R p rastogi introduction to non equilibrium physical chemistry elsevier science (2007)

Equilibrium and non-equilibrium states The important feature of the equilibrium state is that variables such as temperature T , b static equilibrium.. Vapour–liquid equilibrium and chem

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Gorakhpur University, Gorakhpur, India

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First edition 2008

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Dedicated to

Mrs Kamla Rastogi

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2.9 Basic postulates of non-equilibrium thermodynamics close to equilibrium 22

2.11 Application to other disciplines: sociology, economics and finance 23

3 APPLICATIONS TO TYPICAL STEADY-STATES PHENOMENA 27

4.2 Non-equilibrium thermodynamics of electro-osmotic phenomena 59

5.6 Electric potentials generated at crystal interface 89

7.3 Non-linear steady states 104

10.3 One-dimensional chemical waves 166

11.5 Dynamic instability at solid–gas interface [60–68] 213

PART FOUR NON-EQUILIBRIUM PHENOMENA IN NATURE

14.2 Methodology and strategy for study of complex systems 273

14.4 Quantification of relationship between cause and effect 279

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The traditional Physical Chemistry has largely been concerned with equilibrium

laws and paradigms for which Onsager was awarded Nobel Prize in 1968 and Prigogine

in 1977 Their researches gave further impetus to the study of more and more complex

the phenomenon into parts

is such a typical non-equilibrium state These have relevance in Physiology, Geology,

One can have the following type of situations as we move away from equilibrium

and spatio-temporal oscillations −→ More complex situations (chaos, turbulence, pattern

the book

that several monographs have appeared in the last few decades dealing with these

The purpose of this book is to fill this gap so that it can be used as supplementary

in a sequential, coherent and comprehensive manner with greater emphasis on concepts

so that it may be useful from pedagogical angle Additional purpose has been to present

an elementary account to provide an insight into non-linear science and complexity for

I am extremely grateful to Professors A.C Chatterji, B.N Srivastava, M.N Saha,

K.G Denbigh, Karl Popper and Nobel laureate Prof Ilya Prigogine for stimulating my

I am indebted to Lucknow University, Punjab University, Gorakhpur University

and Banaras Hindu University and Central Drug Research Institute where most of the

of funding agencies like University Grants Commission, Council of Scientific and

and Indian National Science Academy for carrying out various projects related to the

acknowledged

I pay my tribute to Prof B.N Srivastava, who was primarily responsible for the

my former Ph.D students and other collaborators who had been involved in theoretical

and experimental studies in the areas covered in the book, whose references appear in

the book

I am particularly thankful to Profs R.C Srivastava (Chapters 1–7, 11), Kehar Singh

(3, 4), Ishwar Das (9, 10, 13), Kalanand Prasad (10) and Dr Pankaj Mathur (8, 12, 14)

for collaborating in writing a couple of chapters

I am also happy to acknowledge the involvement of Dr Ashtabhuja Prasad Mishra,

for specific chapters

I gratefully acknowledge the assistance rendered by Profs A.K Dutt and

A.A Bhalekar related to Appendices II and III

It is a pleasure to acknowledge the stimulating discussions which I had with

Dr Ghanshyam Das in connection with socio-political and financial dynamics

Dr Pankaj Mathur and Mr Ramendra Pratap in connection with the preparation of the

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Frequency (Hz)

(a)

Glass cover Cobalt(II) nitrate 10% Ammonia solution (b)

Chapter 1

INTRODUCTION

1.1 Real systems

now getting oriented to the study of such complex systems From the philosophical angle

the question whether science deals with real world [1a] has been raised The concept of

can serve as a model for economists and biologists Joint effort in understanding complex

1.2 Equilibrium and non-equilibrium states

The important feature of the equilibrium state is that variables such as temperature T ,

(b) static equilibrium Vapour–liquid equilibrium and chemical equilibrium are typical

and rate of vaporization are equal while in the second case, the rates of forward and

In non-equilibrium state, the thermodynamic variables are not the same everywhere

in the system, on account of which gradients of variables (e.g grad P, grad T , etc.)

flow or heat flow (fluxes) Non-equilibrium thermodynamics has the advantage of being

used for identifying cause and effect, i.e forces and fluxes, and also coupling between

the fluxes without a detailed knowledge of the systems However, for real systems

cross-phenomena Further in many systems, additional difficulty arises on account of

lack of knowledge about the nature and magnitude of the coupling coefficients between

The systems can be of three types:

surroundings;

open systems, which exchange both matter and energy with the surroundings

1.3 Open systems

Open system is always in non-equilibrium A closed system can be in non-equilibrium

not be constant in space A typical example of the former type is thermo-osmosis, which

is discussed in Chapter 3, where the two subsystems are separated by a membrane

When the flows and counter-flows in opposite directions are generated by corresponding

are time-invariant states, but in the latter case both flows and gradients are present

Real systems are open systems and may consist of numerous subsystems; global

the subsequent chapters

1.4 Approach to equilibrium

For taking a comprehensive view, it is also desirable to keep in mind the process

of approach to equilibrium Chemical kinetics and kinetic theory of gases have been

the traditional tools Simple reactions have been studied by Monte-Carlo technique or

the computer and employing a criterion that accepts or discards potential conversions

