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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 state or macrostate. Specification of this state for a system at equilibrium
requires only a few variables. The microscopic state or microstate of 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 microscopic variable over all of the microstates that are compatible with the macroscopic
state.
Averages
An average is a value that represents the “central tendency” of a set of values. There are
several kinds of averages. The median of a set is a value such that half of the members of
the set are smaller than the median and half are larger. The mode is the most commonly
occurring value in the set. The mean of a set of numbers w 1 , w 2 , w 3 , w 4 , . . . , w N is
denoted by w and is defined by
w
1
N
(w 1 + w 2 + · · · + w N )
1
N
N
i1
w i
(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 of w are equal to each other we can write Eq. (9.2-1) in a different
way. We arrange the N members of our set so that all of the distinct values are at the
beginning of the set, with w 1 , w 2 , . . . , w M all different from each other. Every remaining
member of the set will be equal to one or another of the first M members. Let N i be
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 state or macrostate. Specification of this state for a system at equilibrium
requires only a few variables. The microscopic state or microstate of 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 microscopic variable over all of the microstates that are compatible with the macroscopic
state.
Averages
An average is a value that represents the “central tendency” of a set of values. There are
several kinds of averages. The median of a set is a value such that half of the members of
the set are smaller than the median and half are larger. The mode is the most commonly
occurring value in the set. The mean of a set of numbers w 1 , w 2 , w 3 , w 4 , . . . , w N is
denoted by w and is defined by
w
1
N
(w 1 + w 2 + · · · + w N )
1
N
N
i1
w i
(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 of w are equal to each other we can write Eq. (9.2-1) in a different
way. We arrange the N members of our set so that all of the distinct values are at the
beginning of the set, with w 1 , w 2 , . . . , w M all different from each other. Every remaining
member of the set will be equal to one or another of the first M members. Let N i be