STABILITY ANALYSIS OF MULTICOMPONi~T SYSTEMS
by
Bruce L. Beegle
S.E., Massachusetts Institute of Technology
(1972)
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August, 1973
Signature of Author:
Department of Chemical Engineering,
August 13, 1973
Certified by:--------------------
Thesis Supervisor
Accepted by:......-.
. .
. .. . . ..... . 4 0
. ..
Chairman, Departmental
Committee on Graduate Theses
JAN 2 8 1974
ABSTRACT
STABILITY ANALYSIS OF MULTICOMPONENT SYSTEMS
by
Bruce L. Beegle
Submitted to the Department of Chemical Engineering
on August 13, 1973, in partial fulfillment of the
requirements for the degree of Master of Science.
The criterion of intrinsic stability for a general
m-component system is developed in terms of derivatives of
U, the total internal energy. This criterion is converted
to equivalent forms in terms of any Legendre Transform of
U.
The corresponding equations which define the critical
point are derived.
Stability and critical point conditions are applied to
pure and multicomponent systems.
Superheat limits and
critical points are calculated using the Redlich-Kwong
equaion-of-state and the Soave modification.
The original
Redlich-Kwong equation predicts that the limit of superheat
for pure materials is at a reduced temperature of about .9,
which agrees well with data.
All other predictions show
much less agreement.
Thesis Supervisor:
Robert C. Reid
Title:
Professor of Chemical Engineering
ACKNOWLEDGEMENT
I am deeply grateful to
Professor Robert C. Reid for
his suggestions and encouragement
thesis.
during the work on
this
TABLE 01E CONTENTS
Abstract
2
Table of' Contents
4
List of Figures
5
List of Tables
5
Summary
6
Introduction
9
I. Derivation of Stability Criteria
II. Derivation of Critical Point Criteria
III. One, Two and Three Component Systems
11
33
43
Binary Systems
51
Ternary Systems
57
IV. Predicting Superheat Limits and Critical Points
61
Superheat Limits
64
Binary Systems
68
Critical Points
73
V. Discussion
79
Appendices
A. Determinant Reduction Formula
85
B. Sum-of-Squares Form
90
C. Legendre Transforms
93
D. Chemical Potential Derivatives
99
E. Redlich-Kwong Derivatives
102
F. Nomenclature
105
G. Literature Citations
109
LIST OF FIGURES
I.
P-V Plot of a Hypothetical Pure Material
II. P-V Plot of a Hypothetical Binary Mixture
III. Limit of Superheat of a Pure Material
IV. Limit of Superheat of n-Octane in n-Pentane
V. Limit of Superheat of Cyclohexane in n-Pentane
VI. n-Heptane-Ethane Critical Temperatures
VII. n-Eutane-Carbon Dioxide Critical Temperatures
VIII. n-Heptane-JEthane Critical Pressures
IX. n-Butane-Carbon Dioxide Critical Pressures
49
56
65
71
72
74
75
76
77
LIST OF1 TAELES
I.
Conditions of Thermal Stability
45
II. Conditions of Mechanical Stability
45
III. Conditions of Diffusional Stability
52
IV. Conditions of Stability for a Ternary System
V. Limits of Superheat of Pure Hydrocarbons
58
67
SUMMARY
The purpose of this thesis is to derive rigorously the
to
criteria of intrinsic stability and of critical points,
what these criteria mean in physical terms, and to
examine
accuracy of
examine the
with
the criteria
using
common
equations-of-state in the prediction of limits of superheat
and critical points.
In
a homogeneous system being
and Nl
which
slightly
splits into two
numbers
and mole
statement is that at constant S, V and Nj,
Consider
total
constant
An equivalent
(S) is maximized.
the total entropy
(NK),
at
volume (V),
(U), total
internal energy
state
equilibrium
a stable
U is
minimized.
held at constant S, V
only
phases, each differing
For a system to be at stable
from the original.
equilibrium, the energy must increase during this change.
The change
Series
in energy
the
about
(SU) is expanded in
original
of
rotentials)
are shown
The
conditions.
U (temperature,
derivatives
pressure
to be constant
a Taylor
and
first
chemical
throughout a stable
equilibrium system.
Except at critical points, the second
derivatives
the
expansion
control
in
determinants
sign
a sum-of-squares
of
the
second
of SU.
Rewriting
form reveals
derivatives
of
the
that certain
U must
positive for the system to be intrinsically stable.
be
One of
these determinants is shown to become zero before or at the
same time as the others, and is thus the first criterion to
be
violated.
This
criterion of
intrinsic
stability is
listed in Section I as Eq. (52).
The criterion
terms
of
the
of intrinsic
Helmholtz
Legendre Transform
Legendre
Energy
of U using the
Transforms derived
this criterion
positive
Free
stability is written
is that
(Eq. (50)).
(A) or
any other
second derivatives
in Appendix C.
a single
second
in
One
of
form of
derivative
be
At critical points, this derivative
and the third derivative are zero.
of Legendre Transforms, these
Using third derivatives
conditions are rewritten
in
terms of U or any of its transforms, including A (Eqs. (62)
and (89)).
Several
stability criteria may be
thermodynamic terms.
condition
of
thermal
constant volume
which
All
is
written in common
stable substances
stability,
positive."
"the
The
satisfy
heat
the
capacity at
stability
criterion
is violated when a pure material becomes unstable is
the condition of mechanical stability.
Equivalent forms of
the mechanical stability criterion are:
"the derivative of
the pressure with
respect to specific
volume at
constant
temperature is negative" and "the heat capacity at constant
pressure
when
the
remains finite."
condition
of
Binary
systems are stable only
diffusional
stability,
"the
derivative of a chemical potential with respect to its mole
fraction at constant temperature and pressure is positive,"
Other equivalent forms of the conditions of
is satisfied.
of stability
conditions
stability,
diffusional
and
mechanical
thermal,
and
are given in
for ternary systems
Tables I through IV.
All second and higher derivatives of A with respect to
Superheat limits of pure and
equation-of-state.
explicit
evaluated using a pressure
numbers may le
volume or mole
multicomponent systems and critical points of mixtures were
Redlich-Kwong
original
the
using
calculated
equation-of-state and the Soave modification.
superheat temperature
l-K equation predicts a reduced
The Soave
agreement with virtually all measured compounds.
produces
equation
Both
instance.
much
less
equations
of
This is in remarkable
9/10 for all pure materials.
about
The original
results
accurate
predict
that
the
in this
superheat
temperature of a mixture is very close to the mole fraction
average
of the
which is partially
pure component values,
reflected in the
data.
Calculations of mixture
equally
points
are
about
Trends
are
predicted
poor
correctly
using
but
equation.
either
actual
critical
values
are
significantly in error.
The
apparent fault in
the equations-of-state used is
their poor handling of mixtures.
Significantly
improved
mixing rules are needed to yield increased accuracy.
INTRODUCTION
Under
certain conditions a liquid
may be heated well
above its boiling point and yet remain in the liquid state.
lWhen vaporization finally occurs, it produces a
explosion"
due to the sudden phase change.
"superheat
The explosion
increases in violence as the liquid is heated further above
its boiling point.
temperature
All liquids at a given pressure have a
above
which they
spontaneously vaporizing.
may
The
not be
degree
of
heated without
superheat
may
strongly influence the violence and/or the possibility of a
superheat
explosion.
Thus
superheat is necessary
the study
in predicting
of such
the
limits of
behavior
of a
rapidly heated liquid.
The
limit of superheat is one
intrinsic stability,
with
may
that is, the stability
regard to spontaneous
undergo
such
aspect of the study of
changes
of a
small changes.
include
Systems which
superheated
subcooled vapors and supersaturated solutions.
problem
of this
intrinsic
Since
thesis is the derivation
stability
the study of
for a general
critical points is
stability phenomena, the
system
liquids,
The central
of criteria of
m-component
system.
closely related to
conditions which define
critical
points are also developed.
The
criteria
of
intrinsic
stability are
derived
starting with the entropy maximization principle.
A system
10
is
whether
equations
and
subsystems,
this
change
infinitesimally different
two
split into
to
assumed
are
developed
proceeds
determine
spontaneously.
of Legendre Transforms are
derivatives
to
The
found and are used
to simplify the equations obtained.
The
examined
stability
as
to
their
multicomponent systems.
equations-of-state
values.
critical
and
in a
predictions
They
conditions
are
pure
and
tested using
two
point
are then
comparison
about
with
experimental
Finally, this treatment is contrasted with that
of several other authors.
I. DERIVATION OF STABILITY CRITERIA
The concept
encountered
in
of
intrinsic stability
chemical
engineering
is not
usually
applications.
For
example, consider the reversible, isothermal compression of
water vapor at 1000 C.
1 atm, liquid water
vessel surfaces or
compressed
Normally, when the pressure reaches
appears.
Condensation begins on the
on impurity motes.
further,
more liquid
As
phase
the system
forms.
is
When the
water is entirely liquid, the pressure rises above 1 atm.
If the vessel surfaces are not "wet" by liquid
water
and no other condensation surfaces are available, the water
will
remain
entirely in the
gas phase
even
though the
pressure is raised considerably in excess of 1 atmosphere.
This
is because microscopic drops
specific availability
detailed
of liquid have a higher
function than
the bulk
phase.
(A
analysis of the availability is not required here
--the important fact is that a potential barrier
nucleation).
The
system
is then
stable with
prevents
regard to
microscopic perturbations (intrinsic stability) even though
it may
be unstable
with regard
(phase instability)
i.e.,
the
to a large
formation
perturbation
of
two unlike
phases with the transfer of mass from certain parts of
system to others.
If
the
This system is termed metastable.
the metastable vapor is compressed further it will
eventually become intrinsically unstable. That is, it will
12
become
unstable
with
point
The
phases.
intrinsically
at
limits will occur.
such
a
which
system
microscopic
into
two
becomes
first
"limit of intrinsic
derived below
to predict
where
Expansions in terms of the Gibbs
Free Energy are
or the Helmholtz
to
separate
unstable is termed the
Formulas are
stability."
even
spontaneously
will
perturbations and
respect
readily evaluated
using
volume or pressure explicit equations of state.
The
criterion
of
intrinsic
stability
for
an
equilibrium state, first derived by Gibbs[I], is that for a
stable,
isolated system,
In other words, for
constant
mole
the total
entropy is maximized.
any possible microscopic variation at
numbers,
total volume
and
total internal
energy (NI, V and U),
J£<0
(1)
Lq. (1) is easily changed into alternate forms.
Consider a two-step
stable
equilibrium state
both steps.
variation
The
first
at constant U.
may then
be
original
value.
reversible process starting
holding V and
step
is any
Ey Eq. (1),
added reversibly
until
at a
NL constant during
small,
reversible
S decreases.
S increases
Heat
to
its
This two-step process is equivalent to a
net variation at constant S.
In the first step U was held
constant while in the second step heat was added and thus U
increased.
The total internal energy then increases during
13
all
small variations around this
stable state at constant
S, V and N(.
A similar
unstable
may
process
equilibrium
be followed
state.
variation at constant
U;
The first
S increases.
U
therefore
decreases
step is
Such a
must exist for the state to be unstable.
removed reversibly until S
starting
this
two
variation
Heat may then be
value.
process.
step
appropriate choices, this may be shown to be equivalent
Lolding
S constant.
Thus there
around this unstable state at
an
a small
returns to its original
in
at
By
to
exists a small variation
constant S, V and NZ where U
decreases.
An
for all
alternate criterion of intrinsic stability is that
variations around
a stable
equilibrium state
at
constant S, V and Nj,
u>0o
(2)
Eq. (2)
is equivalent
to Eq. (1)
when Eq.
(1) is
and is
true
since it
violated
is applicable
when
Eq. (1) is
violated.
The
obtained
V form of the criterion of intrinsic stability is
by
using
similar
reasoning.
The
two-step,
reversible processes are all carried out at constant S
and
Li,.
After the first small variation at constant V, enough
work
energy is
either reversibly
from the system (by
added to
or subtracted
contraction or expansion) to
return U
to
its
original
value.
Assuming that
the
pressure is
positive, the sign of 8V for the two step process is always
the same
another
as the
sign
of
criterion of
U for the
first
step.
Thus
intrinsic stability is that for all
small variations around an equilibrium state at constant U,
S and N1, if P>O
(3)
Sv>o
In some metastable
holds
systems P<O.
The above
except that the sign of SV is changed.
general form of
Eq. (3) is that
argument
Therefore a
for all small
variations
around an equilibrium state at constant U, S and NZ
P V>O
(4)
Using a procedure similar to the above, the criterion
of intrinsic
stability
for
small
variations
around
an
equilibrium state at constant U, S, V, and Nja is
(5)
a,,
%<0
Eq. (5) is not particularly useful since it requires that S
be held constant while N, is varied.
Eq. (2) is
the form
of
stability used in this thesis.
the
transforms
thermodynamic
and
the criterion
It is chosen since most of
derivatives
properties.
of intrinsic
of
U
Any other form
are
common
could be used,
and would yield equivalent results.
