Physical Chemistry , 1st ed.

nN 2 

R

pV

T

0.223 mol N 2

nN 2 O

R

pV

T

0.078 mol N 2 O

Since the total number of moles is 0.223 mol + 0.078 mol 0.301 mol, we

can now calculate the mole fractions of each component:

xN 2 

0

0

.

.

2

3

2

0

3

1

m

m

o

o

l

l

0.741

xN 2 O^0

0

.

.

0

3

7

0

8

1

m

m

o

o

l

l

0.259

(Note that the sum of the mole fractions is 1.000, as required.) We can use

equation 3.23 to determine mixS:

mixS8.314 

mo

J

lK

(0.223 mol ln 0.741 + 0.078 mol ln 0.259)

The mol units cancel and we evaluate to find

mixS1.43 

K

J



Notice that this problem uses two different values for R, the ideal gas law con-

stant. In each case, the choice was dictated by the units that were necessary

to solve that particular part of the problem.

3.6 Order and the Third Law of Thermodynamics

The preceding discussion of the entropy of mixing brings us to a useful gen-

eral idea regarding entropy, that oforder.Having two pure gases on either side

of a barrier is a nice, neat, relatively ordered arrangement. Mixing the two of

them, a process that occurs spontaneously, is a more random, less ordered

arrangement. So this system proceeds spontaneously from a more ordered

system to a less ordered system.

In the mid- to late-1800s, the Austrian physicist Ludwig Edward Boltzmann

(Figure 3.6) began applying the mathematics of statistics to the behavior of

matter, especially gases. In doing so, Boltzmann was able to determine a dif-

ferent definition for entropy. Consider a system of gas molecules that all have

the same chemical identity. The system can be broken up into smaller micro-

systems whose individual states contribute statistically to the overall state of

the system. For any particular number of microsystems, there are a certain

number of ways of distributing the gas molecules into the microsystems. If the

most probable distribution has different ways of arranging the particles,*

(0.550 atm)(3.50 L)



(0.08205 mLoaltmK)(300.0 K)

(0.550 atm)(10.0 L)

(0.08205 mLoaltmK)(300.0 K)

3.6 Order and the Third Law of Thermodynamics 79

*For example, say you have a simple system consisting of two balls and four shoe boxes.

There are 10 possible arrangements for putting the balls in the boxes: four arrangements

with both balls in a single box (the other three are empty), and six arrangements with one

ball in each of two boxes (the other two are empty). The most probable arrangement is one

ball in each of two boxes, and there are six different ways of getting that arrangement.

Therefore, equals 6 in this case. Chapter 17 gives more details on this and other concepts

relating to Boltzmann’s interpretation of entropy.

Figure 3.6 Ludwig Edward Boltzmann (1844–

1906), Austrian physicist. Boltzmann used the

relatively young idea of atoms to develop a statis-

tical mathematical description of matter, which

eventually introduced the concept of order as a

measure of entropy. Although his work is of

profound importance in thermodynamics, the

wrangling over ideas and critiques at that crucial

period in the history of science is thought to have

been a contributing factor in his suicide.

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