Physical Chemistry , 1st ed.

But we can also use equation 17.3 if we first determine the probabilities of

each score in the grades. Since the score of 0 does not appear in the list of in-

dividual scores, we say that P 0 ^07 . A score of 1 does not appear either, so

P 1 ^07 also. But one score of 2 does appear once out of seven grades, so we

can say that P 2 ^17 . Similarly, we have P 3 ^07 ,P 4 ^17 ,P 5 ^07 ,P 6 ^07 ,P 7 ^17 ,

P 8 ^17 ,P 9 ^27 (there are two 9’s), and P 10 ^17 . (You should verify all these

probabilities.) The individual score (out of 10) is represented by uj, so by equa-

tion 17.3 we have for the average score

17.2 Some Statistics Necessities 589

score



10

j 1

ujPj





j

Pj

We get the same answer for the average score. Although for this example the

probability method is more cumbersome, for large numbers of values (here,

“scores”) it will become easier than the simple averaging method.

For a small number of possible arrangements, it is easy to determine the

total number of possibilities by counting them. For example, if Figure 17.3a

represents the possibilities for various arrangements of a system, the total

number of possible arrangements is determined by adding the individual values

represented by the bars in the graph. However, as the number of possibilities

increases, it becomes progressively more time-consuming, and ultimately im-

possible, to determine the total number of possibilities by adding the discrete

number of arrangements.

However, if the distribution of probabilities can be expressed by a smooth

function as in Figure 17.3b, then the total number of possible arrangements—

equal to the area under the curve—is given by the integral of that probability



0 ^07  1 ^07  2 ^17  3 ^07  4 ^17  5 ^07  6 ^07  7 ^17  8 ^17  9 ^27  10 ^17 

 0

7 

0

7 

1

7 

0

7 

1

7 

0

7 

0

7 

1

7 

1

7 

2

7 

1

7 



0  0 ^27  0 ^47  0  0 ^77 ^87 ^178 ^170 

 7

7 



7

1

 7

^479 



1

i

(a) No. of possibilities  ^ P(i)  i

P(i)

i

(b) No. of possibilities  ^ P(i) di

P(i)

Figure 17.3 For a smooth distribution, an integral can be substituted for a summation. This

allows us to use calculus in our derivation of expressions in statistical thermodynamics.