Since N,,
N,, Ns
...
Nm,
V and S completely specify U
in a single phase, the test for stability must involve
the
15
creation
of
phases, or and B,
two
microscopically from the original.
of heat, volume and mass
S, V and all
each
differing only
Differential quantities
may flow between the phases,
NZ are held constant
but
for the entire system.
Therefore,
To simplify
relabelled
dS =-dS'
(6)
dVP=-dV"
(7)
dN =-dNJ
(8)
x, through x, (n=m+2).
mathematically equivalent
order.
V, S and
notation,
N, through N•
Since
they may
are
V, S and NZ are
be relabelled
in
any
For instance, the xi's could be defined (for j>2):
although any other
x,=S
(9)
xz=V
(10)
xj =Nj,
(11 )
ordering would be
satisfactory.
With
any labelling, Eqs. (6), (7) and (8) summarize to
dx =-dxZ
Also
for convenience, partial
(12)
derivatives of U or any of
its transforms (A, G, etc.) are indicated by subscripts:
U t=bx I/ Xx•.I
3t 4
;
1T,Ni
A v= V-V
Since each subsystem is assumed to undergo only
changes,
the total
Taylor Series
through
internal energy
about the
second
order
original
terms
may be
small
expanded in a
conditions.
Expanding
(using a superscript O to
16
indicate that
a variable
is evaluated
at
the
original
conditions)
dxL+4
. ý 4 U.L dx'dx
U'=
U .= dx +2
U
The
(13)
dx dx
(14)
change in the entire system's total internal energy is
the sum of the changes for the two subsystems, or
U= USU+
SU
(15)
Combining Eqs. (12) through (15)
(U.
Since
all
incorporated
constraints
into
on
Eq. (16),
Eq. (16) must therefore
dx,
(Uo++Uo. )dxLdxJ
°U)dxL+
the
each
be true for
through dx,, including
system
dxL
Thus, since dx-
negative, UO
must be equal to UQ
have
been
is independent.
all possible sets
the one where
non-zero dxj.
(16)
of
dxc is the only
may be either positive
or
to prevent SU fror being
negative.
U=UTI
Each
UO is an
intensive variable, being
(17)
either T, P or a
•jA.
The subsystem g may be defined to be any part of the
original system.
Therefore Eq. (17) shows that there are
no temperature, pressure or chemical potential gradients in
a stable equilibrium state.
Since m+1 (i.e. n-1) intensive
variables are sufficient to define
extent)
of a single phase
the state (but not
the
system, all intensive variables
are constant everywhere throughout the original system.
U? , being
with
respect
the derivative
to
proportional
an
of an
extensive
intensive
is
variable,
variable
inversely
to the number of moles in the subsystem under
The product of Uij
consideration.
and the number of moles
is therefore the same for any subsystem.
N'U "=N U
Substituting
Eqs. (17)
and
(18)
(18)
into
Eq. (16)
and
eliminating ULJ
:
U
nU=N
E
The
system
which is
original system.
part
tested for
stability
is the
The subsystem o may be chosen to be
of the original
are dropped
(19)
L
jL(
being
dx dx
when
system.
any
Therefore all superscripts
substituting Eq. (19)
Eq. (2) to
into
yield as an alternate criterion of intrinsic stability
n
n
(20)
ZEULsdx dx>O
A
system
is
intrinsically
stable
if
Eq. (20)
satisfied for all microscopic perturbations.
hand
side
(LHS)
of
Eq. (20)
were
negative
perturbation, the system would be unstable.
Eq. (20) were zero, then Eqs. (13)
is
If the
left
for
some
If the LHS of
and (14) would have
to
18
be
include third (and possibly higher) order
expanded to
Following the above developement, Eq. (20)
terms.
include third order terms.
then
would
the signs of all the
If
dx4's were reversed, then then the sign of the second order
terms would be unchanged while the sign of the third
order
Thus when the LHS of Eq. (20) is
terms would be reversed.
zero, the change in U may be either positive or
negative,
also zero.
Usually,
unless
the
order terms
third
therefore, when
of Eq. (20)
LHS
the
system becomes unstable.
third order
terms are
Eq. (20) when
are
zero,
the
At critical points, however, the
also zero.
the LHS
becomes
This
special case
is zero is discussed
of
further in
Section II.
LHS
The limit of intrinsic
stability is reached when
of Eq. (20) is zero.
Since
each dxL
the
may be either
positive or negative, it is desirable to express the LHS of
Eq. (20) in a sum-of-squares
form.
Then the sign of
the
expression will be controlled by an appropriate combination
of
U L s.
The
sum-of-squares
form
is
derived
in
Appendix B. Eq. (20) may then be written as
Dk dZ2 >0
(21)
where
dZk=)C, k JdxjDk
and
Ckkj are
defined
as in
Appendix
(22)
B, with
Zk of
19
Appendix E written as dZk.
Ckkj =
U,,
Ua
U.1L
U 2. 0-.
U0k
U .kU*
Uk, Uki ---
Ukki
..-
Uk
k
U,1 U,~.
000
Ulk-,
U2 1 U1
2
...
U
Uk, Uk 2
Uj
,k- U2j
*.. Ukk-1
Ukj
I
Eq. (21) is the
basic equation from
which all
criteria of intrinsic stability will be derived.
simplified
using
the
Legendre
Transforms
It may be
discussed
Appendix C.
Following the notation of Appendix C, y
function
x, through
of
x n and
T
is a
Transform from x, space to E, space.
indicate
partial
derivatives
with
corresponding variable.
y=y(x,,
x ...
xY
other
in
is a
partiel Legendre
Subscripts on y or Y
respect
to
the
20
first let y be the total internal energy, U.
12,,
yII
3,,
Rewriting Dk
*
YTz
I
Dk= Y31 Y Y3 3
32
Yk2 Yk3
Vl
lactoring y,,
*'
"'
(23)
Y3 k
Ykk
from colum n 1,
1
y2. y3
Y,
Yk
Y - Y2 Y; 3 .
Yzk
YIl
Dk=Yl
Y31 Y 32
y,,
LL... Yk2 Yk;3·
The
first column
multiplied by y,,
the
and so on.
The end result is:
by y,3
That
Eq. (24) is then
the second
and subtracted from
column is multiplied by
ith column.
Yk
determinant in
first column is multiplied
the third column;
first
of the
(24)
y" k
Y~
33
column;
and subtracted from
is, for all i>1,
the
y,L and subtracted from the
0
0
""
Z
Yk--Y,•
Y, k
y3,
Y,
y,3
IY1
yII
YkI•,I
k YeI
Y,,
Y,,
(25)
**" Yk-Y~ y,
yI
yl,
"'"
k
Yk3
Ykk--Y'•IY
y.'
k
Y1"
Simplifying,
•.Y
Y• k--Y
Y32
fY k
Y,,
yl
• "Y3k-Y YLk
Dk=y, I
Y,"
y
Yk I
,
-y
form yYy~jy,•
y, Yl',-I
form
Appendix
derivative
C),
-y 1 k Yi2
ALJYL
Y,,3
S."
Y11l
the determinant
'
the
Y1
of Eq. (26)
where i>2 and j>2.
each term
of
•
3y
Each term in
(26)
yl
is equal to
Legendre
is of
the
By Eq. (C-25) (from
Y] , a second partial
Transform,
I.
UsinL
this
substitution, Eq. (26) becomes
I'32
V33
(27)
D k=y I
Skk
Applying Eq. (27) to the ratio of terms found in Eq.
(21)
22
4Jr12
S
'T33
''' V
'Fk3
"'
0
kk
(28)
Dk =
Dkj I
T3 13
Tk-I 1 Tk-,
by
reduced
be
derivatives of y have,
in Eq. (21)
old
eliminated
form
in
contain
of
partial
second
The
second
however, been replaced with
T is the Legendre Transform of y
The first row and first column
from x, space to 6, space.
which
-kk-
order.
one
partial derivatives of 1.
of the
"S
each of the determinants (except LD)
Thus
may
k-j
'
the
determinant,
the reduced
derivatives
form, are
with
have
which
been
the row and column
respect
to
x,,
the
transformed variable.
All of
reduced
the determinants
by another order by
(except Do and D,)
repeating the process used to
generate Eq. (27) on each of the determinants in
That
is, defining VT ý(
x. space into F, space
be
may
Eq. (28).
as the Legendre Transform of T from
(also termed
the second
Legendre
Transform of U from x, and x. space into E, and g, space)
='f)(&,,
E , x, ...
xn)
=yl(@, x. .-- xn)-e,•=U(x, ... xj)-8,x,-gx,2
(29)
23
a may be defined as a partial derivative of either U or ?,
i .e.,
(30)
The
two derivatives in
Eq. (30) may be
shown to be equal
either by differentiating the definition of T, or by
using
Eq. (C-22) (from Appendix C).
Reducing
each
of
the determinants
in Eq. (28) and
cancelling the term that was factored out, Y,,, yields
(a)
S-k-
Eq.
(21)
has been
(a)
(2)
k-I
reduced
by
(2.)
(a)
Tk-i
3
Y'k-I L
two orders,
and the second
'k-i k-1
Thus
each has
of been
the determinants
in
Eq. (21)
reduced by two(except
orders, Doandandthe ID)
second
partial derivatives of y are now replaced by second partial
derivatives of T (
)
**.
.
This stepwise procedure is continued until the
of
Dk
to
Dk ,
derivative.
Legendre
In
is reduced
to
general,
T (p
Transform of
U from
a single
is defined
ratio
second partial
as
the
x, through xp space
pth
to E1
through Ep space, that is,
(P) (E,,
Xp+,,
g ...
9p,
Xp+
*.x)=
p
u(x,, x1 ...
xn)-
(32)
ELxL
where
Using the notation of Eq. (32), y (or U) is written as ~ (0
'
and Y is written as
generate
.
Repeating the procedure used to
the determinants in Eq. (28)
Eq. (28) on each of
k-2 times
4k
D
(k
kk
(34)
Eq. (34) is the final reduction of the determinants in
Eq. (21).
identical
the
Following
procedure,
the
determinants in Eq. (22) are reduced to
j
The RHS of Eq. (35) is
%
=.=
(k-i)
k
/(k-
-
is the transformed
Appendix
C) shows that
B3k•)
.
By Eq. (32),
is
from x k space to gk space.
the Legendre Transform of Y~(•)
Since xk
(35)
variable,
/
-'
Eq. (C-29)
is equal
(from
to V,,
IRewriting Eq. (35) (for j>k)
C
-=T (k)
(36)
25
rtom the definitions of Ckkj and Dk,
if j=k
,j=k
,k'
C
(37)
Eq. (34) is substituted into
Eqs. (36)
Eq. (21) and
and (37) are substituted into Eq. (22) to yield the reduced
form of the criterion of intrinsic stability
-
dZ
>0
(38)
k=j
d
where
dZk =dk
dxj
(39)
j=k+,
(k-i)
is shown equal to ýk either by differentiating the
definition of
(k- '
(from Appendix C).
(Eq. (32)) or
Therefore
k-) =
Similarly,
t)
differentiating
is
the
xkk
Xk E,.shown
k-,)
equal
of
using
Eq. (C-22)
simplifies to
k-I,, , k+i
definition
Eq. (C-21) (from Appendix C).
by
(40)
xn
to
-xk
either
(k) or by
by
using
T ( k simplifies to
F.M -X
I xk
For
material.
Eqs. (9),
(41)
example, the system under test may contain a pure
Then, if
x,,
xa
(10) and (11), the
by Eqs. (21) and (22) is:
and x 3 are
defined
as
in
criterion of stability given
26
Uss
Us
UIss
dZ, +
Uvs Uvv Uvp
USV
UVs Uvv
Usv UsN
dZa +
UWs UNV UAw
1UssI
Uss
Usv
Uvs
UVV
1
dZ3 >0
(42)
where
(43)
UsNdN
dZ =dS+s UVS
UssdV+
Uss
(44)
Uvs Us~
Uvv
UuV
dZ 3 =dN
(45)
through (45) may be
The determinants in Eqs. (42)
reduced
to single terms using the Legendre Transform methods above.
This
alternate
form
of
the
criterion
stability, given by Eqs. (38) and (39),
(0o)(dS+
- 131 dN)•
II(1) dV+LI_
Since x,,
'1 (dV+Y'
V - A3
of
intrinsic
is
d2)a 33 (dN) >0 (46)
x. and x3 were ordered to represent S, V and
N respectively, the Legendre Transforms
of U as used here
are
° =A(T, V, N)=U-TS
)---G
=_(T, -P,
N)=U-TS-(-P)V
27
The second derivatives in Eq. (46) are
=Uss
=1)
=
T
STNN
() =A v=
I= 1
T,P
N
SS
=v=()T,P
=
';
Substituting the above formulas into Eq. (46) yields
'
Uss (dS-+ATvdV+AT)dN)
+A(+G(dV+GpCdN)+GNN(dN) >0
(47)
or equivalently,
d bS
dV(dS+-Plb'l(dV-
bT
TN
+67I
Eq. (48) may
Eqs.
also
be
-
dNi);
TP
obtained
dN)'+
from
(dl') '>C (48)
T
Eq.
