..^9

. 87
..^6
. 54
..^3
. 2 1.

Chapter 9 Reaction Energetics

191

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State

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