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THE
PRINCIPLES
OF
CHEMICAL
EQUILIBRIUM
WITH
APPLICATIONS
IN
CHEMISTRY
AND
CHEMICAL
ENGINEERING
BY
KENNETH
DENBIGH,
F.RS.,
'"

FOURTH
EDITION
'
CAMBRIDGE
.
UNIVERSITY
PRESS
PUBLISHED
BY
THE
PRESS
SYNDICATE

OF
THE
UNIVERSITY
OF CAMBRIDGE
The
Pitt
Building, Trumpington Street, Cambridge
CB2
1RP, United Kingdom
CAMBRIDGE
UNIVERSITY
PRESS
The Edinbu1·gh Building, Cambridge
CB2
2RU, United Kingdom
40 West 20th Street,
New
York, NY 10011-4211, USA
10 Stamford
Road, Oaldeigh, Melboume 3166, Australia
©Cambridge University Press 1966,
1971,
1981
This book is in copyright. Subject
to
st-atutory exception
and
to
the
proYisions

of
relevant collective licensing agreements,
no reproduction
of
any
part
may take place without.
the written
permission
of
Cambridge University Press.
First
published
1955
Reprinted 1957, 1961, 1964
Second edition 1966
Third edition
1971
Reprinted 1973, 1978
Fourth
edition
1981
Reprinted 1986, 1987, 1989, 1992, 1993,
1997
Britiah Library cataloguing
in
publiration data
Denbigh, Kenneth George
The principles
of

chemical equilib1ium
-4th
edition
1.
Thermodynami~.\1
2.
Chemi~.alreactiollS
I. Title
541'.369 QD504 8()-40925
ISBN 0
521
23682 7 hardback
ISBN
0
521
28150 4 paperback
Transferred
to
digital
printing
2002
PREFACE
TO
THE
FIRST
EDITION
My aim baa been to write
a.
book on
the

general
theory
of
chemical
equilibrium, including ita statistical
d~velopment,
and
displa.ying
its numerous
pra.ctica.l applioa.tions,
in
the
laboratory
and
industry,
by
means
of
problems.
It
is hoped
that
the
book
may
be
equa.lly
useful to students
in
their final years

of
either a chemistry
or
a.
chemica.l engineering degree.
Thermodynamics is
a subject which needs to
be
studied
not
once.
but
several times over
at
advancing levels.
In
the
first round, usually
taken
in
the first
or
second year
of
the
degree,
a.
good deal
of
attention

is given
to
calorimetry, before going forward
to
the
second
la.w.
In
the
second
or
third
roun~ch
as I
a.m
concerned with
in
this book
-it
is assumed
that
the
student
is a.lready very familiar with
the
concepts
of
temperature
and
heat,

but
it
is useful once again to go
over
the
basis
of
the
first and second laws, this time
in
a more
logioa.l
sequence.
The student's confidence,
and
his ability
to
apply thermodynamics
in novel situations,
oa.n
be greatly developed
if
he
works
a.
consider-
able number
of
problems which are both theoretioa.l
and

numerical
in
character. Thermodynamics is a quantitative subject
and
it
can
be
mastered,
not
by
the
memorizing
of
proofs,
but
only
by
detailed
and
quantitative applioa.tion
to
specific problems. The
student
is therefore
advised
not
to
a.iin
at
committing anything

to
memory.
The
three
or
four
basic equations which embody
the
'laws',
together with
a.
few
defining relations, soon become familiar,
and
all
the
remainder
can
be obtained from these as required.
The problems
at
the
end
of
each chapter have been graded from
the
very easy
to
those
to

which
the
student
may
need to
return
several times before
the
method
of
solution occurs
to
him.
At
the
end
of
the
book some notes are given on
the
more difficult problems,
together with numerioa.l answers.
Questions marked
C.U.C.E.
are
from
the
qualifying
and
final

examinations
for
the
Cambridge University Chemioa.l Engineering
degree, and publioa.tion is
by
permission.
The
symbols which occur
in these questions are
not
always quite
the
same
as
in
the
text,
but
their meaning is made clear.
In
order
to
keep
the
size
of
the
book within bounds,
the

thermo-
dynamics
of
interfaces has
not
been included.
The
disOUB8ion
of
gal-
vanic cells
and
the
activity coefficients
of
electrolytes is
a.Iso
rather
brief.
iv
Preface
Part
I conta.ins
the
ba.sis
of
thermodynamics developed on tradi-
tional lines, involving
the
Ca.mot cycle. Pa.rt

II
conta.ins the
ma.in
development in
the
field
of
chemica.! equilibria.,
and
the
methods
a.dopted here have been much influenced
by
Guggenheim's books, to
which I
am
greatly indebted. Pa.rt
III
contains
a.
short introduction
to
sta.tistica.l mechanics along
the
lines
of
the
Gibbs ensemble
and
the

methods used
by
R.
C.
Tolman in his Principle8
of
StatiBtical
M
ec.1uJnic8.
It
is
a.
great plea.sure to acknowledge
my
gratitude
to
a.
number
of
friends.
In
particular,
my
best
thanks are due to
Dr
Peter
Gray,
Professor N.
R. Amundson,