The methodology adopted for the study of simple set of simultaneous reactions has been

network which may simultaneously involve (i) electron transfer reaction, (ii) free-radical

and inorganic species For such type of systems, a new methodology called non-linear

reaction

for which the equilibrium constant K is of the order of 218 × 10−2 at 763.8 K The

the ratio of the true temperature coefficient of the yield to the temperature coefficient

of the yield, which would be predicted on the assumption that the system is at equili­

the corresponding factor is as much as 2.8

In a similar manner, cooling rate at the rocket nozzle throat used to be computed by

1.5 Non-equilibrium states

work of Professor Prigogine [6] and his school in Brussels stimulated a good deal

of interest in the field of non-equilibrium statistical mechanics Formal solutions of

a theoretical expression for electrical conductivity tensor, which easily leads to classical

used for estimating the coefficients in homogeneous conductors and thermo-couples

Onsager relations are satisfied, showing that free electron gas theory is consistent

with thermodynamic theory The free electron theory correctly predicts the temperature

of thermo-osmosis of gases on the basis of non-equilibrium thermodynamics and kinetic

1.6 Complex non-equilibrium phenomena

The departure from equilibrium occurs primarily on account of appearance of gra­

and subsequently leading to a specific non-equilibrium state Earlier in the first instance,

the subject of investigation Discussion of such processes has been given due attention

in conventional Physical Chemistry texts However, complex and exotic phenomena

in the non-equilibrium thermodynamics provide a good tool for understanding such

phenomena

Far from equilibrium, one comes across exotic phenomena as pointed out earlier

The study of these involved novel theoretical approaches and novel experimental studies

on a variety of phenomena to check the validity of phenomenological relations, Onsager

ena attracted good deal of attention from theoretical and experimental angle in view of

1.7 Scope

The great importance of thermodynamics and hydrodynamic methods lies in the

fact that these provide us with a reduced description in simplified language to describe

tal, in the advancing field of knowledge related to complex phenomena from equilibrium

to far from equilibrium region From this angle, it is reasonable to expect that the

real systems in nature and social surroundings Recent developments in non-equilibrium

The book is divided into four parts Part One, which consists of six chapters, deals

with basic principles and concepts of non-equilibrium thermodynamics along with dis­

for open system, identification of fluxes and forces and development of steady-state

ized by minimum entropy production Under these circumstances, fluctuations regress

uids and gases along with thermo-osmotic concentration differences Correlation with

in a system containing two subsystems separated by a membrane Relationship with

ent, concentration gradient and potential gradients without any barrier are involved In

have also been described

may be expressed in terms of same independent variables as if the system were at

in Appendix I At no moment, molecular distribution function of velocities or of relative

for the application of thermodynamics method Some new developments related to alter­

and III

In Part One, steady state corresponding to the situation when linear phenomeno­

has also been discussed These aspects have been discussed in Chapter 7 (Part Two)

Bifurcation from steady states to different types of dissipative structures takes place

with experimental and mechanistic studies for different types of situation

tive structures involving time order and space order Chapter 9 is concerned with

time order involving chemical oscillations Around 1968, the credibility of Belousov–

well established after the discovery of Brusselator model [10] This triggered detailed

The field of oscillatory reactions has been covered in Chapter 9

When phenanthroline is added to the B–Z system, in a test tube, travelling red and

blue bands are observed displaying spatio-temporal oscillations This feature denotes a

been discussed in Chapter 10 Turning patterns, mosaic structure, precipitation patterns

Part Three deals with complex non-equilibrium phenomena, which occur very far

from equilibrium (Chapters 11–13)

(ii) liquid–liquid interface, (iii) solid–liquid and liquid–liquid interface together, (iv)

In Chapter 12, special attention has been given to non-periodic oscillations of var­

ious types, including deterministic chaos and random motion (noise) Mathematical

In Part Four, attempt has been made to point out in what way the concepts developed

in previous chapters could be utilized to analyse and get an insight into the behaviour

of real systems, including socio-economic, socio-political and biological systems This

is the need of the hour, which could promote Synergetics (constructed from Greek

effort in different fields It may be noted that experimental studies of particular physico­

can provide appropriate models and mechanisms for the analysis of dynamics of real

References

1977

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Part One

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Chapter 2

BASIC PRINCIPLES OF NON-EQUILIBRIUM

THERMODYNAMICS

2.1 Introduction

we move away from equilibrium or far away from equilibrium, we immediately enter

the region of complexity For analysis of such situations, new procedures have to be

of the subject in this chapter with special reference to steady states close to equilibrium

with the surroundings at a lower temperature, there would be transfer of heat from con­

may consist of various sub-systems, through which exchange of matter and energy can

take place between one another We may call such systems as discontinuous systems,

ical potential � and electrical potential � may differ in different systems, but may be

the same in each system On the other hand, we can have continuous systems, where

such variables can vary from point to point

of many sub-systems between which interaction can take place In the world of today,

we have continents and nation states between which continuous interaction takes place

