librium, in terms of macroscopic measurable quantities that do not refer at

all to atomic details. Statistical mechanics links the thermodynamic quan-

tities to appropriate averages over atomic details, thus establishing the ulti-

mate coarse-graining approach. Both theories have something to say about

non-equilibrium systems as well. The logical exposition of the link between

atomic and macroscopic behavior would be in the sequence:

(i) describe atomic behavior on a quantum-mechanical basis;

(ii) simplify to classical behavior where possible;

(iii) apply statistical mechanics to average over details;

(iv) for systems in equilibrium: derive thermodynamic; quantities and

phase behavior; for non-equilibrium systems: derive macroscopic rate

processes and transport properties.

The historical development has followed a quite di¬erent sequence. Equilib-

rium thermodynamics was developed around the middle of the nineteenth

century, with the de¬nition of entropy as a state function by Clausius form-

ing the crucial step to completion of the theory. No detailed knowledge of

atomic interactions existed at the time and hence no connection between

atomic interactions and macroscopic behavior (the realm of statistical me-

chanics) could be made. Neither was such knowledge needed to de¬ne the

state functions and their relations.

423

424 Review of thermodynamics

Thermodynamics describes equilibrium systems. Entropy is really only

de¬ned in equilibrium. Still, thermodynamics is most useful when processes

are considered, as phase changes and chemical reactions. But equilibrium

implies reversibility of processes; processes that involve changes of state

or constitution cannot take place in equilibrium, unless they are in¬nitely

slow. Processes that take place at a ¬nite rate always involve some de-

gree of irreversibility, about which traditional thermodynamics has nothing

to say. Still, the second law of thermodynamics makes a qualitative state-

ment about the direction of processes: a system will spontaneously evolve

in the direction of increasing excess entropy (i.e., entropy in excess of the

reversible exchange with the environment). This is sometimes formulated

as: the entropy of the universe (= system plus its environment) can only

increase. Such a statement cannot be made without a de¬nition of entropy

in a non-equilibrium system, which is naturally not provided by equilibrium

thermodynamics! The second law is therefore not precise within the bounds

of thermodynamics proper; the notion of entropy of a non-equilibrium sys-

tem rests on the assumption that the total system can be considered as the

sum of smaller systems that are locally in equilibrium. The smaller systems

must still contain a macroscopic number of particles.

This somewhat uneasy situation, given the practical importance of the

second law, gave rise to deeper consideration of irreversible processes in the

thirties and later. The crucial contributions came from Onsager (1931a,

1931b) who considered the behavior of systems that deviate slightly from

equilibrium and in which irreversible ¬‚uxes occur proportional to the devia-

tion from equilibrium. In fact, the thermodynamics of irreversible processes,

treating the linear regime, was born. It was more fully developed in the

¬fties by Prigogine (1961) and others.1 In the mean time, and extending into

the sixties, also the statistical mechanics of irreversible processes had been

worked out, and relations were established between transport coe¬cients

(in the linear regime of irreversible processes) and ¬‚uctuations occurring in

equilibrium. Seminal contributions came from Kubo and Zwanzig.

Systems that deviate from equilibrium beyond the linear regime have been

studied extensively in the second half of the twentieth century, notably by

the Brussels school of Prigogine. Such systems present new challenges: dif-

ferent quasi-stationary regimes can emerge with structured behavior (in time

and/or in space), or with chaotic behavior. Transitions between regimes of-

ten involve bifurcation points with non-deterministic behavior. A whole

new line of theoretical development has taken place since and is still active,

1 see, for example, de Groot and Mazur (1962)

16.2 De¬nitions 425

including chaos theory, complexity theory, and the study of emergent be-

havior and self-organization in complex systems. In biology, studies of this

type, linking system behavior to detailed pathways and genetic make-up,

are making headway under the term systems biology.

In the rest of this chapter we shall summarize equilibrium thermodynamics

based on Clausius™ entropy de¬nition, without referring to the statistical in-

terpretation of entropy. This is the traditional thermodynamics, which is an

established part of both physics and chemistry. We emphasize the thermo-

dynamic quantities related to molecular components in mixtures, tradition-

ally treated more extensively in a chemical context. Then in Section 16.10

we review the non-equilibrium extensions of thermodynamics in the linear

regime. Time-dependent linear response theory is deferred to another chap-

ter (18). Chapter 17 (statistical mechanics) starts with the principles of

quantum statistics, where entropy is given a statistical meaning.

16.2 De¬nitions

We consider systems in equilibrium. It su¬ces to de¬ne equilibrium as the

situation prevailing after the system has been allowed to relax under con-

stant external conditions for a long time t, in the limit t ’ ∞. Processes that

occur so slowly that in all intermediate states the system can be assumed to

be in equilibrium are called reversible. State functions are properties of the

system that depend only on the state of the system and not on its history.

The state of a system is determined by a description of its composition, usu-

ally in terms of the number of moles2 ni of each constituent chemical species

i, plus two other independent state variables, e.g., volume and temperature.

State functions are extensive if they are proportional to the size of the sys-

tem (such as ni ); intensive properties are independent of system size (such

as the concentration of the ith species ci = ni /V ). An important intensive

state function is the pressure p, which is homogeneous and isotropic in a

¬‚uid in equilibrium and which can be de¬ned by the force acting per unit

area on the wall of the system. Another important intensive thermodynamic

state function is the temperature, which is also homogeneous in the system

and can be measured by the pressure of an ideal gas in a (small, with respect

to the system) rigid container in thermal contact with the system.

Since a state function (say, f ) depends on the independent variables (say,

x and y) only and not on processes in the past, the di¬erential of f is a total

2 Physicists sometimes express the quantity of each constituent in mass units, but that turns out

to be very inconvenient when chemical reactions are considered.

426 Review of thermodynamics

or exact di¬erential

‚f ‚f

df = dx + dy. (16.1)

‚x ‚y

The line integral over a path from point A to point B does not depend on

the path, and the integral over any closed path is zero:

B

df = f (B) ’ f (A), (16.2)

A

df = 0. (16.3)

If the second derivatives are continuous, then the order of di¬erentiation

does not matter:

‚2f ‚2f

= . (16.4)

‚x‚y ‚y‚x

As we shall see, this equality leads to several relations between thermody-

namics variables.

A thermodynamic system may exchange heat dq, work dw (mechanical

work as well as electrical energy) and/or particles dni with its environment.

We shall adopt the sign convention that dq, dw and dni are positive if heat

is absorbed by the system, work is exerted on the system or particles enter

the system. Both dq and dw increase the internal energy of the system. If

the work is due to a volume change, it follows that

dw = ’p dV. (16.5)

Neither dq nor dw is an exact di¬erential: it is possible to extract net heat

or work over a closed reversible path. We see, however, that ’1/p is an

integrating factor of dw, yielding the exact di¬erential dV of a state function

V . Similarly it can be shown that a function β exists, such that βdq is an

exact di¬erential. Thus the function β is an integrating factor of dq. It can

be identi¬ed with the inverse absolute temperature, so that a state function

S exists with

dq

dS = . (16.6)

T

The function S is called the entropy; it is an extensive state function. The

entropy is only de¬ned up to a constant and the unit for S depends on the

unit agreed for T . The zero point choice for the entropy is of no consequence

for any process and is usually taken as the value at T = 0.