Dr
J.
F. Davidson
and
Dr
R.
G.
H.
Watson,
for helpful criticism
and
suggestions,
and
to
Professor T. R.
C.
Fox,
for stimulating
and
friendly discussions on thermodynamics over
several
years. Finally I wish
to
express
my
appreciation
of
the
good
work

of
the
Cambridge University Press,
and
my
thanks to Messrs
Jonathan
and
Philip Denbigh, for help with the proof correcting, and
to
my
wife for help
in
many
other ways.
CAMBRIDGE
Ocloba- 1954
K.G.D.
PREFACE
TO
THE
FOURTH
EDITION
My work for
this
edition has been mainly
a.
revising
of
the

text
in
the
light
of
recent contributions
to
the
literature. Many new references
have been added,
and
there
are
also certain changes
of
emphasis.
The difficulties in
the
way
of
establishing chemical thermo-
dynamics
in
a fully rigorous manner have been described afresh
by
:Munster
in
his
GlassicoJ,
Tkerrrwdynamic8 (1970). As

he
has said,
the
1
'laws'
do
not
constitute a complete
set
of
axioms, especially in
the
case
of
systems having variable composition.
As
regards entropy, one way
of
dealing with these difficulties is
simply
to
postulate its existence, rather
than
seeking
to
prove it.
However
this
method seems
to

me
not
sufficiently satisfying for
the
student.
Far
better, in my view,
to
put
forward
the
classica.l
arguments
as
well
a.s
they
can
be put,
and
to
develop simultaneously
the
statistica.l interpretation
of
the
second law, so
a.s
to create a
linkage

of
thermodynamics
with
the
rest
of
physics and chemistry.
This
leaves my previous scheme for Chapter 1 essentially
unchanged.
But
I have become
better
aware
than
previously,
especially from
Popper,
that
there
is a certain hazard in using
the
statistical argument, even
at
the
elementary level
of
the
present
volume.

If
the
argument is
put
forward in terms of
'lack
of
informa-
tion'
about
micro-states,
this
may well create
the
impression,
Preface v
although quite unwarrantably,
that
thermodynamics contains
very
subjective elements. Some
of
my
re-writing has been intended
to
correct
that
impression.
Although
'the

information theory
approach'
is very helpful,
especially in
an
heuristic sense, I believe
it
has also somewhat
obscured
the
central issue, relating
to
the
second law, of how
irreversible phenomena
can ever occur. The fact
that
thermodynamic
systems are incompletely specified is only
part
of
the
story, although
an
important
part.
One has also
to
ask questions
about

de
facto
initial conditions,
and
how they can arise. These questions can only
be
answered, in my view,
by
referring
to
the
pervasive irreversibility
within
the
total environment.
Apart
from these points concerning Chapters 1
and
11, various
footnotes have been added
and
improvements have been made
to
Chapter
14,
and
to
the
section in Chapter 6 which deals with lambda
transitions.

In
earlier editions I expressed
my
indebtedness
to
Professors
Guggenheim, Peter Gray and
John
Row Iinson for suggesting various
improvements to
the
text. I should now like
to
express
my
gratitude
to
Professors
J.
A.
Campbell, T. W. Weber
and
N. Agmon for
providing me
with
substantial lists
of
errors
and
misprints.

Many
other
correspondents have
sent
very
helpful
remarks,
and
to
these
also I offer my
best
thanks.
Aprill980
K.
G.
D.
CONTENTS
Preface
to
the First Edition
Preface
to
the Fourth Edition
List
of
Symbols
V
aluu
of

Physical Constants
PART
I:
THE
PRINCIPLES
OF
THERMODYNAMICS
Chapter
1:
First
and
Second
Laws
1·1
Introduction
vii
page
iii
iv
xvii
xxi
3
1·2
Thermodynamic
systems
5
1·3
Thermodynamic
variables
6

1·4
Temperature
and
the
zeroth
law
9
1·5
Work
14
1·6
Internal
energy
and
the
first
law
15
1·7
Heat
18
1·8
Expression
of
the
first
law
for
an
infinitesimal

process
19
1·9
Adiabatically
impossible processes 21
1·10
Natural
and
reversible processes
1·11
Systematic
treatment
of
the
second
law
1·12
Final
statement
of
the
second
law
1·13 A ct:iterion
of
equilibriwn.
Reversible
processes
1·14
Maximwn

work
1·15
The
fundamental
equation
for a closed
system
1·16
Swnmary
of
the
basic
laws
1·17
Natural
processes
as
mixing
p:r:ocesses
1·18
The
molecular
interpretation
of
the
second
law
Problems
23
25

39
40
43
45
46
48
56
60
viii
Contents
Okapter 2: Auxiliary Functions and Conditions
of
Equilibrium
2·1
The
functions
H,
A
and
G
11age
63
2·2
Properties
of
the
enthalpy
2·3
Properties
of

the
Helmholtz
free
energy
2·4
Properties
of
the
Gibbs
function
2•5a Availability
2·5b
Digression
on
the
useful
work
of
chemical reaction
~·6
The
fundamental
equations
for a closed
system
in
terms
of
H,
A

and
G
2·7
The
chemical
potential
2·8
Criteria
of
equilibrium
in
terms
of
extensive properties
2·9
Criteria
of
equilibrium
in
terms
of
intensive properties
2·10
Mathematical
relations
between
the
various
functions
of