(4.6)
using
(40) and (41).
The
leading coefficient of the
is the derivative
potential),
of an intensive
last term in Eq. (48)
variable (the
evaluated. while holding
two
intensive
variables
completely
specify
chemical
other intensive
variables (temperature and. pressure) constant.
,
by
the
Since
state
single component system, this derivative is equal to
The result is generalized below to any system.
two
of a
zero.
28
Eq. (38)
intrinsic
is the
reduced form
The
stability.
leading
coefficient.
derivative
final
Using
En, holding
of
all
Since each ýL is either T, -P or
variables.
Since
n-1
(i.e.
of
the
criterion of
term
has
Y
as a
-n)
is a
Eq. (40),
other n-1
8g's constant.
yJ, all 8's are intensive
m+1)
intensive
variables
equal
is
,,
completely specify the state of a system,
to zero.
The
LHS of Eq. (38) must be greater than zero for all
permutations around a stable
*
equilibrium state (except at
critical points, where it may be equal to zero).
There is
an
and
apparent
contradiction
between
this
fact
the
preceding paragraph, since the dxt's in a variation may
selected
all the
dZk's except
dZn are
Since the coefficient of dZ, (-in
zero.
be
so that
equal to
) was shown to
equal to zero, the LHS of Eq. (38) is equal to zero for
this variation.
The contradiction
is resolved by
the nature of this particular variation.
mole
numbers, total volume
noting
It is a chance in
and total entropy
by the same
proportions, or simply a shift in the boundary between
two
be
subsystems.
property, or
system.
There
in any
Therefore
is no
extensive
this
in any intensive
change
property
variation
the
of
the
is actually
entire
not a
variation in any measurable, physical sense.
n-1 independent
intensive
variables
can
always
be
29
found
for a stable single phase system.
For example, the
temperature, pressure and mole fractions of all but one
the
components
variables.
could
be
the
independent
system (c plus P) are
these n-1
system
variations cause
is stable.
an increase
dZf through dZn,
dZI through dZ_, (-1<
a stable phase.
then all of
in U,
since
the
Thus the coefficients of
) must be
positive
The "limit of intrinsic stability" is
(except
('-' ) becomes zero.
to Dk- , if
The ratio of Dý
reduced k-1 times
using
Legendre Transform methods discussed earlier, is shown
by Eq.
D k-
the
correspond to linear
through
reached when any T' k-
the
S, V and N of
If
held constant,
combinations of these variations.
in
intensive
Therefore the subsyster c has r-1 independent
variations which do not change N .
entire
of
(34)
to be equal to )(k
.
If the ratio of Dk
to
is reduced only k-2 times, it is shown equal to a form
involving only derivatives of \Y -
k- i
k-i
Since
phase,
both
-•
kk
Vk
kk
and
must be
k-i k-i
are
positive in
positive as well.
k[
)
kk
the coefficient of dZý_,
if the ordering of xk-
reversed.
(k
Therefore
without limit as
k
is assumed
approaches
zero.
to
a stable
would
be
and Xk was
not increase
If
k
is
30
also assumed
(k-1)
4)
not to
becomes zero
kk
Both
be
zero, then
at the
Eq. (52)
same time
of these assumptions are
shows that
W -2-
or before
used throughout the rest of
this thesis.
Since
k
Vk-(k-1)
k-I I
-kkk
--
any cther
becomes zero at
-2
becomes zero
6-I
(except
-)
,-')
the same time or
at the same
).
before
time or before
Therefore
the
final
criterion of intrinsic stability is
-, >0
(50)
Rewri ting Eq. (50) in terms of xj's and 9ý's
>0
)811,-
(51)
Using Eq. (34), Eq. (50) is equivalent to
(52)
D., >0
Using Eqs. (34) and (28), Eq. (50) is equivalent to
S
.
i2 3
...
•_,
.
_, •
3_,..
A.,
_, ,_, (the ratio of Dy-, to Dn-z)
ratio
of two
VY).
Using Eq. (50)
>0
.
(53)
,-I
may be expressed as the
determinants whose terms are derivatives of
L•>O
where LL is defined for any given n and i
(54)
T(L)
C.#I
ý42
L#/
'LF
LfZ n-I
If.
and (53)
Eqs. (52)
ii-i
L+2
(55)
'i-1
are Eq. (54)
with
i=O
and
i=1,
and if
i=O,
respectively.
(x,,
If
... xn)=(S, Ni, N;L ... Nw, V)
X,
then Y " ) =U and Eqs. (54) and
Uss
USN,
Usv,
U•s
UIV,
UN
UNm
UN N4
Uh N a
55) become
Usum
...
0rjM
(56)
>0
..UNJ
•
Derivatives are defined as before:
UN N.(7
Eq. (56)
FT
)
PN,ý(j
is a criterion of intrinsic
stability stated
Gibbs[2] .
The example of a system containing a pure material
again
employed.
Eqs. (9),
(10) and
Values
of x
(11).
is
again defined
as in
Eqs. (52), (50) and (51)
then
are
become
Usv
Uvs
Uvv
AV,>0O
>0
(57)
(58)
32
b(- P )
Eqs. (57),
(58) and
(59) are
equivalent
criterion of intrinsic stability for a system
pure material.
(59)
TN>0
forms
of the
containing a
Forms which arise from different orderings
of x,, x. and x,,
as well as
considered in Section III.
multicomponent systems
are
33
II. DERIVATION OF CRITICAL POINT CRITERIA
The
"limit
of intrinsic
stability" is
reached when
Eq. (50) is first violated,
,_,
i _,.
=0
(60)
or equivalently, when Eq. (51) is first violated
(61)
ben-=0
or when Eq. (54) is first violated (for any i)
LL=O
The locus of
(62)
the points
is called the
system
reaches
(62)
which satisfy
Eq. (60), (61)
"spinoidal curve."
the
spinoidal
or
In general, when a
it
curve,
becomes
intrinsically unstable and spontaneously separates into two
(or more) phases.
Consider
through
Kn-z
equivalently,
This is demonstrated below.
a system which is
and x,.
Eq. (50))
In
a
is
being held at constant t,
stable phase,
true.
In
Eq. (51)
other
(or
words, U
increases for all small variations at constant S, V and Ný.
for
certain values of E, through
-,_2, the locus of points
formed by varying x,_, will intersect the spinoidal
curve,
where Eq. (60) becomes true.
Eq. (50) was based on the assumption that second order
terms
would be sufficient to determine whether
U would be
negative for some variation or positive for all variations.
This assumption is not valid on the spinoidal curve,
since
34
Eq. (50) predicts
U is non-negative for all variations and
is zero for at least one variation.
analogous to Eq. (50),
fourth)
but including
order terms is necessary
of a system on the
Therefore an equation
third (and possibly
to examine the stability
spinoidal curve.
Such an equation
is
derived by examining again the two subsystems, o and g.
A
variation possible to the system described above is
holding ý, through 8,-, and x, constant in each of the
subsystems
and
varying
x,-, and
then V-~z
Y)-I ,-I
of
the
total x,.,
Assume that the subsystem o increases
allocated to each.
in
the fraction
two
If
that p decreases.
will become
positive and
n-s
n-
positive
- n-I
will
become
negative.
o will now be in a stable region, but p will be
unstable.
At this point, an additional transfer of
from p to a will take place.
splitting
into
two smaller
becoming part of oc.
In effect, the subsystem
subsystems,
one of
well
equilibrium
is
Since p is in an unstable region, this
as driving 9 further
Thus this process
,
which is
will result in a decrease in U, the total internal
as
xn-,
into the
is spontaneous and
is reached, with at
energy,
unstable region.
will continue
until
least two distinct phases
formed.
If
still
--
is
negative then
the
above
applies, with the roles of a and f reversed.
system to
be stable
and to
lie on
the spinoidal
argument
For a
curve,
35
S_•_
must be zero
(63)
n-i n-i r-i
Rewriting Eq. (63) in terms of xL's and E•'s
,
For
=0
a system on the spinoidal
addition to
Eq. (63)
(or
(64)
curve to be stable, in
(h-a)
(64)),
n-t
n-(
n-
must
be
positive
iI
If
Eq. (65) is not
zero.
(65)
satisfied, then the
be positive, and all
must
>0
derivative of y(n-.)
non vanishing
i
1
lowest even order
with
respect to
xn,-
lower order derivatives must be
This condition is necessary to insure that after a
small
variation inside
the entire
system, all subsystems
are still stable.
x,,-
Varying
spinoidal curve
will
and
different values
a
system
remain stable
of g, through Fn-,
values
allow
to
only when
are held constant.
are chosen
then the
touch
the
particular
If slightly
system will
either
rass through the unstable region or else miss the spinoidal
curve
entirely.
curve lie
(where
on
two
the
phases
Thus the stable
points on the spinoidal
boundary between
are
changes are continuous.
formed) and
the
unstable
a region
region
where all
These points are called "critical
points ."
For
example, a pure
material may have
x,, x, and x,
36
Then the conditions
defined as in Eqs. (9), (10) and (11).
of the critical point (Eqs. (60) and (63)) are
Av,=O
(66)
AV V=0
(67)
or equivalently, Eqs. (61) and (64)
(68)
P ITN=O
S-)T,N
pure
of the
the conditions
Other forms of
well as
materials, as
critical point
for
examples using multicomponent
systems are presented in Section IV.
The section below derives a general form of
analogous
to Eq. (57).
the determinant LL
The
ML is defined as
The determinant
(Eq. (58)) with the
jth term in the last
Eq. (64),
last row
changed.
row becomes the derivative of LL
with respect to x;,j.
(C1-
4)L4
c+2
IA-
L3
Ll
IA
; Li-
-i
L+1
÷÷.
Lý+a C4--1
n-i
0-ý
n-
L
t=
(70)
•
(;1
-,2 L+I
rl
_L
LL
•X
L,
u J(L
LCL1
Trn-' La
At
L+LL
•X-~
•
n-
L+3
XL÷-
n- 1
...
X
__L_
-
t
The first determinant considered will be Mo
37
H7
i.o=
7
hf 3
Y2
(71)
....
>ýx
I
T
x,
Vx,
Y
The same sequence of operations used to produce Eq. (25)
applied to Mo.
by y,,
while
Each element of the first column is divided
the entire determinant
quantity.
same
is
is multiplied by
Then, for all j from
column is multiplied
by y,y
the
2 to n-1, the first
from the
and subtracted
jth
column.
1
M•=,i
Y j,
.
f#Lo
.
Y,
a
I
3Lo
y',
...
0
ex
Y,
O
1
.
Y1ý
(,
3Lo-~
ZYI
(72)
Y, ,
y1
L
ax,-
.. .
aL, -y,-
y,x,
ax,
y
is defined (for j between
,
t
Lo
x
2 and n-1) as the jL•
term in the last row of the determinant in Eq. (72)
ftLo = Lo-y _ŽLo
)x3 y1,
(73)
bx,
Defining [#] as the derivative operator in brackets, below
[W]=[ X -y
1 _]
)xj y1
Eq. (73) is rewritten
bx,
(74)
38
ýTLo=[#]L,
(75)
Multiplying and dividing Lo by y,,
j Lo=[#](y,, L,)
(76)
jLo=y,, [1#]( Lo)+( Lo)[#y]y,
YI!
Y11
(77)
Expanding Eq. (76)
If Eq. (77) is considered only
on the spinoidal curve,
Lo
is equal to zero
.L =y, [
Repeating
1I,
used
the procedure
Eq. (72) on L. over y,,
1
(78)
Lo)
L#o](
above
to
generate
0 0
y IY
YY,
Y1I
Lo-
(79)
Y1,
...
*
y,
Y•,
y
A 'derivative
- -,-,jI n,
f, Y
y•,L
Yi
operator
Y1I
applied
to
a,
determinant
evaluates as the sum over all k of the determinant with the
derivative operator applied to each element of the kth row.
Simplifying Eq. (79) and substituting it into Eq.
(78)
39
·' Xn"- -X
YI
17-f
[k-]Y
[t]Yk-,-L-yLk-1-r~~
n-1--Yl kYI yr
***
ye'
k=a
yll
Jz-ye'
L
y,,
(80)
yII
YI,
Y~cI
;;LLLt+
ye'
• -- Y-, .- -Y n-#, Yl A-,
Y7e
Evaluating
mth
the
Y1I
element
kth row
in
the
(80)
determinant in Eq.
xjj
S
the
of
YI,
Yl,
Y3r
ik+Y
,
b•(Yk m-YYII
Yo
bx,
)
If
+
,
(81)
Yik
y3
I;L
Ie
Rewriting Eq. (81)
SYM
(YkM-Y,
k Y,
Yek )
+Yj Y, k )Y,-I
+YJ Ya kmn
=Ykm)-(Yi kY nj +Yim Ym
tj
r
+(Y k Ylm Ytij
+Yj
Yim
Yj
k
)Y.,
+yk Y, Y,,1
-Y7k
Using
equal
Yl, Y, • Y, ,,, Y,1
Eq. (C-29) (from Appendix C), the RHS of Eq.
to
Tkm, 9
a third derivative
of
I,
the
(82)
(82) is
Legendre
Transform of y from x, space to £, sp)ace.
