Physical Chemistry Third Edition

384 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

9.1 Macroscopic and Microscopic States

of Macroscopic Systems

A system of many molecules has both macroscopic and microscopic (molecular)

properties. The state of the system involving macroscopic properties is called the

macroscopic stateormacrostate. Specification of this state for a system at equilibrium

requires only a few variables. Themicroscopic stateormicrostateof a macroscopic

system requires information about every atom or molecule in the system, a very large

amount of information.

A General Postulate

We now assert that the macroscopic state of a system is determined by its microscopic

state. However, if a macroscopic system is at macroscopic equilibrium, its macrostate

does not change over a period of time, but the molecules are moving and the microstate

changes continually. Many different microscopic states must correspond to the same

macrostate, and we have no way of knowing which one is occupied at any instant. We

state a general postulate that connects the macrostates and microstates:The equilibrium

value of a macroscopic variable corresponds to an average of the appropriate micro-

scopic variable over all of the microstates that are compatible with the macroscopic

state.

Averages

Anaverageis a value that represents the “central tendency” of a set of values. There are

several kinds of averages. Themedianof a set is a value such that half of the members of

the set are smaller than the median and half are larger. Themodeis the most commonly

occurring value in the set. Themeanof a set of numbersw 1 ,w 2 ,w 3 ,w 4 ,...,wNis

denoted by〈w〉and is defined by

〈w〉

1

N

(w 1 +w 2 + ··· +wN)

1

N

∑N

i 1

wi (9.1-1)

We use the standard notation for a sum, a capital Greek sigma (Σ). The index for the

first term is indicated below the sigma and the index for the last term is indicated above

the sigma. Unless otherwise stated, any average value that we discuss will be a mean

value.

Probability Distributions

If some of the values ofware equal to each other we can write Eq. (9.2-1) in a different

way. We arrange theNmembers of our set so that all of the distinct values are at the

beginning of the set, withw 1 ,w 2 ,...,wMall different from each other. Every remaining

member of the set will be equal to one or another of the firstMmembers. LetNibe