class="bi x0 y0 w0 h1"
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
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

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

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,

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

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

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

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,

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

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

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.

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

8·11
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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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,

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

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, 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


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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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