(83)
.xj
40
of
the
determinants in Eq. (80) is simplified using Eq. (83).
All
The
of
row
involving
the
[#]
the other terms in the
in
each
determinants are shown equal to
'f,using Eq. (C-25) (from Appendix
second derivatives of
C).
operator
(80) is
The simplified form of Eq.
•*0**
0
0
ýxs
bI'2j2
0r
***
n-0
Y1
3Ik.
k=a
Z
flk-I
n-I
** "•
(84)
n-I
0kk
'' '
Y/-I
o,.
n-I
The RHLS of Eq. (84) is the formula for the derivative
of the determinant LO with respect to xj
(85)
,L.=~Lx
determinant in Eq. (72) is simplified
The last row of the
using
Eq. (85),
and
the
simplified using Eq. (C-25).
rest
of
the
terms
are again
Eq. (72) becomes
**o
y'2n-I
I
(86)
Mo =
,I
V
,.l
0x
'x
n-i
0XA-
All the terms
in the determinant
in Eq. (86),
except
for
the last row, are identical to the terms in L,.
For all j,
the
of L
jth term in
respect to
the last row
xl+j.
Therefore,
Eq. (86) is equal to M,.
is a derivative
by
Eq. (70),
the
with
RHS
Multiplying by y,2
Mo=ya M,
The
procedure used to
Mo , could start with ML.
(87)
derive Eq. (87), starting with
The result is
+I1L-+1i+,
(88)
Repeatedly applying Eq. (88) demonstrates
equal
of
that if
Mj
is
to zero, then ML is equal to zero, for all i<j.
Eq. (70),
Tn-j
is zero
alternate
is
at
Mn_,.
critical
form of the second
Eq. (63)
points.
By
shows
that
Therefore
an
equation for critical points
is (for i<n-1)
ML--=O
Eqs. (62) and (89)
(89)
(or (60) and (63), or
(61) and
(64))
define the critical points of any substance.
If
and
the xL's are defined as in Eqs.
i is chosen to be
(9),
(10)
and (11),
2, then the Legendre Transform in
Eqs. (55) and (70) is G.
The conditions of the
critical
point given by Eqs. (62) and (89) then become
where
L2=0
(90)
M=--O
(91)
42
N,
IN
G
N I N2
GN; N,
G
(92)
G
Nm-r
0ntW
a,
GNI ,
,N
GN;. N,
GN
GNm-a N,
0zW
0Nn7
(93)
~
S GNm-. Nm-;
... •
L•2
bTT
Eqs. (90) through (93) are conditions of the critical point
stated by Gibbs[3].
43
III. ONE, TWO AND THREE COMPONENT SYSTEMS
In
Sections I and II formulas were derived to predict
limits of
intrinsic stability
general m-component system.
special
and critical
points for a
This section will consider the
cases of pure materials
and of binary and ternary
systems in more detail.
Eq. (46) gives a criterion of intrinsic stability
a pure material.
F3
,2),
was shown
for
The leading coefficient of the last term,
equal to
to be
Therefore a pure
zero.
miaterial is stable if the leading coefficients of the first
two terms are positive:
<
>0
(94)
S>0
(95)
Since the leading coefficient of the second term was
to
become zero before or
first, Eq. (95)
examined
is the
to determine
at the same time
only criterion
the limit
shown
as that of the
that needs
of stability.
to
be
However,
iqs. (94) and (95) each contain useful information.
The specific form that Eq. (94) takes is dependent
on
the ordering of x,, x, and x 3 . If x, is defined as S, then
Eq. (94) becomes
Uss >o
(96)
Since Us=T, Eq. (96) is equivalent to
•_bT
>0
(97)
Lefining Cv,
the heat capacity at constant volume
C(=T bS =TIb\
(98)
bTV 757TIV,N
Substituting Eq. (98) into Eq. (97)
T >0
(99)
Since T and K are always positive, Eq. (99) reduces to
(100)
CV>O
Eq. (94),
from which Eq. (100)
for rulticomponent
as
systems
well
was derived, is valid
as
pure
materials.
Therefore Eq. (100) states that for an intrinsically stable
the heat capacity at
substance,
constant volume is always
thermal
Eq. (100) is termed the "condition of
positive.
stability."
forms of Eq.
Other
x 3 are
may be obtained if
ordered differently.
Table
S,
V and
possible
orderirgs
correspondinr
be
(94)
of
form of Eq. (94).
limit of intrinsic
violated,
and
all
1 lists all
N, each
stability is reached
forms
of
the
of
the
with
All forms of Eq. (94) must
stable equilibrium state.
satisfied in any
x,, x, and
Eq. (94)
Since the
when Eq. (95)
is
are automatically
satisfied up to that point, the label "condition of thermal
stability" is herein applied to
all of the forms in
Table
1, not just the first.
Eq. (95) is that condition of intrinsic stability (for
a pure material) which is violated first.
If (x,,
xZ, x 3 )
45
TABLE I ORDERING OF (x,
Sor
, V,
CCONDITIONS OF THERMAL STALILITY
x,, X3 )
___
To >0 FORM
hT
SS>0
T,9
or
UvV >0
N)
or
N,
S, V
V,
TAELE: II
U, ,>O
>0
bP
<0
>O
i-,ý*) s 7v
CONDITIONS OF MECHANICAL STABILITY
--
ORDERING OF (x,,
~T
SVIN,
TV) S,N
v)
o,
LERIVATIVE FORM
x;,
x3 )
>O FORM
LERIVATIVE FORM
(S, v,
A,V>O
bP
<0
bTV T,N
(S, L1,v)
ANM >0
>0
(V, s,N)
(V, h, S)
(N, S, V)
(N, V, _S)
HA
>O
H,>O
( ) T,V
>0
>0
>0
(P I4 <0
S
\VIp,
Ix,_
ý7 1,_.s
46
are
Multiplying each side of Eq. (59) by
equivalently, (59)).
C,
Eq. (95) becomes Eq. (58) (or
ordered (S, V, N), then
the total
an expression
moles, yields
involving only
intensive properties:
(101)
V)T<0
Eq. (101)
is
termed
"condition
the
of
mechanical
stability."
are possible, and are
Other forms of Eq. (95)
in
Table 2.
necessary
and
Since
any of the
sufficient
to
forms in Table
2 is both
establish
intrinsic
the
stability of a pure material, they must all be
Therefore
listed
equivalent.
of mechanical stability" is
the term "condition
herein used as a label for any of the forms of Eq. (95).
If
Y'
(x,, xZ, x 3 ) are
ordered
as
(V, S, N)
then
=H=_U-(-P)V and Eq. (95) becomes
(102)
H55>O
Since Hs=T, Eq. (102) is equivalent to
bT
>O
(103)
Defining Cp, the heat capacity at constant pressure
C-=T
P=(! I))T
)PN
(104)
Substituting Eq. (104) into Eq. (103)
T >0
(105)
Since T and N are always positive, Eq. (105) reduces to
47
Cp>0
Eq. (106) shows
(106)
that the
heat capacity
at
ccnstant
pressure is always positive for a stable equilibrium state.
Eq. (105) also shows that as a pure material approaches its
limit
of
increases
equivalent
intrinsic
without
stability
limit.
forms of the
(spinoidal
Eqs. (101)
curve),
Cp
(105)
and
are
criterion of mechanical stability
even though one involves a heat capacity and the other uses
cnly P-V-T properties.
which determine critical
Eqs. (60) and (63),
are
easily evaluated for
a pure material.
points,
Rewriting the
critical point conditions for n=3
If
(x1 , x , x 3 ) are
(1)=0
(107)
(t)=0
(108)
again
defined
as
(S
, N)
then
Eqs. (107) and (108) become Eqs. (66) and (67), or, in
derivative
form,
Eqs. (68) and
(69).
Each
the
side
of
Eqs. (68) and (69) is multiplied by N, the total moles
k T=0
(109)
2Pi =0
(110)
7)JT
Eq. (109)
defines the
spinoidal curve,
points which are on the limit of stability.
simply
Eq. (101)
equality (=).
with the
inequality
the locus of
Eq. (109)
(<) changed
Other equations which define the
is
to an
spinoidal
48
curve
are obtained
from the
last column
chanting an inequality to an equality.
the
second
entry
in
the last
of Table
2, by
For example, using
column of
Table
2, the
spinoidal curve is defined by
=0
(111)
TV
Critical points are defined as points on the spinoidal
satisfied to insure
the
intrinsic stability for
any point
It is obtained from
curve.
spinoidal
must be
equation which
is an
Eq. (110)
variations.
small
to all possible
stable with regard
curve which are
Eq. (109) by
changing the first derivative to a second derivative.
procedure may be
spinoidal
curve
used on
and
any equation
is of
This
which defines
the
For
form of Eq. (61).
the
on
example, the first derivative in Eq. (111) is changed to a
second derivative
(112)
TV0
and (112) define the
Eqs. (111)
critical points in a pure
material, and may be used in place of Eqs. (109) and (110).
The significance
apparent
when examining
pressure
versus
hypothetical
specific
isotherms plotted
volume.
Such
material with a critical
atm, a critical
volume
(109) and
of Eqs. (101),
temperature (Ta) of
(110)
is
on a graph of
a plot,
for a
pressure (Pc) of 26
5000 K and a critical
(V ) of .4 liters/g-mole, is presented in Figure 1.
49
FIGURE I -
P-V PLOT OF A HYPOTHETICAL PURE MATERIAL
P
+
40
30
(OOOK
20
5000K
10
0
o40OK
V
/g-mole)
-10
-20
-30
E
N
ION
-40
50
assumed
is
material
The
to
follow
Redlich-Kwong
the
The
equation-of-state, which is discussed in Section IV.
isotherms drawn inside the metastable region are valid only
If such surfaces
if no nucleation surfaces are available.
exist, or
curve,
substance is brought to
if the
the
spinoidal
the material will separate into two phases, each on
a boundary between the metastable and stable regions.
The
area below the critical temperature (5000 K) and to the left
of
the
point
critical
area to the
corresponding
is the
region;
liquid
right of the
the
critical point is
the vapor region.
The
Three isotherms are drawn on Figure 1.
at
600 0K always
has
a negative
therefore always satisfied
stable,
single
points.
Even
region
phase
isotherm reaches zero
and the
at
the
Eq. (101)
slope.
material remains
all
times.
slope (the spinoidal
though
isotherm
If
the
temperature
is low
is always
is
exist as a single phase.
enough,
e.g.
400 0K, the
metastable liquid may exist under a negative pressure.
a gas
two
predicts an
isotherm running through the unstable region, the slope
positive and the material cannot
in a
The 4000K
curve) at
equation-of-state
is
positive.
The
The
critical
pressure
of
isotherm
(5000 K) touches the spinoidal curve at one point.
Since both the slope and the curvature are zero, Eqs. (109)
and (110) are satisfied.
51
Other
Figure 1 shows only the liquid-gas transition.
transitions, for instance solid-liquid, will show identical
effects
except
thesis
is
mairnly
the
(liquids in
critical
for
not having
concerned
a
critical point.
with
superheated
metastable region)
points.
with
and
This
liquids
liquid-gas
These topics will be considered further
in Section IV.
EINARY SYSTEMS
In a
satisfied
binary
by
all
Eqs. (94)
system
and
equilibrium
stable
(95)
states.
are
still
However,
to obtain the condition
Eq. (50) is rewritten with n=4
of
intrinsic stability which is violated first:
If
-
(113)
>O
33;
(XI, xP,X,, x,) are again
ordered (S, V, N_, Nb)
then
0=G
and Eq. (115) becomes
G >O
(114)
or equivalently,
)
(115)
P >0
S T,P,Nb
LRewritinL Eq. (115) in
terms of x ,
the mole fraction
of
component a,
(ixx
7X
Eq. (116)
stability."
is
termed
(116)
>0
T,P
"the
condition
of diffusional
Other orderings of (x,, x., x,, x,) will yield
other forms of Eq. (113).
