Physical Chemistry , 1st ed.

function over all possible arrangements. For balls in boxes, summing the prob-

abilities is easily done. But for gaseous systems, in which the number of parti-

cles can be on the order of 10^20 , we will favor the approach of integrating a

smooth function. This perspective implies that the mathematics of calculus

will be useful to us in understanding the statistical behavior of our system.

17.3 The Ensemble

One of the ways that statistical thermodynamics tries to understand the ther-

modynamic state of a large macroscopicsystem is by separating it into tiny, or

microscopic,parts. These parts are called microsystems.The state of each mi-

crosystem is called the microstate.Understand that each microsystem may have

a different individual microstate, including volume, pressure, temperature, en-

ergy, density, and so on. All of the microstates of the system combine statisti-

cally to generate the overall state, or macrostate,of the system: its overall tem-

perature, pressure, volume, energy, and so on. This is a basic postulate of

statistical thermodynamics. In order to understand how these microstates

combine, we will first have to separate our system into microsystems and de-

termine the microstates of the microsystems.

There are several ways to do this. A convenient way will be chosen for our

purposes. We define the term ensembleas a collection of an undetermined

number of microsystems that collectively make up our macroscopic system.

Figure 17.4 illustrates one way of mentally separating a macroscopic system

into an ensemble of microsystems. Each microsystem in the ensemble has its

own characteristic microstate, defined by a particular number of particles, en-

ergy, volume, pressure, temperature, and so on. It is common to use the terms

“microsystem” and “microstate” interchangeably, although technically the mi-

crostate is the set of conditions that define the state of the microsystem.

A canonical ensembleis an ensemble separated into jindividual microstates

such that the numbers of particles in each microstate Nj, the volumes of the

microstates Vj, and the temperatures of the microstates Tjare the same.* As

extensive variables, particle numbers and the volumes are additive over the

microstates, whereas the temperature, an intensive variable, is notadditive

over the ensemble. Another way of saying this is by defining the total number

of particles N, the system’s total volume V, and the system’s overall tempera-

ture Tas

N

j

NjjNj (17.4)

V

j

VjjVj (17.5)

TTj (17.6)

Consider the energy of the particles in an ensemble. From quantum me-

chanics, we recognize that energy can have only certain values for the elec-

590 CHAPTER 17 Statistical Thermodynamics: Introduction

*There are other ways to define ensembles. For example, a microcanonical ensembleis de-

fined as a set of microstates in which the volume, number of particles, and energies of each

microstate are the same. So although equations 17.4 and 17.5 are still valid for a micro-

canonical ensemble, equation 17.6 is not. Rather, for a microcanonical ensemble we have

Esystem

j

EjjEj

Grand canonical ensembleshave V,T, and (chemical potential) the same for all microsystems.

An ensemble of microsystems,

whose overall thermodynamic

properties are determined from

the combined states of the

constituent microsystems

A single

microsystem

p, V, T,

N,...

Figure 17.4 This is one hypothetical way of
dividing a large system into an ensemble of
smaller microsystems. The individual states of the
microsystems combine to determine the overall
state of the system.