Burdge/Overby, Chemistry: Atoms First, 2e Ch14 | Page 4

S E C T ION 14.2 ?Entropy

two molecules, the number of possible arrangements is equal to the number of cells squared. (Note
that a cell may contain more than one molecule.) Each time we increase the number of molecules
by one, the number of possible arrangements increases by a factor equal to the number of cells. In
general, for a volume consisting of X cells, and containing N molecules, the number of possible
arrangements, W, is given by the equation

W = XN

573

Student Annotation: The number of
possible arrangements is sometimes called
the number of microstates.

Equation 14.2

Figure 14.2 illustrates this for a simple case involving just two molecules. We imagine the container is divided into four cells each with volume v. Initially, both molecules are confined to the
left side, which consists of two cells. With two molecules in two cells, there are 22 = 4 possible
arrangements of the molecules [Figure 14.2(a)]. When the barrier is removed, doubling the volume
available to the molecules, the number of cells also doubles. With four cells available, there are
42 = 16 possible arrangements of the molecules. Eight of the sixteen arrangements have the molecules on opposite sides of the container [Figure 14.2(b)]. Of the other eight arrangements, four
have both molecules on the left side [as shown in Figure 14.2(a)], and four have both molecules on
the right side (not shown). There are three different states possible for this system.
1. One molecule on each side (eight possible arrangements)
2. Both molecules on the left (four possible arrangements)
3. Both molecules on the right (four possible arrangements)
The most probable state is the one with the largest number of possible arrangements. In this
case, the most probable state is the one with one molecule on each side of the container. The same
principle applies to systems with larger numbers of molecules. Increasing the number of molecules
increases the number of possible arrangements, but the most probable state will be the one in
which the gas molecules are divided evenly between the two sides of the container.

Student Annotation: The state with the
largest number of possible arrangements
has the greatest entropy.

(a)

(b)

Figure 14.2 (a) Before the barrier is removed, the molecules are both in the left side of the container, which we imagine is divided into two cells of equal
volume. There are four possible arrangements of two molecules in two cells. (b) When the barrier between the two sides of the container is removed, the volume
(and the number of cells) available to the molecules doubles. The new number of possible arrangements is 42 = 16, eight of which have the molecules in opposite sides of the container—the most probable outcome.

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