.
state
2·11
Measurable
quantities
in
thermodynamics
2·12
Calculation
of
changes
in
the
thermodynamic
functions
over
ranges
of
temperature
and
pressure
2·13
Molar
and
partial
molar
quantities
2·14 Calculation
of
partial

molar
quantities
from
experimental
data
Problem8
PART
II:
REACTION
AND
PHASE
EQUILIBRIA
Okapter 3: Thermodynamics
of
Gases
3·1 Models
3·2
The
single
perfect
gas
3•3
The
perfect
gas
mixture
3·4
Imperfect
gases
3·5

The
Joule-Thomson
effect
63
66
67
70
72
76
76
82
85
89
94
98
99
104
106
Ill
Ill
ll4
119
120
Contents
ix
3•6
The
fugacity
of
a.

single
imperfect
ga.s
page
122
3·7 Fuga.cities
in
a.n
imperfect
ga.s
mixture
125
3·8
Temperature
coefficient
of
the
fuga.city a.nd sta.nda.rd
chemical
potential
127
3·9
Ideal
ga.seous solutions a.nd
the
Lewis
and
Ra.nda.ll
rule
128

Problem8
130
Chapter 4:
Equilibria.
of
Reactions
Involving
Gases
4·1
Introduction
133
4·2
The
stoichiometry
of
chemical rea.ction
133
4·3
Preliminary
discussion
on
reaction
equilibrium
135
4·4
Concise discuBBion
on
reaction
equilibrium
139

4·5
The
equilibrium
constant
for
a.
ga.s rea.ction
140
4·6
The
temperature
dependence
of
the
equilibrium
constant
143
4·7
Other
forms
of
equilibrium
constant
for
perfect
ga.s
mixtures
146
4·8
Free

energies a.nd entha.lpies
of
formation
from
the
elements
148
4·9 Some
examples
149
4·10
Free
energies
of
formation
of
non-ga.seous
substances
or
from non-ga.seous
elements
153
HI
Preliminary
discussion
on
rea.ction equilibria. involving
ga.ses
together
with

immiscible
liquids
a.nd solids
156
4-12
Concise discussion
on
rea.ction equilibria. involving
ga.ses
together
with
immiscible
liquids
a.nd solids
159
4-13
Example
on
the
roa.sting
of
galena.
161
4·14
Mea.surement
of
the
free
energy
of

rea.ction
by
use
of
ga.lva.nic cells
163
4-15
Alternative
discUBBion
of
the
ga.lva.nic cell
167
4·16
Number
of
independent
rea.ctions
169
4·17
Conditions
of
equilibrium
for
several
independent
rea.ctions
172
4·18
General rem&rks

on
simultaneous
reactions
4•19 General
remarks
on
maximum
attainable
yield
Problems
Chapter 5: Phase Rule
5·1
Introduction
5·2
The
phase
rule
for
non-reactive
components
5·3
The
phase
rule
for
,rea.ctive components
5·4
Additional
restrictions
5·5

Example
of
the
application
of
the
phase
rule
5·6
Alternative
approach
5·7
Two
examples
from
the
zinc
smelting
industry
Problems
Chapter 6: Phase Equilibria.
in
Single Component'Systems
page 173
175
177
182
184
187
188

188
191
191
104
6·1
Introduction
196
6·2
The
Clausius-Clapeyron
equation
197
6·3
The
enthalpyofvaporizationtanditstemperature
coefficient 200
6·4
Integration
of
the
Clausius-Clapeyron
equation
202
6·5
The
effect
of
a
second
gas

on
the
vapour
pressure
of~
liquid
or
solid 203
6·6
Lambda
transitions
207
Problems
213
Chapter
7: General Properties
of
Solutions
a.nd
the
Gibbs-
Duhem
Equation
7·1
The
Gibbs-Duhem
equation
215
7·2
Pressure-temperature

relations
216
7·3
Partial
pressure-composition
relations 221
7·4
The
empirical
partial
pressure
curves
of
binary
solutions 222
7·5
Application
of
the
Gibbs-Duhem
equation
to
the
partial
pressure
curves
232
Contents
7·6 Application
of

the
Gibbs-Duhem
equation
to
the
total
pressure
curve
7·7
The
Gibbs-Duhem
equation
in
relation
to
Ra.oult's
and
xi
page
235
Henry's
laws 236
7•8
The
Gibbs Duhem
equation
in
relation
to
the

Margules
and
van
La.a.r
equations 240
Probkma
Chapter
8: Ideal Solutions
8·1 Molecular aspects
of
solutions
8·2 Definition
of
the
ideal
solution
8·3 Ra.oult's
and
Henry's
laws
8·4
Imperfect
vapour
phase
8·5
The
mixing properties
of
ideal
solutions

8·6
The
dependence
of
vapour-solution equilibria
on
temperature
and
pressure
8•7
Nernst's
law
8•8
Equilibrium between
an
ideal
solution
and
a
pure
crystalline component
8·9 Depression
of
the
freezing-point
8•10
Elevation
of
the
boiling-point