These forms are listed in
52
TABLE III -- CONDITIONS OF DIFFUSIONAL STABILITY
ORDERING (x,, XI,
or
or , V Nao,
7V, -9, Nc,
5, V, Nb,
or
x, )
Nb
Nb)
'N _I, s,
or
(TS, s,
or
(NS,
or
Nbs
V, Na,
S,
or
g,
or
NgV,N Nb
Nb
l,Nb
V,
>0
Ga- >0
[N) T,P,Nb
>0
)) T,P,Na
>0
V)bi
No_
5, V, N, )
, S,N,N
DERIVATIVE FORM
( PT,4 ,N1
Nb
TNT
,V
FORM
<0
V,
S, Na,
S,
S(a>0
Nai
, Nb, 1No-
_,
or
or
X3 ,
T,
,V
<0
(P
\7 IT,jlbNo
>0
V_-,
Nb
(bT
>0
>0
() P,aIN
>0
6,Sb
P,
yP,1 ,Na
V,
Na,
or
,, NV,
or
Nr,N
Nb , N I
or
N b, Na.,q
PI la ySN>0
D
5)
E, i)
Nb , V, S
")
bT
>0
bP
ý ,,
s
<0
53
Table 3.
to
Since any form is both necessary and
determine the stability of
sufficient
a binary mixture, all forms
are given the label "condition of diffusional stability."
Eq. (116) is often not
the most convierient form
applications to real materials.
P-V-T
data.
This data may
Free
Usually one desires to use
be in the
explicit equation-of-state.
for
In this
form of a pressure
case, the
Helmholtz
Energy is particularly useful, as it is a function of
temperature, total volume, and mole numbers.
Eq. (53) is of the
ordering
as
required form.
(Eqs. (9),
above
(10)
Using
and
the
same
(11)), Eq. (53)
becomes for n=4
Avv >0
Amv Aa
(117)
or, in the expanded form,
A vv A,-A
Eq. (118) may be evaluated
equation-of-state.
directly
using only a pressure
AV=-P,
computable.
(118)
>0
and
Appendix
thus
Avv
D derives
and
a
explicit
A,
are
formula for
evaluating AO, Eq. (D-9).
The conditions
of the
critical
point for
a binary
mixture are Eqs. (60) and (63), with n=4.
Eqs. (119)
and
(120)
=0
-)
(119)
333-0
(120)
may
be
evaluated
directly.
54
Alternately,
conditions of
stability may
be derived from
stability criteria, as was done with a pure material.
Eq. (116) is a condition of stability
Therefore
is defined as
the spinoidal curve
for a. linary.
the locus of
points where Eq. (116) is first violated.
(121)
( 0,P=0
For a binary on
the
spinoidal curve
second derivative of
to be
stable,
the
with respect to x, must be zero, as
well as the first.
Other
(122)
T,P =0
\bx
forms of the conditions of the critical point may be
obtained
stability
in
the
same
in Table
fashion
3, or
from
directly
the
conditions
of
from Eqs. (119) and
(120).
The critical point conditions may also be expressed in
Helmholtz Free Energy
Eqs. (62) and (89)
forms.
become,
for n=4 and i=1,
L,=O
(123)
M,=O
(124)
where
L,=
A
A
SAoV Ao-0
M,=
(125)
(126)
UL, bL,
ýVjT-W;
Expanding Eqs. (123) and (124)
Avv Aao-A--O
SA
Eqs. (127)
and (128)
(127)
A +3A
(128)
may be evaluated
using any pressure
explicit equation-of-state.
The P-V-T diagram of a hypothetical binary mixture
is
presented in Figure 2. The binary is assumed to follow the
Redlich-Kwong
equation-of-state
The mixture composition is:
in
Figure
liters/g-mole;
and
atm and
liters/g-mole.
Tc and Vc are approximately
componet
atm
material
and
VC=.4
20% a substance with T,=700 K, Pc=20
The binary has
Pc=30 atm and Vc=.45 liters/g-mole.
pure
Section IV.
80% the hypothetical
Tc=500 0 K, P==26
1, with
V,=.7
discussed in
values.
Pc
T,=5600 K,
The mixture values of
mole fraction averages of
of the
mixture,
the
however, is
considerably larger than either of the pure component Pc s.
A binary system becomes
pure material.
This is because for a mixture, Eq. (118) is
violated before Eq. (58).
be
unstable more readily than a
calculated
using
The unstable region which would
Eq. (58) is contained
within
the
unstable region indicated in Figure 2. This is verified by
noting
that
the
isotherms
in Figure
2 always
have a
negative slope.
Different mixture
compositions
would
produce
P-V-T
plots differing slightly from Figure 2. Therefore, for a
56
FIGURE II
--
P-V PLOT OF A HYPOTHETICAL BINARY MIXTURE
P
(atm)
40
30
20
,O0 K
'D°K
10
V
(1/g-mole)
0
-10
-20
-
ISOTHERMS
URVE
INT
-30
GION
REGION
-40
57
complete
description
needed.
P-V-T plots of
of
a mixture,
a P-V-T-xo
plot is
approaching 0
mixtures with xCL's
and 1 will approach the P-V-T plots of pure component b and
pure component a, respectively.
TERNARY SYSTEMS
Ternary
systems are
pure and binary systems.
analyzed by the
same methods as
The intrinsic stability criterion
for a system with n=5 is, using Eq. (50),
(129)
4 >0
Table 4 presents
the fundamentally
forms of Eq. (129).
while
different
derivative
Each form involves taking a derivative
holding at least one ul constant.
forms are difficult
to evaluate
and
Therefore, these
are not
useful
in
calculations with real materials.
Again
(11).
the xZ's are
Then
defined as in
-_=A
and Eq. (54), an
Eqs. (9), (10) and
alternate form of
the
criterion of intrinsic stability becomes (for i=1)
L,>O
(130)
where
Av, AvO Avb
L,=
Aav Aaa Aab
(131)
,A Ab Abi
AV,,
A w, and
A,b are
evaluated directly
from a pressure
explicit equation-of-state.
Aaa, Abb and Aab are evaluated
using Eqs. (D-9) and (D-10),
from Appendix D.
58
TABLE IV -
CONDITIONS CF STABILITY FOR A TERNARY SYSTEM
ORDERING OF (xl,
DERIVATIVE FORM
xI, xS, x0, X)
x
N, N, Nc )
(S, v,
T,P,ýU,Ne>O
(v, NO, Nb,
V, Ne)
V)
No,
(S, No,
Nb,
(V, No,
Nb , S, Nj)
1bT\
>0
(V, NO,
N b , Nc
(&\
>0
>0
(N,9Nb , NC ,
,
S)
V)
bp
_
(N , N, N , V, S)
,TV
,
<o
NOTES:
1.
Any orderings of x,,
only
in the arrangement
order of
the first
x,, x 3, x.
and x s which
of N., Nb and
three variables
differ
Ne and/or in the
are not
considered
different and are not listed separately above.
2.
Since
no third
Legendre Transforms of
U have common
names or symbols, no condition of intrinsic stability
the Vý3
L4
4I·I>0 form is listed above.
in
59
Critical
points are handled
the same ordering
(1),
of (x,,
the critical
in the same way.
Using
x2 , x 3 , x4, x5 ) and the
same i
point conditions of Eqs. (62) and (89)
then become
L,=O
(132)
M,=O
(133)
where L, is defined in Eq. (131) and
Avv Aw, Avb
M,= Aav Ao As
bL
( 14)
ŽL
, )L,
A ternary P-V-T plot at a given x, and xb will
approximately the same as Figure 2.
appear
The unstable region of
a ternary is larger than that predicted by Eq. (113) (which
is used in Figure 2), but is of a similar shape.
Systems
the same
with four or more
procedures
components are analyzed by
used above.
If
the
equations
are
always chosen to be in the Helmholtz Free Energy form, then
they
may
be
evaluated
equation-of-state.
second
condition
considerably
using
only
a pressure explicit
Although the number
of
the
critical
of terms in
point
the
increases
with an increase in the number of components,
this is not a significant difficulty if a computer is used.
The equations derived above may be used to locate
the
spinoidal curve and critical points of any substance, given
a suitable equation-of-state.
Section IV demonstrates this
60
using the equation of Redlich and Kwong (and also the Soave
modification)
with
nulticomponent systems.
several
pure
materials
and
IV. PREDICTING SUPERHEAT LIMITS AND CRITICAL POINTS
The equations derived in Sections I, II and III may be
materials
the
if
appropriate
data
is
applied to
real
available.
Yor example, equations involving the Gibbs Free
Energ.y,
such
volume
explicit
are readily
as Eq. (90),
equation-of-state.
evaluated using a
Pressure
explicit
equations-of-state, as already noted, enable the evaluation
of equations which use the Helmholtz Free Energy.
Perhaps the most used two parameter
equation-of-state
is that of Redlich and Kwong[4]:
P=RT -
aI
(135)
7'-U T'3 V(V+b)
K is the Gas Constant and
for each substance.
a' and b are constants selected
Defining a as the ratio of a' to T 5'
(136)
a= a'
T77-
Combining Eqs. (135) and (136)
P=RT - a
(137)
P-E V(V+bJ
Lq. (137) is the form of the Redlich-Kwong equation used in
this thesis.
The term "a"
This
modifying
permits
changing the
is a function of
the
temperature
temperature.
equation-of-state
dependence
of
by simply
"a" (Eq. (136))
without changing Eq. (137).
The
reduced temperature, T., is defined as the ratio
of the temperature to the critical temperature
62
(138)
T,=T
The constant a" is defined as the ratio of a' to T 5
a"= a'
(139)
Combining Eqs. (136), (138) and (139)
a-:a "
0(140)
Substituting Eq. (140) into Eq. (137)
P=RT The constants
using
(141)
Eq. (137) are
b in
a" and
point conditions for
the critical
Eqs. (109) and (110).
V=V, .
a"
evaluated
a pure material,
At the critical point T=Tc, T,=1 and
simultaneously
Solving Eqs. (109), (110) and (137)
for a", b and V. yields
a"=@R "TR
(142)
b=6 bRTc
(143)
VC=RTe
(144)
@a= .427480
(145)
@b=.086640
(146)
3r7
where
Thus
in the Redlich-Kwong equation
the constants
evaluated
for
temperature and pressure.
Lq. (144)
yields
compressibility.
material,
pure
any
of
value of
The
value
given
of
1/3
for
the
may be
critical
Vc predicted
the
by
critical
This is somewhat higher than the actual
63
value for all known compounds.
The reduced pressure, P., is defined as the ratio
of
the pressure to the critical pressure
P4=P
(147)
V', a type of reduced volume, is defined as one-third the
ratio of the volume to the critical volume predicted in
Eq. (144)
(148)
V'=VPc
Combining Eqs. (141),
(142),
(143) and (147)
vP-•,T
a(V-'
(149)
Eq. (149) is the completely reduced form of Eq. (135).
It
gives PR as a
function of TR and
easier to use in calculations
last
term
by partial
V%.
A form that is
is created by expanding
fractions,
and using
the
the identity
Av=-P.
-P,=Av=-
Ts +T@-
where
@C
(150)
VW
where
@C=
(151)
@Q
Ce is a function of temperature.
Using Eq. (148), the definition of VO, and taking all
derivatives at constant T and N
AVV=
Wi
= FAv)
c IMy\(1
(152)
Combining Eqs. (150) and (152)
NRTr A,=
T
@c+
77JT0VP2
Eq. (153)
is used
with
(153)
_@c
(V 7+-3j
Eq. (58) to
determinine
the
stability of any pure material, given T, and V(.
SUPERHEAT LIMITS
A liquid
at 1 atm pressure
temperature is raised
will vaporize
its
above
normal
boiling
in
temperature
stability
heated
with
contact
may
be
surfaces,
any nucleation
raised
to
the
limit
of
the
normal
"superheated" liquids,
boiling
temperature at
and the
Liquids
are
point
the
intrinsic
(spinoidal curve) before it vaporizes.
above
point,
If the liquid is
assuming that nucleation surfaces exist.
not
when the
termed
which
the
liquid, reaches the spinoidal curve is termed the "limit of
superheat."
gives
Eq. (66)
a defining
equation
of
the
spinoidal curve for a pure material.
Combining
(153) yields
Eqs. (66) and
a criterion of
the spinoidal curve in terms of T, and VT
T
- @e+
@C
=0
(154)
A liquid at the limit of superheat will satisfy
Eq. (154).
In addition, if the pressure is 1 atm,
P,=1 atm
(155)
p.
Solving Eqs. (154) and (155) simultaneously
for VO and TR
yields the limit of superheat for any material, in terms of
65
[FIGURE III -
LIMIT OF SUPERHEAT OF A PURE MATERIAL
LIMIT OF
SUPERHEAT
T7
.94 -
•93 -
Lu=.
.92 -
.91
~
~rc·
W=c
.90
'.Ow
.89
-
I-l
I
--
-
I
-
I
I
I
.65
.10
20
10
I
I
I
~'~
.15
6.66667
- ORIGINAL REDLICH-KWONG
SOAVE MODIFICATION
Pc (atm)
the reduced temperature.
of
Figure 3 is a graph of the limit
superheat (as T,) as a function of PR (or P,).