Even such nations can have various sub-units as we have in the United States, which

can act for separate sub-systems for a limited purpose The interaction between such

in basic understanding of non-equilibrium steady states close to equilibrium and also

in the region far from equilibrium The theoretical and experimental studies can serve

as a model for studies of similar type of real systems including biological, social and

Inflow of reactants

Outflow of the products

In this chapter, the main objective is to present the basic principles of NET in a

of theory and experiment have been discussed in subsequent chapters

Open systems are typical non-equilibrium systems, which display a variety of exotic

and the living state

with the surroundings through the following types of metabolism as an example,

(a) Ingestion → Glucose (in blood) → Utilization for energy (b) Carbohydrates utilization → (i) waste CO2, (ii) storage of fats and (iii) other

products

Discontinuous systems (membrane phenomena), where two or more homogeneous

S

flow occurs from chamber II to chamber I due to the pressure difference �P = P2−P1

The two flows continue to proceed till a steady state is reached

Continuous systems (with no barrier): In this case, the intensive variables are not

only functions of time but also continuous functions of space coordinates Such a typical

The following effects are observed in such a case

Thermal diffusion/Soret effect: Establishment of steady concentration gradient due

to fixed temperature gradient

Dufour effect: Establishment of steady temperature gradient due to fixed concen­

ring in the system Typical examples are vapour–liquid, liquid–liquid, solid–liquid

and chemical equilibria However, time-invariant non-equilibrium steady states

are also possible when opposite flows are balanced and gradients are maintained

constant

2.2 Second law of thermodynamics for open systems

In an open system, entropy production dS is made up of deS, entropy exchanged

with the surroundings, and diS, the internal production of entropy within the system

for open systems, it can be positive, negative or zero For a closed system at equilibrium

the help of Gibbs equation with the objective to estimate internal entropy production

by a barrier but maintained at different temperatures T1 and T2 In the present case,

heat flow only occurs on account of force generated due to temperature difference

It should be noted that since T2> T1, heat will flow from chamber II to chamber I

The sum of the first two terms on the right-hand side of the above equation would

be equal to deS, while the third term would be equal to diS Thus, the rate of entropy

� = diS/dt = �dQ/dt���1/T �I− �1/T �II� (2.7)

For a general case X = grad T/T (thermal force) For a more general case where

2.3 Law of conservation of mass, charge and energy

In any process, there has to be conservation of mass, charge and energy, and

(a) discontinuous systems and (b) continuous systems for fluxes and forces with the

help of Gibbs equation; one has to use law of conservation of mass, charge and density

from this angle to deduce explicit relations

Let us consider a system as indicated in the figure Let us suppose the internal transfer

of mass energy and charge from one sub-system to other The law of conservation of

mass would take the form,

For continuous systems, the expression for mass and energy are little complex,

separately

(a) Conservation of mass

If �k is the density (mass per unit volume) of k and vk velocity of species/

left-hand side is equal to the sum of negative divergence of a flow term and a source

term giving the production and destruction of the species k The divergence of

the flow has the simple physical meaning of giving per unit volume, the excess

of the flow, which leaves a small volume to the flow, which penetrates into it

The energy equation for u, the energy per unit mass worth the exclusion of bar

be written as

and Nk the number of molecules

It may be noted that volume and entropy are extensive properties, while temperature,

2.4.2 For continuous systems

In most situations, we may assume that equilibrium thermodynamic relations are

When this is done, all thermodynamic variables become functions of position x and

time t, so that

The intensive properties are replaced by densities s, u and nk defined as

Thus Gibbs equation is assumed to be valid for small elements with

may be noted that Gibbs equation provides a simple route for identifying fluxes and

even in non-equilibrium close to equilibrium

2.5 Phenomenological equations for single flows

Some heat flows in connection with entropy production are associated with other ther­

of some of these flux force relations (J = LX) Here, “L” is called phenomenological

2.6 Phenomenological equations for coupled flows

heat flow and mass flow takes place, since a membrane separates two compartments

Table 2.1 Fluxes and forces in non-equilibrium systems

coefficients

coefficient

but no coupling can take place between vectorial and scalar forces (Curie–Prigogine

principle)

Thermo-osmosis

difference

2.7 Onsager reciprocity relation

(ii) fluctuation theory and (iii) the assumption that decay of fluctuations follows ordinary

We consider the fluctuations in two variables ai�t� and aj�t+ �� where � is the

time interval The fluctuation in the average value of the product of the two variables

On subtracting aj�t� and ai�t� from both the sides and on dividing by �, we get

or

form where Ji = �ai/�t and Xk is the force Substituting Eq (2.18) into Eq (2.22) we

obtain

In principle, the results based on fluctuation theory and principle of microscopic

for the assumption of linear relation between fluxes and forces One has to understand

also the serious limitation of phenomenological linear laws The condition is that the

the time required to establish a steady flow in hydrodynamics, and �r is the regression

time of fluctuations, i.e a time in which the deviation from equilibrium is appreciably