8·11
The
osmotic pressure
of
a.n
ideal
solution
8·12
The
ideal solubility
of
gases
in
liquids
8·13
The
ideal solubility
of
solids
in
liquids
Problems
Chapter
9: Non-Ideal Solutions
9·1 Conventions for
the
activity
coefficient
on
the

mole
242
244
249
249
252
252
255
256
257
260
261
262
264
266
267
fraction scale
270
9·2
The
activity
coefficient
in
relation
to
Ra.oult's
and
Henry's
laws
271

9·3
The
use
of
molality
and
concentration
scales 274
9·4
Convention
for
the
activity-coefficient
on
the
molality
scale 276
9·5
The
effect
of
temperature
and
pressure 278
xii
Contents
9·6
The
determination
of

activity
coefficients page
281
9·7
The
Gibbs-Duhem
equation
applied
to
activity
coefficients 284
9·8
The
calculation
of
the
activity
coefficient
of
the
solute 284
9·9
Excess
functions
of
non-ideal solutions 285
9·10
The
activity
9·11

The
osmotic coefficient
Problem&
0/w,pter 10: Reaction Equilibrium
in
Solution. Electrolytes
287
288
288
10·1
Reaction
equilibrium
in
solution 292
10·2
Free
energy
of
formation
in
solution. Convention
concerning
hydrates
295
10·3
Equilibrium
constants
expressed
on
the

molality
and
volume
concentration
scales 298
10·4
Temperature
and
preBBure dependence
of
the
equilibrium
constant
299
10·5
Ratio
of
an
equilibrium
constant
in
the
gas
phase
and
in
solution
301
10·6
Notation

for
electrolytes
:102
10·7
Lack
of
significance
of
certain
quantities
303
10·8
DiBBocia.tion
equilibrium
and
the
chemical
potential
of
the
electrolyte 304
10·9
Activity
coefficients 305
1 0·1 0
Phase
equilibrium
of
an
electrolyte. Solubility

product
307
10·11
Equilibrium
constant
for
ionic reactions 309
1 0·12
Magnitude
of
activity
coefficients
of
charged
and
uncharged
species 310
1 0·13
Free
energy
of
dissociation 312
10·14
The
hydrogen
ion
convention
and
the
free energies

and
entha.lpies
of
formation
of
individual ions 314
10·15
Activity
coefficients
and
free energies
as
measured
by
the
use
of
the
galvanic
cell 316
10·16
Activity
coefficients
by
use
of
the
Gibbs-Duhem
equation
10·17

Partial
preBBure
of
a.
volatile
electrolyte
10·18
Limiting
behaviour
at
high
dilution
ProblerrUJ
322
324
:125
327
Contents
PART
III:
THERMODYNAMICS
IN
RELATION
TO
THE
EXISTENCE
OF
MOLECULES
Chapter
II:

Statistical Analogues of Entropy and Free Energy
11·1 Thermodynamics
and
molecular
reality
page 333
11•2
The
quantum
states
of
macroscopic
systems
333
11·3
Quantum
states,
energy
states
and
thermodynamic
states
334
11·4
Fluctuations
335
11·5 Averaging
and
the
statistical

postulate
336
11·6 Accessibility 337
11·7
The
equilibrium
state
338
11·8 Statistical
methods
339
11·9
The
ensemble
and
the
averaging
process 340
11·10
Statistical
analogues
of
the
entropy
and
Helmholtz
free
energy 345
11·11 Comparison
of

statistical
analogues
with
thermodynamic
functions 350
1 1·12 Thermal
and
configure,tional
entropy
353
11·13 Appendix
I.
Origin
of
the
canonical
distribution
356
11·14 Appendix
II.
Entropy
analogues 359
Problem
360
Chapter
I2: Partition Function of a Perfect Gas
12·1 Distinguishable
states
of
a.

gas
and
the
molecular
partition
function
12·2 SchrOdinger's
equation
12·3 Separability
of
the
wave
equation
12·4
Factorization
of
the
molecular
partition
function
12·5
The
translational
partition
function
12·6
The
internal
partition
function

12·7 Thermodynamic properties
of
the
perfect gas .
12·8
The
Maxwell-Boltzmann
distribution
361
365
367
3'71
372
376
377
383
xiv
Contents
12•9 Dist1ibution over translational
and
internal
states
12·1 0
Number
of
translational
states
of
a.
given energy

12·11
The
Maxwell velocity distribution
12·12 Principle
of
equipa.rtition
12·13 Appendix. Some definite integrals
Problems
Chapter
13: Perfect
CeystaJ&
a.nd
the
Third Law
13·1 Nornnalco-ordinates
13·2
The
Schrodinger
equation
for
the
crystal
13•3
The
energy levels
of
the
ha.rnnonic oscillator
13•4
The

partition
function
13·5
The
Ma.xwell-Boltzma.nn distribution
13·6
The
high
temperature
approximation
13·7
The
Einstein
approximation
13·8
The
Debye
approximation
13·9 Comparison
with
experiment
13·10
Vapour
pressure
at
high
temperature
13·11
The
third

law-preliminary
13•12
Statement
of
the
third
law
13·13
Tests
and
applications
of
the
third
law
Problema
Chapter 14: Configurational Energy
a.nd
Entropy
14·1
Introduction
14·2
Example
1:
the
lattice
model
of
mixtures
14·3