For
materials with critical pressures above 10 atm, the reduced
temperature at
the limit
of
superheat predicted
by
the
original Redlich-Kwong equation is between .894 and .904.
Limits
of
calculated
using
Table 5 under the
Table 5
are
superheat
for
Eqs. (154)
ten
and
(155),
columns labelled R-K.
obtained
experimental values
Kremsner and Blander[5].
are listed
in
Also listed
in
from Eberhart,
The calculated values are in good
agreement with the experimental data, the average
deviation
being
20K.
as
hydrocarbons,
Since
absolute
Redlich-Kwong equation
the
always predicts a reduced limit of superheat of about .9, a
Lood rule-of-thumb for pure materials is that the limit
of
superheat is nine-tenths of the critical temperature.
A modification of the Redlich-Kwong equation is used
in this thesis.
Soave proposed[6] retaining Eq. (137), but
changing the definition of "a".
Instead of Eq. (140), "a"
is defined by
a=a"[1+( .480+1 .57w-.176
where w
is the acentric factor
defined by Eqs. (142) and (143).
)(1-T;'
and a" and
(156)
b are still
If the definition of @c
is changed to
@c=@a[1+( .480+1 .57w-.176w2)(1-T 5)]
(157)
67
TABLE V --
~F PURE HYDROCARBONS
LIMITS OF' SUPERHEAT O
SUPERHEAT LIMIT
(DEGREES K)
HYDROCARBON
-
R-K
--
REDUCED
SUPERHEAT LIMIT
SRK
EXP
R-K
- --
EXP
SRK
--
n-butane
381
388
378
.897
.911
.890
n-pentane
421
430
420
.897
.915
.896
n-hexane
455
466
455
.898
.918
.899
n-heptane
485
498
486
.898
.921
.902
r-octane
511
525
513
.898
.924
.902
n-nonane
534-
551
538
.898
.926
.906
2,-3-dimethylpropane
389
395
386
.897
.911
.891
2,2,4-trimethylpentane
488
500
488
.898
.919
.898
cyclohexane
496
505
493
.897
.912
.891
1-pentene
417
426
417
.897
.917
.898
2
11
.004
.020
AVERAGE ABSOLUTE ERROR
Notes:
R-K=Calculated using the Redlich-Kwong equation-of-state
SRK=Calculated using the Soave modification to the R-K
EXP=Experimental values[5]
68
Eqs. (152), (153) and (154) are usable with the Soave
then
modification.
Figure 3 and Table 5 also have calculations using
Soave-Redlich-Kwong
In
equation.
3,
Figure
the
the
Soave
equation predicts that the limit of superheat is a function
of critical pressure
span a much larger range
values
of
The
experimental
to the
much closer
Table 5 are
superheat in
predicted
(TR from .89 to .94) than
do those of the original Redlich-Kwong.
limits
The
and acentric factor.
original Redlich-Kwong prediction of T,=. 9 .
Therefore, the
limit
recommended for superheat
Soave modification is not
calculations.
EINAEY SYSTEMS
original Redlich-Kwong
The
equation-of-state and the
Soave modification are used for multicomponent systems,
mi-aking a' and b functions of composition.
of the
mixture
constants,
are
constants,
and
the
square root
fraction average
of
terms
"mixing rules."
termed
mixing rules, still
in
widely used, the
of a' are each
of the
by
The definitions
pure
In
component
the original
mixture values
computed as
pure component values.
of b
a mole
Writing
these rules for a binary system,
a =xa.
where
af, a", b, and bb
+bab
(158)
b=x b-+xbbb
(159)
are the pure component values and
69
x, and xb
(159)
are mole
in terms
Rewriting Eqs. (158)
fractions.
of mole
numbers, using
and
Eq. (136) to note
that the mixing rules for "a" are the same as those for a',
a=1 (Na' 5 +Nba •5)2
(160)
(161)
b=1 (Nb,+N b, )
Eq. (137), in either
the original
or the
Soave form,
is
applicable to binary mixtures using Eqs. (160) and (161).
The
most
convenient
equations
which
define
the
spinoidal curve and critical points of a binary mixture are
in terms
expressed
Hielmholtz Free Energ:y.
of A, the
make Eq. (137) compatable with A, it is rewritten in
To
terms
of the total volume, V
-Av=P=NRT
where
-@G+ @d
C=Na=(Naa
@where
U-
+Ia
(162)
)2
(163)
Nbb+Nbb6
@ =Nb=NO+Nbb6
Cd
and
4
are
functions of
(164)
composition,
and
@d
is a
function of temperature.
Eq. (127) defines
mixture.
The terms Avv,,
the Redlich-Kwong
(E-6)
the
spinoidal curve
A,
equation)
(from Appendix E).
a binary
and AL are evaluated (for
using Eqs. (E-1),
Solving
simultaneously yields the limit
material.
of
Eqs. (127)
(E-2)
and
and (155)
of superheat for a binary
The required data are the criticel temperature,
70
pressure
and
each component
of
mole fraction
(plus the
Figures 4 and 5
&centric factor, for the Soave equation).
gLive the calculated and experimental[5] limits of superheat
for
and cyclohexane in
n-octane in n-pentane
the systems
r-pentane, as a function of composition.
of
limits
The calculated
for
superheat,
both
the
original Redlich-Kwong equation and the Soave modification,
are
almost exactly nole fraction averages of the superheat
well
not agree
This does
pure compounds.
limits of' the
the data, as the experimental values in Figures 4 and
xwith
The original
5 deviate somewhat from a straight-line plot.
much better than the
Redlich-hwong matches the data
is to be expected, since the original
This
modification.
Soave
equation is also better with pure materials.
The reason
disappointing
that
results may
using
calculated
derivative
Eq. (D-9)
number.
region,
the
produces
A,
A_.
Appendix
(from
with
respect
cannot exist
in the
to
is
which
D),
the unstable region of
pressure
Since substances
the
a mole
unstable
any equation-of-state in that region must be based
on extrapolations.
pressure
of
the term
lie in
involves an integration across
second
treatment
the theoretical
Taking
the second
derivative of
the
with respect to a mole number is a severe test of
mixing
rules
(Eqs. (158)
and
(159)).
thernodynamic calculations take at most the first
Most
FIGURE IV -
LIMIT OF SUPERHEAT OF N-OCTANE IN N-PFNTANE
SUPERHEAT
LIMIT (OK)
560
520
480
440
400
MOLE FRACTION N-OCTANE
- - - CALCULATED USING THE REDLICH-KWONG EQUATION
CALCULATED USING THE SOAVE MODIFICATION
A EXPERIMENTAL DATA[5]
72
FIGURE V -
LIMIT OF SUPERHEAT OF CYCLOHEXANE IN N-PENTANE
SUPERHEAT
LIMIT (oK)
560
520
480
440
400
MOLE FRACTION CYCLOHEXANE
- -
- CALCULATED USING THE REDLICH-KWONG EQUATION
CALCULATED USING THE SOAVE MODIFICATION
Z
EXPERIMENTAL DATA[5]
73
derivative.
Much more refined mixing rules are required to
produce accurate predictions of the limit of superheat.
CRITICAL POINTS
the second condition of the critical point
Eq. (128),
is also evaluated using Eqs. (E-1)
for a binary material,
through (E-7) (from Appendix E).
involves
of P with
a third derivative
integration does
respect to VN,
across the unstable
not extend
the critical point
Therefore
Even though the term A 0aa
the
region.
calculations may be expected
to produce more accurate results than those of the limit of
superheat.
The critical point
are
conditions, Eqs. (127) and
solved simultaneously for the critical temperature and
volume, and
critical
pressure.
n-butane
composition.
pressures.
is used
the equation-of-state
in
to obtain
the
Figures 6 and 7 give the calculated
critical temperatures for the
and
(128),
carbon
systems n-heptane in
dioxide,
as
ethane
a function
of
Figures 8 and 9 do the same for the critical
The results of both the original Redlich-Kwong
equation and the Soave
with experimental
modification are plotted, together
data collected
by Spear,
Robinson
and
Chao[7].
Both equations-of-state yield values in fair agreement
with
the experimental data.
As in
the limit of superheat
calculations, the mixing rules are suspected of causing
74
FIGUFE VI -
N-HEPTANE-ETHANE CRITICAL TEMPERATURES
CRITICAL
TEMPERATURE (eK)
600
00
400
300o
MOLE FRACTION N-HEPTANE
-
- - CALCULATED USING THE REDLICH-KWONG EQUATION
-- CALCULATED USING THE SOAVE MODIFICATION
A
EXPERIMENTAL DATA[7]
75
FIGURE VII -
N-BUTANE-CARBO-7 DIOXIDE CRITICAL TEMPERATURES
CRITICAL
TEMPERATURE (OK)
q1s
qoo
395
300
MOLE FRACTION N-BUTANE
--
- CALCULATED USING THE REDLICH-KWONG EQUATION
CALCULATED USING THE SOAVE MODIFICATION
A EXPERIMENTAL DATA[7]
76
FIGURE VIII -
N-HEPTANE-ETHANE CRITICAL PRESSURES
CRITICAL
PRESSURE (ATM)
100
80
60
40
20
.6
.8
MOLE FRACTION N-HEPTANE
-
-CALCULATED
USING THE REDLICH-KWONG EQUATION
CALCULATLD USING THE SOAVE MODIFICATION
A EXPERIMERTAL DATA[7]
0
77
FIGURE IX -
N-BUTANE-CARBON DIOXIDE CRITICAL PRESSURES
CRITICAL
PRFSSURE (ATM)
100
80
60
40
20
1 .0
MOLE FRACTION N-BUTANE
- -- CALCULATED USING THE REDLICH-KWOIKG EQUATION
-----CALCULATED USING THE SOAVE MODIFICATION
A EXPERIMENTAL DATA[7]
78
most of the inaccuracy.
in
the
mixing
predictions.
The use of interaction parameters
rules
These
undoubtably
would
parameters
being determined from data already
the
use of
are
improve
usually empirical,
Therefore,
available.
interaction parameters
the
is more
a correlative
The results obtained using the original
Redlich-Kwong
than a predictive technique.
equation
and the
limits and
Their
Soave modification
critical
moderate
points
success,
are
to predict superheat
somewhat
however, leads
disappointing.
to
hope
that
improved equations-of-state or more likely, improved mixing
rules will yield significantly higher accuracy.
79
V. DISCUSSION
The study of intrinsic stability, as well as virtually
other branch
every
first major developenents from
article
"On
got its
of thermodynamics,
the
J. Willard Gibbs.
Equilibrium
of
and
shows
its
principle.
minimization
u=U(s, V, N, ...
to
equivalence
Then
the
In
his
Heterogeneous
Substances"[83], Gibbs introduces the entropy
principle
start and
maximization
the
energy
fundamental
ecuation
Nm) is developed, as well as the forms
in
terms of A, G and H.
Working
with
only
the
fundamental
equations,
uniformity of temperature, pressure and chemical
in
a
system
equations
(x,
...
at equilibrium
Dk>O
,)=(S,
which is
for
N, ...
violated
all
is
deduced.
k<n,
in
listed
the
as
potential
The stability
Nm, V) are then found.
first is
the
form
where
The equation
Eq. (56).
These
criteria are transformed into a single equation in the form
of Eq. (51), with the same ordering of x, through x,.
The
corresponding conditions
Eqs. (60)
and
intensive
properties of coexistent
point.
(63),
are
of the
developed
by
critical point,
considering
phases near a critical
A consideration of the stability of the coexistent
*phases yields the alternate criteria of Eqs. (62)
Swith i=O and the same ordering
The
the
forms
with i=2
are also
and (89),
of x, through x, as
stated,
and are
above.
listed as
80
Eqs. (90) through (93).
The above paragraphs demonstrate that Gibbs
every significant area of stability.
his work is its extreme lack
can
be
The major problem in
of readability.
Often
the
Much of later authors'
step-by-step logic is not apparent.
work
developed
considered a clarification
rather
than an
extension of Gibbs.
The separation of one
is
used
by Prigogine
phase into two separate
and Defay[ 9]
as
calculation of the stability of a system.
is much more intuitive,
as it models
of a pure component
studying
the
micrcscopic new
original.
entropy
phase
This
in
treatment
unstable.
is determined
increase
for the
the actual physical
process which occurs when a system becomes
stability
a basis
phases
the
differing only
The
by directly
formation
slightly
of a
from
the
Multicomponent systems are considered only at
constant temperature and pressure.
Prigogine and Defay derive the conditions of
mechanical
and
which conditions
diffusional
are
stability.
violated
first
The
is not
thermal,
question of
discussed,
except that the condition of diffusional stability is shown
to
be the violated before mechanical for a binary mixture.
The critical
however,
point conditions
are stated
without
proof;
a fairly extensive treatment of critical behavior
in solutions is presented.
In deriving the
S(n ,
the nth
conditions of
stability, Gibbs
Legendre Transform
of U
uses
(transformed with
respect to all of its variables), as well as the transforms
between
these
functions, the
relationships
the
discusses
he
Although
H.