Example
2:
the
Langmuir
isotherm
Chapter 15: Chemical Equilibrium
in
Relation
to
Chemical
Kinetics
15·1
Introduction
15·2 Kinetic
speci~
page 386
387
390
392
394
396
397
400
401
402
405
406
408
409
411

414
416
421
424
427
429
432
436
439
440
Contents
XV
15•3
Variables
detennining
reaction
rate
page 441
15·4
Forward
and
backward
processes 442
15•5
Thermodynamic
restrictions
on
the
form
of

the
kinetic
equations
444
15·6
The
temperature
coefficient
in
relation
to
thermodynamic
quantities
449
15·7
Transition-state
theory
450
15·8
The
equilibrium
assumption
453
15·9
The
reaction
rate
455
Appendix. Answers
to

Problems and Comments
460
IM ex
487
xvii
LIST
OF
SYMBOLS
Definition
Equation
Page
itt
Activit.y
of
ith
1:1pecies
287
A
Helmholtz
free energy
of
a.
system
2·1
63
Gt
Molarity
of
ith
species

9·11 275
0
Compressibility factor
of
a.
gas
3·51
124
c
Number
of
independent
components
of
a.
171, 184.
system
187
c,
Molar
heat
capacity a.t
constant
pressure
2·87 96
c,
Heat
capacity
of
system

a.t
constant
2·86
95
pressure
Gy
Molar
heat
capacity
at
constant
volume
2·87
96
c
Heat
capacity
of
system
a.t
constant
2·86 05
volume
e
Symbol for
a.n
electron
E
Any
extensive

property
of
o.
t~ystem
8
E
Electromotive
force
75, 164
E
Total
energy
of
a
ByHtem
17
Et
Energy
of
the
ith
quantum
state
of
a.
342
macroscopic system
f
Fugacity
3·45

and
122, 125
3·56
f
Molecular
partition
function 12·9
364
F
The
Faraday
75
F Degrees
of
freedom
of
system
185
G Gibbs free energy
of
system
2·3
63
6./G~
Standard
free energy
of
formation
from
the

148
elemep.ts a.t
temperature
T
6.G~
Standard
free energy
change
in
reaction
4·17
142
a.t
temperature
T
a•
Excess free energy 285
h
Planck
constant
h
Enthalpy
per
mole
of
a.
pure
substance
100
H

Enthalpy
of
system
2·1
63
xvili
Lisl
of
Symbois
Definition
Equation
Page
H,
Partial
molar
enthalpy
of
ith
species
2-104:
101
a
1
H'l'
Standard
enthalpy
of
formation a.t
tem-
149

perature
T
f.HO
Enthalpy
change
in
reaction
under
condi-
4:·22
144,300
tions
where
the
species obey
the
perfect
gas
laws
or
the
ideal
solution laws
llH
0
An
integration
constant
having
the

di-
4·26
145
mansions
of
an
enthalpy
change
k
Velocity
constant
of
reaction
441
k
Boltzmann
constant
345,378
K
An
equilibrium
constant
4·12, 10·4 141,293
and
10·7
K,
Equilibrium
constant
expressed
in

4·16
142
fugacities
K.,
Equilibrium
constant
expreBSed
in
4-12
141
partial
pressures
K'
Partial
equilibrium
constant
4·50
157, 159
fl
K,
Henry's
law
coefficient for
ith
species 8·17
and
250,271
9·5
L
Enthalpy

('
latent
heat
')
of
phase
change
2·94and
98,198
6·7
m,
Molality
of
ith
species
9·7
274
Mi
Chemical
symbol
of
ith
species
n,
Amount
(mols)
of
ith
species
in

system
N,
Number
of
molecules
of
ith
species
in
system
L
The
Avogadro
constant
378
N
Number
of
species
in
system
187
p,
Partial
preBBure
of
ith
species
3·20
115

p:
V a.pour preBBure
of
pure
ith
species
223
p
Total
preBSure
on
system
p
Number
of
phases
in
system
184
P,
A
probability
340
q
Heat
taken
in
by
system
18

Q
Partition
function
of
closed system
at
1H2
345
constant
temperature
and
volume
R
Number
of
independent
reactions
in
169
system
R
Gas
constant
111
Lisa
of
Symbola
xix
Definition
Equation

Page
8
Entropy
per
molrJ
of
pure
substance
2·99 98
8
Entropy
of
system
1-13 32
s,
Partial
molar
entropy
of
ith
specie ;~
2·104 99
8'
8':
-k~P,InP,
11-14
34:3
8"
8":klnll
11-15

34:3"
T
Temperature
on
thermodynamic scale
1-12
31
u
Internal
energy
per
mole
of
pure
substance 2·99 98
u
Internal
energy
of
system 17
u,
Partial
molar
internal
energy
of
ith
species
2·104
99

"
Volume
per
mole
of
pure
substance
2·99
98
v
Volume
of
system
v,
Partial
molar
volume
of
ith
species 2·104
99
to
Work
done
on
system 14
to'
Work
done
on

system,
not
including
that
66
part
which is due
to
volume change
to,l
Potential
energy
of
a
pair
of
molecules
of
243,430
types
i
andj
a:,
Mole fraction
of
ith
species
in
condensed
phase

y, Mole fraction
of
ith
species
in
vapour
phase
%+,%-
Charges
of
positive
and
negative ions
73, 163,
respectively in
units
of
the
proton
300
charge
«
Coefficient
of
thermal expa.nsivity
2·8tl
94
fJ
A statistical parameter
11-10

342
y,
Activity coefficient
of
ith
speciet~
9·2, 9·3
269,274
and9·16
y
Surface tension
A
Sign indicating excess
of
final
over
initial
value
e,
Energy
of
the
ith
quantum
state
of
a
molecule
(}
Temperature

on
any
scale
11
IC
Compressibility coefficient
2·89
94
"''
Chemical potential
of
ith
species
2·39and
76,77
2·41
"''
Gibbs free energy
per
mole
of
pure
sub-
Ch. 3
stance
at
unit
pressure
and
at

the
same temperature
as
that
of
the
mixture
under
discussion
n
n,
Li6c
of
Symbols
Gibbs free energy
per
mole
of
pure
sub-
stance
at
the
same
temperature
and
pressure
as
that
of

mixture
under
discussion
Chemical
potential
of
ith
spe6ies
in
a
hypothetical
ideal solution
of
unit
molality
at
the
same
temperature
and
pressure
as
solution
under
discussion
Joule-Thomson
coefficient
Stoichiometric
coefficient for
ith

species
in
a reaction
Numbers
of
positive
and
negative ions
respectively formed
on
dissociation
of
one
molecule
of
electrolyte

::

++~'-
Extent
of
reaction
Continued
product
operator
sign
Osmotic pressure
Density
Created

entropy
Summation
operator
sign
Potential
energy
Fugacity
coefficient
of
ith
species
Degeneracy
of
the
ith
molecular energy
level
Number
of
accessible
quantum
states
of
a macroscopic
system
of
constant
energy
and
volume

Ditto
as
appli.ed
to
the
particular
energy
state
E,
Denotes
an
approximate
equality
Denotes
a
very
close approximation
Used
where
it
is
desired
to
emphasize
that
the
relation
is
an
identity,

or
a
definition.
Definition
Equation
Page
Cbs.
8
and9
276
3·42 120
134
302
10·34 305
4·2
135
141
8·54 263
}ol5
39
17,
87
3·54 125
361, 367
335,338
353, 367
VALUES
OF
PHYSICAL
CONSTANTS

Ice-point
temperature
Boltzmann
constant
Planck
constant
Avogadro
constant
Faraday
constant
Charge
of
proton
Gas
constant
TiCIJ
=
273.15
K
k =
1.380
54 X
IO-U
J
K-1
h =
6.625
6 X
I0-81
J s

L =
6.022
52 x IOta
mol-1
F = 96 487 C
mol-1
e =
1.602
10 X
IO-lt
C
R=
8.3143JK-tmol-1
=
1.
987 2
cal
K-t
moi-
1
j
=
82.06
em•
atm
K-t
mol-1
THE
SI
UNITS

The
basic
SI
units
are
the
metre
(m),
kilogramme
(kg),
second
(s),
kelvin
(K;
not °K), mole (mol),
ampere
(A)
and
candela. (cd).
The
mole
is
the
unit
of
'amount
of
substance'
and
is defined

as
that
amount
of
substance
which
contains
as
many
elementary
entities
as
there
are
atoms
in
0.012
kg
of
ca.rbon-12.
It
is
thus
precisely
the
same
amount
of
sub-
stance

as, in
the
c.g.s.
system,
had
been
called
the
'gramme-molecule'.
Some
of
the
SI
derived
units
which
are
important
in
the
present
volume,
together
with
their
symbols,
are
as
follows:
for

energy

the
joule
(J);
kg
m•s-1
for force

the
newton
(N);
kg
m
s-
1
= J
m-1
for pressure

the
pascal
(Pa.);
kg
m-1
s-•
= N
m-•
for electric
charge

the
coulomb
(C); A s
for electric
potential
difference

the
volt
(V);
kgm•s-aA-
1
= J
A-
1
s-
1
In
terms
of
SI
units
two
'old-style'
units
which
are
also
used

in
this
book
are:
the
thermochemical calorie (cal) =
4.
184 J
the
atmosphere
(a.tm) =
101.325
kPa. = 101 325 N
m-
1
PART
I
THE
PRINCIPLES
OF
THERMODYNAMICS
CHAPTER
1
FIRST
AND
SECOND
LAWS
1·1.
Introduction
One reason why

the
study
of
thermodynamics is so valuable
to
students
of
chemistry and chemical engineering is
that
it
is a
theory
which can be developed in its entirety, without gaps in
the
argument,
on
the
basis
of
only a moderate knowledge
of
mathematics.
It
is
therefore a self-contained logical structure,
and
much
benefit and
incidentally much
pleasure-may