A, G and
general nature
of Legendre
Transforms is not developed.
the use of
Callen presents
Legendre
Transforms
in
A, 0, H and 4/'P) , the pth transform of
thermodynamics[IO].
U, are defined and analyzed in terms of a general theory of
Legendre
Transforms.
For
instance,
the
various
first
derivatives of transforms are derived.
The sum-of-squares form of the expansion of the change
in U is obtained
directly in the
(39).
,Eqs. (38) and
In an
formulation in terms of
significance
explained,
the
of
however.
reduced
appendix,
the
use
of
of
alternative
determinants is developed.
The
is
not
criteria of
stabilitly
Callen's treatment, although Eeneral
and highly mathematical, is easy to follow.
his
notation
Legendre
Transforms,
This is due to
a
very
important
developement.
Munster covers
equations
Legendre Transforms
in somewhat more detail[C1].
the stability criteria is
several
different
similar to Callen's.
ways.
and
fundamental
The derivation of
also extensive and presented
in
treatment
is
Otherwise,
the
82
The
idea
of
a subsystem
and
diathermal, non-rigid
introduced
by
Modell
possible variations
and
which
is
enclosed
membrane
totally permiable
Reid[L2].
This
is
all
permits
the subsystem and
between
by a
the
main
system, and insures complete generality.
Second derivatives of Legendre Transforms are obtained
a Taylor Expansion.
using
the U form of the
form
stability criteria to the A form,
to the G form, and so on until one derives the
form.
are
This permits the conversion of
the A
A_,,•,
Thus the stability criteria of Eqs. (54) and
deduced, with
x, through
(55)
as in Eqs. (9),
x, defined
(10) and (11).
Other authors
critical
point
present
conditions
derivations
which
are
previous works[Lg] or are unclear[si].
suffers
from not
having defined
of
stability
or
similar to
either
The last reference
the change
which occurs
when a system becomes unstable.
This thesis considers the
any
formation of two phases
of
size, not necessarily one large and one small, from an
original homogeneous phase.
Thus this model of
intrinsic
instability corresponds more closely to the actual physical
processes.
The
third
derivatives
of
Legendre
Transforms were
found in order to show the equivalence of the various forms
of the second condition of the critical point.
This, plus
83
the
general representation of
variables used, enables the
expression of the criteria of stability and of the critical
point in terms of any Legendre Transform of U.
Redlich and Kister derive critical point criteria
a
binary
system,
usable
with
an equation-of-state[15].
in terms of volume, some
Some of their derivatives are
in
Although formulas are given to convert
terms of pressure.
suitable for any equation-of-state,
to the form most
for
this
treatment is awkward and unnecessary.
Given a pressure explicit equation-of-state, the forms
of
the stability and critical point criteria in terms of A
are the most convenient, A being a function of T, V and NZ.
are
They
in this
derived
for
an
the
condition
of
equation-of-state
to
pure materials
and
thesis
arbitrary
m-component system.
results
The
stability
and
the
poor
for
using
Redlich-Kwong
predict superheat limits
fairly
by
obtained
are good for
mixtures.
For this
and
other reasons
discussed in Section IV, the mixing rules are thought to be
causing most of the error.
The Soave modification produces
quite poor superheat limit predictions.
Critical points of binary
well
using either
are suspect.
The
mixtures are not
equation-of-state.
calculation
of
predicted
Again mixing rules
superheat
limits
and
critical points involves taking second or third derivatives
84
of the pressure with respect to a mole number.
This is an
extreme test of mixing rules, and it is not surprising that
the results are poor.
The
accurate
prediction
of
superheat
critical points may only be obtained if
limits
and
equations-of-state
and associated mixing rules are greatly improved.
In fact,
obtaining this accuracy may be regarded as an advanced test
of
an equation-of-state, probably not satisfied by any now
in common use.
85
APPENDIX A
DETERMINANT REDUCTION FORMULA
This appendix
determinant
smaller.
a
proves
formula
of one
to determinants
which
relates
order and
in Appendix B in
The relation is used
any
two orders
deriving
the sum-of-squares form for a general quadratic expression.
The
formula is presented here as a separate appendix since
For
it may have other uses.
used
to prove
which of
example, it could have
the conditions
been
of stability were
violated first, except that other methods were easier.
if i<k and j<m then
The desired formula is:
B BLjk,,=BL Bk,-B4,B j•
B is defined as an n
the
it" row
and jtý
EBSkm as B with the
(A-1)
by n determinant, EiL
with
column removed
(the minor
of U;j ),
i ' h and kth rows
and the je
and
columns removed, etc.
U1, U12
...
Uln
U
...
Uan
UA
U*
U,
B
as E
*
U,2
...
U2
U31 U3 3 Us
I
..
Ua
...
U
.
UI
Un
Unn
.
Unr
mth
86
U33 U3•
B I 9 =B120
=BII1•
, =B~1 ,I =
In the first step
Uqs
-
U3n
Uq'
...
U
Unz Us
...
Us
of the proof
Eq. (A-i) is assumed
true for a particular n with (i, j, k, m)=(1, 1, 2,
of i, j, k, and m.
is then shown true for any set
2) and
B' is
defined identical to B but with the second and mrh columns
interchanged (m>2).
U,,
B
.. UIM-,
Ulm U,
UMP
U, 11,
U
UUM ...
...
Uln
Uni UnM U
'=
.
*3
..
UnfM-I
U*1
Relabling 2 as m' and m as 2' inthe second and m
Un,
Ui,
UZ
i3
.**
U I M-
UjM' Ulr...
00..
U•
U.1-' U-r.l.-U2n
I1
*
U*'
if true
U*0
•
4
Uns
Uh;' Unt
Eq. (A-1),
0
columns
*00
U4 K-1
Uni
Unm
0+..
Un
for (i, j, k, m)=(1, 1, 2, 2), applied
to B except that
to B' (which appears identical
certain
subscripts are primed) yields
B'B'Pj I' =B', B'P, -Bf' , B ,
(A-2)
Irom determinant column exchange rules,
B'=-B
B it
F=-B
(A-3)
; B'1 =-BE
'2
21
(A-4)
87
; B,=(-l)A+Bam
B,2,=(-)m+,
(A-5)
(A-6)
E',,, ,=(-1)"' Bleam
Combining Eqs. (A-2) through (A-6) and factoring yields
B BEI,m=Bi, Bm-B•mBl
(A-7)
Equation (A-7) is Eq. (A-i) with (i,j,k)=(1, 1, 2),
but with m arbitrary.
The same proof applies if i, j, or k
instead of m were changed to a different value.
if Eq. (A-i)
were
(i,
true for
j, k, m)=(1,
Therefore,
1, 2, 2)
it
must be true for any (i, j, k, m).
The
Eq. (A-i)
second
step
to be true
of
the
for n=2.
proof
involves
Since EB,1,
showing
as a 0 by 0
determinant, is defined equal to 1,
B=U,, U2-U, U., =U,~Ul, -U.,
Uja
(A-8)
transforms to
B B11 ,
=B 1 ,B
1 B-B1 ,B,,
(A-9)
Thus Eq. (A-1) holds for n=2.
In the third
true
step of the
proof Eq. (A-i) is assumed
for an n-1 by n-1 determinant, B11 , and is then shown
true for B, the n by n determinant.
By the above assumption, if k>2 and i>2 then
B,,
hB12 k =E, zBkt-B,
Lach term is multiplied by
and rearranged, giving
(-1)kUk,,
1
B ,ka
(A-10)
summed from k=3 to n
88
kz3
(-1)
B
k -3
k=3
Expanding
(A-11)
U k Baitk I -EIa2 Z(-1 )k Uk1jl
B',,
E,i
and
B3j42a
by
minors along
the first
column
n
E2 =U;, B
2
1I; =U2 Bl(21
+
-
· C3
lk.=
Combining eqs. (A-11)
(A-12)
)kU i l3ik,
(A-13)
+ ka3(-1
-1)
Elza =:
)k Uki B12k/
(A-14)
Ukj BE ie22
through (A-14)
and simplifying
-E, Bif,IC22
-22=--E--B
Ila
/
,a,LB,,
by (-1)'u,,
Each term is then multiplied
(A-15)
summed from
i=3
to n, and rearranged, giving
-L
~i
(-1)
U,; B22c, =
L= 3
rl
S1122
-~
(-1) U1I i -b
'r7
Ia ~(-l
(A-16)
)L ~
0= 3
Expanding B, B2, and B9, by minors along the first row
B=UE 1 ,, -U,
SB,_- (-1 )ýUI,ý F
Cc'?
E 1 =UB
I 1
UIBL
iL23
,I
(A-17)
(A-18)
89
B,~=U
13
+(A-19)
Combining Eqs. (A-16) through (A-19) and simplifying
B B,,,=B, Bý2-B,19B
(A-20)
I
By the first part of this appendix, Eq. (A-20).implies
Thus, if Eq. (A-i) is true for determinants of
Eq. (A-i).
order n-1, it is true for determinants of
ixq. (A-i)
is
true for
determinants of
order n.
Since
order 2, we have
showr by induction that Eq. (A-i) is true for all n_2.
90
APPEIiDIX B
SUM-OF-SQUARES FORM
This appendix
general
derives the
sum-of-squares form
when the
differential
in
form is
The
quadratic expression.
energy goes
for a
used to show
negative
and
the
system in question therefore becomes unstable.
Assuming
that Uký=
and that
Uj k
all denominators are
non-zero, the desired result is
DkZ
Uks x k=j
j
k-•
where
Zk
k:
C "_xj
j=k
Dk is
matrix
the k
by k
in the k'h
(B-2)
<
principle sub-determinant
of' the coefficients of
with all U k, in the k t h
the quadratic.
U,
U,2
U2 1
U
Uki
U1
...
U11
Ckkj
of
Ckij
row replaced with U., and all
column replaced with U]J.
Dk=
(B-1)
k-[
UýI
U , U
...
Ulk
...
Ukkk
U 1 k-I Ulj
as0
UZ
-. Ukk~
Uký
k-i
2
the
is
Dk
Uk
U2 '
Ckj =
U,,
•..
U22
..0. Uak-
Uk-t_•
U a
*** Uk-, k-,
** UL k-i
U.
.
J
k-,
ULj
derivation comes from working with the expression
UL
The
UIk-.j
·...
·
;
-
E. defined
Since
(B-3)
Since C,,,=3Mz,
E,= D (xXm+•Xl
+2xr
-
C
I)mM1)+
my+I
X IX,
xn
Lxy
R
(B-4)
*/h~
J1M+
Adding and subtracting terms to "complete-the-square"
n
CMIxj+(
.D,-•
E = D. [x'+2xZ
SSimplifyikx
CM Ixje) ]
j-M
I = M+1
ts\+
S'
I
DM-
C kI xkxj
Xj +
Tim -I
(B-5)
Simplifying.
C 'x'
EM- DM (x,,+
T,,
3fl\tI
/1:
C ,kCmI-C
Mk CcAj
D•D•_
1
(B-6)
If B (of Appendix A) is defined as C -, jk v then the
determinant reduction formula shows that
C,,.jk Dr-, =C•kcC
rAmrA-C
kmCi,~j
(B-7)
92
Substituting Eq. (B1-7)
CM,, kt
3t;--yv
+
E,= Dm (• C
D/
Dm
J=ni
Substituting Eqs.
(B-6)
into Eq.
(B-2)
2
kmn
and (B-3)
into Eq.
(B-8)
(B-9)
E,= Dm Z +E
•-Ie
Applying Eq.
(B-9)
(B-8)
to E, n-1 times
(B-10)
n-I
El=yT
D Z Eh
From Eq. (B-3)
E
Using Eq.
(B-11)
kx
(B-2)
(B-12)
Eh= Dn Z
D,-r
Substituting Eq.
(B-12)
into Eq. (B-10)
EEl
D
Dk_
Z
(B-13)
Since UkS=CI kj and Do=1, Eq. (B-3) shows that
(B-14)
k=, S--,
Combining
Eqs.
(B-13)
Eq. (B-I), the desired result.
and
(B-14),
one
obtains
93
APPENDIX C
LEGENDRE TRANSFORMS
In
derivatives
of
expressed in
terms of
enable
the
T,
the
Legendre
derivatives
interconversion
rapid
transforms (A, G, etc.).
arising
second and third partial
this appendix the first,
of y. These
of U
any
and
In particular, the
stability and
from the
of
Transform
y,
are
formulas
of its
determinants
critical point conditions
are easily simplified.
Throughout this derivation, all terms that will appear
only in
higher order
fourth or
Subscripts
on
y
and
derivatives are
W indicate
corresponding
dropped.
partial
derivatives.
y is the given
y=y(x 1, x 2, x 3 ...