be obtained from its study. Another
reason is
that
it
is one
of
the few branches
of
physics or chemistry
which is largely independent
of
any
assumptions concerning
the
nature
of
the fundamental particles.
It
does
not
depend
on
'mech-
anisms', such
as are used in theories
of
molecular structure
and
kinetics, and therefore
it

can often be used as a check on
such
theories.
Thermodynamics is also a subject
of
immense practical value.
The
kind of results which may be obtained
may
perhaps be summarized
very
briefly as follows:
(a)
On the basis
of
the first law, relations
may
be establisherl
between quantities
of
heat
and
work,
and
these relations are
not
restricted
to
systems
at

equilibrium.
(b)
On the basis
of
the first
and
second laws together, predictions
may
be
made concerning
the
effect
of
changes
of
pressure, tem-
perature and composition on a
great
variety
of
physico-chemical
systems. These applications are limited
to
systems
at
equilibrium.
Let
X be a
quantity
charactertstic

of
an
equilibrium, such as
the
vapour pressure
of
a liquid,
the
solubility
of
a solid,
or
the
equilibrium
constant
of
a reaction. Then some
of
the
most useful results
of
thermo-
dynamics are
of
the
form
(
o
In
x)

=
(A
characteristic energy)
oT
p RT2 •
(
olnx)
=(A
characteristic volume)
op t
RT
.
The present volume is mainly concerned with
the
type
of
results
of
(b)
above. However, in
any
actual problem
of
chemistry,
or
the
chemical industry,
it
must always be decided,
in

the
first place,
whether the essential features
of
the
problem
a,.re
concerned
with
equilibria or with rates. This point
may
be illustrated
by
reference
to
two well-known chemical reactions.
Principles
of
Chemical Equilibrium
[1·1
In
the
synthesis
of
ammonia, under industrial conditions,
the
reaction normally comes sufficiently close
to
equilibrium for
the

applications
of
thermodynamics to prove
of
immense value. t Thus
it
will predict
the
influence
of
changes
of
pressure, temperature
and
composition on
the
maximum
attainable yield.
By
contrast
in
the
catalytic oxidation
of
ammonia
the
yield
of
nitric oxide
is

determined,
not
by
the
opposition
of
forward
and
backward reactions, as
in
ammonia synthesis,
but
by
the
relative speeds
of
two independent
processes which compete
with
each other for
the
available ammonia.
These are
the
reactions producing nitric oxide
and
nitrogen respec-
tively,
the
latter

being
an
undesired and wasteful product. The useful
yield
of
nitric oxide is
thus
determined
by
the
relative
speeds
of
these
two reactions
on
the
surface
of
the
catalyst.
It
is therefore a problem
of
rates
and
not
of
equilibria.
The

theory
of
equilibria, based on thermodynamics, is much
simpler,
and
also more precise,
than
any
theory
of
rates which has
yet
been devised.
For
example,
the
equilibrium constant
of
a reaction
in a perfect
gas can be calculated exactly from a knowledge only
of
certain macroscopic properties
of
the pure reactants and products.
The
rate
cannot be so predicted with
any
degree

of
accuracy for
it
depends
on
the
details
of
molecular
structure
and
can only
be
cal-
culated,
in
any
precise sense,
by
the immensely laborious process
of
solving
the
Schrodinger wave equation. Thermodynamics, on
the
other
hand,
is independent
of
the

fine
structure
of
matter,t
and
its
peculiar simplicity arises from a certain condition which
must
be
satisfied in
any
state
of
equilibrium, according
to
the second law.
The foundations
of
thermodynamics are three facts
of
ordinary
experience. These
may
be expressed very roughly as follows:
(I) bodies are
at
equilibrium with each
other
only when
they

have
the
same degree
of
'hotness';
(2)
the
impossibility
of
perpetual motion;
(3)
the
impossibility
of
reversing any
natural
process in its entirety.
In
the
present
chapter
we shall be concerned with expressing these
facts more precisely,
both
in
words
and
in
the
language

of
mathe-
matics.
It
will be shown
that
(l),
(2)
and
(3)
above each gives rise
to
the
definition
of
a
certain
function, namely, temperature, internal
energy
and
entropy
respectively. These
have
the
property 'of being
entirely determined
by
the
state
of a body

and
therefore
they
form
exact
differentials. This leads
to
the
following equations which contain
the
whole
of
the
fundamental
theory:
t
The
en·or
in
using
thermodynamic
predictions,
as
a
function
of
the
extent
to
which

the
particular
process
falls
short
of
equilibrium,
is discussed
by
Rastogi
and
Denbigh,
Chem.
Eng.
Science, 7 (1958), 261.
t
In
making
this
statement
we
are
regarding
thermodynamics
as
having
its
own
secure
empirical

basis.
On
the
other
hand,
the
laws
of
thermodynamics
may
themselves
be
interpreted
in
terms
of
the
fine
structure
of
matter,
by
the
methods
of
statistical
mechanics
(Part
III).
1·2] Firsr and Second Laws

5
dU=dq+dw,
dS=dqfT,
for a reversible change,
d8
~
0, for a change in
an
isolated system,
dU=Td8-pdV+IfL
1
dn
1
for each homogeneous
part
of
a system.
Subsequent chapters of
Parts
I
and
II
will be concerned with
the
elaboration
and
application
of
these results. The
student

is advised
that
there is no need
to
commit
any
equations
to
memory;
the
four
above, together with
a
few
definitions
of
auxiliary quantities such
as
free energy, soon become familiar,
and
almost
any
problem
can
be
solved
by
using them.
In
conclusion

to
this introduction
it
may be remarked
that
a new
branch
of
thermodynamics has developed during
the
past
few
decades which is
not
limited in
its
applications·
to
systems
at
equi-
librium. This is based on
the
use
of
the
principle
of
microscopic
reversibility

as
an
auxiliary
to
the
information contained
in
the
laws
of
classical thermodynamics.
It
gives useful and interesting results
when applied
to
non-equilibrium systems in which there are coupled
transport processes,
as in the thermo-electric effect
and
in thermal
diffusion.
It
does
not
have significant applications in
the
study
of
chemical reaction or phase change
and

for this reason is
not
included
in the present volume.
t
1·2. Thermodynamic systems
These may be classified as follows:
Isolated
system~~
are those which are entirely uninfluenced
by
changes in their environment.
In
particular, there is no possibility
of
the
transfer either
of
energy or
of
matter
across
the
boundaries
of
the
system.
Closed
system~~
are those in which there is

the
possibility
of
energy
exchange with
the
environment,
but
there is no transfer
of
matter
across
the
boundaries. This does
not
exclude
the
possibility
of
a.
change
of
internal composition
due
to
chemical reaction.
Open
system~~
are those which can exchange both energy
and

matter
with their environment.
An
open system is thus
not
defined in terms
t
For
an
elementary
Bl'count
of
the
theory
see
the
author's
Thermodynamics
of
the Steady State (London, Methuen, 1951). Also Prigogine's Introduction
to
the Thermodynamics
of
Irreversible Processes (Wiley, 1962), Callens' Thermo-
dynamics
(Wiley, 1960),
Fitt's
Non-Equilibrium Thermodynamics (McGraw-Hill,
1962),
van

Rysselberghe's Thermodynamics
of
lrreversible.Processes
(Hermann,
1963) a.nd
de
Groot
and
Mazur's Non-Equilibrium Thermodynamics
(North
Holland
Publishing
Co., 1962).
For
criticism
see Truesdell's Rational Thermo·
dynamics
(McGraw-Hill, 1969).
6
Principles
of
Chemical Equilibrium
[1·3
of
a.
given piece
of
material
but
rather as

a.
region of space with
geometrically defined boundaries across which there is the possibility
of
transfer
of
energy
and
matter.
Where
the
word body is used below
it
refers either
to
the
isolated
or
the
closed system. The preliminary theorems
of
thermodynamics
all refer
to
bodies,
and
many
of
the
results which are valid for them

are
not
directly applicable
to
open systems.
The application
of
thermodynamics is simplest when the system
under discussion consists
of
one or more parts, each
of
which is
spatially uniform
in
its properties and is called
a.
phase. For example,
a system composed
of
a liquid
and
its vapour consists almost entirely
of
two homogeneous phases.
It
is true
that
between the liquid and
the

vapour there is
a.
layer, two or three molecules thick, in which
there is
a.
gradation
of
density,
and
other properties,
in
the direction
normal
to
the
interface. However, the effect of this layer on
the
ther-
modynamic properties
of
the
overall system can usually
be
neglected. .
This is because
the
work involved in changes
of
interfacial area, of
the

magnitudes which occur
in
practice, is small compared
to
the
work
of
volume change
of
the
bulk phases. On
the
other hand,
if
it
were desired
to
make
a.
thermodynamic analysis
of
the
phenomena.
of
surface tension
it
would be necessary
to
concentrate attention on
the

properties
of
this layer.
Thermodynamic discussion
of
real systems usually involves cer-
tain
approximations which are made for
the
sake
of
convenience
and
are
not
always
stated
explicitly.
For
example,
in
dealing with
vapour-liquid equilibrium,
in
addition
to
neglecting the interfacial
layer,
it
is customary

to
assume
that
each phase is uniform throughout
its
depth, despite
the
incipient separation
of
the
components due to
the
gravitational field. However, the latter effect can itself be treated
thermodynamically, whenever
it
is of interest.
Approximations such
as
the
above are
to
be distinguished from
certain idealizations which affect the validity
of
the
fundamental
theory. The notion
of
isolation is an idealization, since
it

is never
possible
to
separate
a.
system completely from its environment. All
insulating materials have
a.
non-zero thermal conductivity and allow
also
the
passage
of
cosmic
rays
and the influence
of
external fields.
If
a.
system were completely isolated
it
would be unobservable.
1•3.
Thermodynamic
variables
Thermodynamics is concerned only with
the
macroscopic properties
of

a body
and
not
with its atomic properties, such as the distance
between
the
atoms
in
a.
particular crystal. These macroscopic pro-
perties form a large class
and
include
the
volume, pressure, surface