',
x,)
the Legendre Transform of y, is defined
,- Y-=y
(C-1)
='=Yf((,, x,, x ...x)=y-I,x,
(C-2)
Defining S9, and eSx
K,=9,-
(c-3)
1,
gx Z=xý -xL°
(C-4)
Using Eqs. (C-3) and (C-4) in Eq. (C-2)
xp=y-
ST
•oux,-xysu,doxy-
• x,
Expanding y around yO using a Taylor Series
(C-5)
94
b
n
n
jki
nCZ-I
yIoLxLL
.xcx
+L
1
SXx Sxix I
Yj
(C-6)
DO, D1 and D2 are defined independent of x,
V%
Sx. 9j; xk (C-7)
DO=E
Dl=
(C-8)
y
L=• j=2
i z
n
D2=y
%
(0-9)
nx
2=2
Combining Eqs. (0-6) through (C-9)
y=y +DO+(y'+D1)Sx,+7-(y
-1 +D2)Sx>+-y
,
1
• "il
1
From Eq. (c-4)
Using Eq. (C-11)
X,1 3
d(x, )=dx,
(C-10)
(C-11)
to differentiate Eq. (C-10) with
respect
+D2)gx , ,+.y
ay I
(C-12)
to x,
Y, =Y y=y°+D1+(yo
Dl+(
Substituting Eqs. (C-1)
and (C-3) into Eq. (C-12)
c,=D1l+(yo,, +D2)gx,+,-.y,,
Solving
x
Sx
Ex,"
(C-13)
for 9x, and choosing the positive root of the
quadratic
x,= {-y
-D2+ [ (y,
(D-
+D2) -2y,
y0-
,)
(C-14)
Using the binomial expansion
U
x,=-(y, +D2)
(D1-
• ,)y,
Using the geometric series
(y
+D2)
(D1-
(C-15)
95
y,0 x,=-(1-y,
,
D2)(D1-
,)-'y,,, yo-(Dl -
,)
(C-16)
(C-5) and Eq. (C-16)
Substituting Eq. (C-10) into Eq.
into both yields
T=y-O
Ex , - x Or
DO ,
-2(D2-y,
,,)2
)(D1-
(-9)
(-), -8)an(DEqs. (C-2),
Substituting
and
(C-8)
(C-7),
(C-17)
(G-9)
into
Eq. (C-17)
O~
-x0s, i
+
y
7 oox,Y g• •-•Y 0 •
y
(y
'di,
+t
z
JE(Y
kY,
2 Lk
Rearranging
Y"gx
,
2+Y10
-I
IL
yoo
I yiI/
Eq. (C-18) so
IYj"
+
•.Yi
-1,2
YC Y/]
1,, . ) x, x;
YIit
o
Y
,
the terms which
L
I/k
ýxj sxk
(C-18)
are summed are
symmetrical with respect to the summation indices
96
ySx
Y=V -xS
yo-I
-
0o
o-
,
(-2
LZ
ya.
-
-y
x
'7ý y.YO )ýX~dxx:
Y0r
-IiYr0 y3
+
y
0-I
o
o
0-3
1
Y1 O ) 5•,x.
,: -YYl
3=
VL
(Y 01
y,.j
-Y
2
[YO Y
y,.
+,-3
+Y "0Y,
y i, I
+y 0
0-C,
-Y1
I=2 jAt kh
-Y,,
[Y,•
Yrgk
+Y
J
y,
Y,
) °, x
xx
Y jj)Esx
o
jo
YO
7got7jo7
-y,,
1
0
I
Tl
7A Fi' 70
(C-19)
y,,
yx0x3
,xky,; y•. y-3 )
Expanding T around ~to using a Taylor Series
Iz
ITIISel
o
0
o
L
n0
,6 111
Sxi x +
Comp
•.
ja
SE7Lt=
3+1E 10
se Ix
x
jk Sx , 1 x
i
(1X
LgX
(C-20)
Eqs.
Comparing Eqs.
(C-19) and
(C-20) term-by-term yields
(for i>2, j22 and k>2)
(C-21)
(C-22)
If
=Y, LL Y,
Y =Y ! -Y, LY,
(C-23)
(C-24)
(C-25)
97
(C-26)
,"L7,1-y
(C-27)
Y,7-YI! Y1,, Yj I
YTjKk =Yz; Y1, -(Y,1
+Y'I Y,I )Y ; +Y,' Y, Y11,
Y,,I
Yj I
(C-28)
-3
(C-29)
TY~jk =Yljk -(YiI yiSk +Yifr
Y,zk+Yj, Yj ýj)Y-1
+(YY
J
+
2
+Y,j Y, Y,, )y,7
Iy
+
Eqs.
(C-21)
through
(C-29)
are
If
Y,
kY
the
desired first,
second and third derivatives of the Legendre Transform.
For example, let
y(x,, x2, x3 )=U(S, V, N)
Y(
I,,x2,
x3)=A(T, V, N)=U-TS
dy=dU=TdS-PdV+ dN
dt'=daA=-SdT-PdV+AdN
Derivatives are defined as before:
A,=b~)TN
all derivatives
In this example
are taken
holding N
constant.
4/1
=AT=-S
=-x I =-S
=1I =
S,=ATV=-T•T
i -- TTT--V
;YI- =USVU=S
T
(Ž)T -
I2T'
=/)2T
I'"Y"=U555 S _-PT iv
-3-3=
3
_j
98
,
- -TVV)
-I
T
2-
-I
2
-,2
-I
-3
a
=Usv Uss-2USvUs5vUs +UsvUsssUss
S NT
2T (-bT
-2 ýbT S T (T +T
sVS
_
S
"9Ž V iS
V
So te:
In
this appendix only
the first Legendre Transform
Eq. (32)
of y from x, space to 6, space was considered.
defines
(P), the
xp
through
• (-i)
T
(,
since
...
...
through
may be
x,)
p-,,
y from x,
,p
space.
defined recursively
Transform
Legendre
the
as
8,
to
space
4p, xp.,
v P)(8, ...
p-h Legendre Transform of
of
xn) from x, space to Cp space,
xp ...
(o) is defined as y.
(t
Eqs. (C-21) through
third
derivatives
of
(C-29) give
first, second
and
'V in terms of derivatives of y.
They may also be used to give the derivatives of 4(P)
in
terms of derivatives of W(P-' ) , if y is changed to
F (P ' 2V
,
is changed to
P) and all subscript 1's are
changed
to subscript p's.
In
other words, Eqs. (C-21)
through (C-29) are usable with any Legendre Transfcrm, as
long as the subscript "1" means a derivative with respect
to the transformed variable.
99
APPENDIX D
CHEMICAL POTENTIAL DERIVATIVES
In
this
appendix a formula
derivative of
mole
the Helmholtz
numbers,
for
Free Energy
is derived.
Aab is the
A,b,
a second
with respect
to
derivative of a
chemical
potential
constant
temperature, total volume and other mole numbers.
with
respect
to
a mole
number
The formula requires pressure to be given as a function
temperature,
therefore
total
volume
usable
with
equation-of-state.
and
numbers,
mole
any
pressure
and
at
of
is
explicit
These derivatives are used in finding
the limits of intrinsic
stability, given such an
equation
of state.
The derivation begins with the Maxwell Relation
T,N
V
Taking
the
=-IP
(D-1)
7 T,V,NI
derivative of
each side
with respect
to Nb,
holding T, V and other NL constant
I)!2p1
____D
(D-2)
Multiplying each side by dV and integrating from V=co
(with superscript
*'s
indicating
that
the
to V
variable is
evaluated in the limit of infinite total volume)
T,,N
Tv.Y2N;ýb-Pb
T9V *INI-*
b=-f,,
dV (D-3),NTV,
T VI I'l L,a b
100
All materials (at a given temperature) behave as ideal
gases if the specific volume is sufficiently
is,
T are held constant,
if L' and
large.
then a substance will
approach ideal gas behavior in the limit of infinite
volume.
Since the fugacity of an
gas
definition of fugacity
(using the
total
ideal gas is equal to
of an
the chemical potential
the partial pressure,
That
ideal
with N,(T) being a
function of temperature only) is
,~,=RTln(PNa)+
Taking the
derivative
(T)=RTln(NaRT)+
side
of each
,(T)
(D-4)
with respect
to
Nj
(holding T, V and all other Ný constant) and evaluating at
the limit of infinite total volume
=RT
Repeating the
last step
but taking
(D-5)
the derivatives
with
respect to Nb instead of N, (valid only for afb)
S=0
(D-6)
T,V ,Ngb
Substituting Eq.
(D-5) into Eq.
rV
(D-3)
- AP
.ikz
(D-7)
dV+E T
Substituting Eq. (D-6) into Eq. (D-3) (valid for a-b)
V
=-)P
T,V_,Nl'
To
simplify
the
b
6C0 -W -)V
notation of
dV
(D-8)
Eqs. (D-7)
and (D-8),
T,_VNý-ý
subscripts on A are again defined as partial derivatives
101
AV=(b)
;
\TV_ IT,N
ATN=
\
}
/bT,V,N ¢
Eq. (D-7) simplifies to
V
a=jAv_ dV+RT
A=
(D-9)
Eq. (D-8) becomes (for a4b)
A•=
AfAbdV
(D-10)
Eqs. (D-7) and (D-8) (or (D-9) and (D-10)) may le used
to find the second derivative of the Helmholtz Free
with
Energy
respect to mole numbers, given pressure as a function
of temperature, total volume and mole numbers.
102
APPENDIX E
REDLICH-KWONG DERIVATIVES
equation-of-state
Redlich-Kwong
stability and
point
critical
used
in
the
evaluating
the
a
of
conditions
derivatives in
The required
mixture.
of
derivatives
the
derives
appendix
This
binary
terms of A, the
Helmholtz Free Energy, are Avv, Av,,A , Avv
Avv,
Av
and A,,,.
Eq. (162)
derivatives
gives A v in terms of V, T, N_ and Nb.
The
Avaa are
thus
Avv,
Avv
Avv,
Av,
and
evaluated directly:
(F-1)
,= NRT -Qd+ G
A- RT -@
,.1Tj
1 +0@4
NRT
AV,,7+fb@4
_
J-e•
V1-(@j
~X )v+(
Avvv =- 2NRT +2Q@+
Avva=
RT
+2 @41
NRT+2
(E-2)
(E-3)
-2@t_
NRT - (i)1
+
Av,,a-[ W"
@4
+1W)I
RT]
Q_/_
1
-2
2
-2
(E-4)
NRT +2@d) 1
\) @d
(E-5)
103
Eq. (D-9 ) (from Appendix D) gives a formula for A,_
of
terms
A v.
A,,,
and
A,
calculated
are
in
using
Eqs. (D-9) and (L-5)
i
I2 ln( V )+[ --
Ao=
A
9
')+ In((
)+[
NR•
f2T+
+3[
2
]
)@
@)+2
(
-a[(@
+
1 RT] 1 +NRT
NRT+2
+W@ 2
(. \NRT+3 (2@ IRT] 1
@I+
ET
Eqs. (165)
d_
expressed in terms
(164).
and
I]
+2
1
-2
d( and Cf are
@d +RT (E-6)
NRT
•
-RT
(E-7)
of mole numbers
in
and. (164)
to
Using Eqs. (163)
evaluate the derivatives of @d and @r
t@) =2a. N a• ,+N a -b (N.a +Na)')
4
2ab,
'
ýN•IT)
Na
1a~.b
NY b,~+Nb
@= -6q®
+12a•
a~ (N•b)
•
+Na.+2b2 (N,,a +Na' )
bb+Nbbb)
ko(N
(Nb,+Nkb
,a
+N'a - 6 a(N
(7bE bbr)
j h f=b
]ý
af +N
(E-8)
(E-9)
(E-0)
(,b,+N•bý
(E•l)
104
(E-12)
throu
Eqs. (E-1)
through
(E-7),
(E-13)
which
through (E-13), are the desired formulas.
use
Eqs. (E-8)
105
APPENDIX F
NOMENCLATURE
LETTERS
A
--Helmholtz Free Energy=U-TS
a
--a'/T "
a'
--constant in the Redlich-Kwong equation-of-state
a"
--a /Tc
B
--n by n determinant
EBiJ
--B with the i"4 row and jPA column removed
Pjkm -- Bj with the k h row and m -4 column removed
with the second and m"1 columns interchanged
E'
-B
b
--constant in the Redlich-Kwong equation-of-state
CkLj
--Dk with all Ukm
in the kt
row
changed to U ,
all U,~ in the ktA column changed to UMj
Cp
--heat capacity at constant pressure=TIS)
CV
--heat capacity at constant volume=T ()
Dk
-k
\ýTT P
by k determinant with UL;
h ,
--
G
-- Gibbs Free Energy=_U-TS+PV
-total
--
LL
C kt
the ith by j th term
xkx 3
enthalpy=U_+PV
-- the determinant
-0
,
"-
+.
and
106
Mt
-LL
with the jth term in
the last row replaced with
the derivative of Lt with respect to xej
m
--the. number of components in the system
S.1 --the total number of moles
n
--the number of independent variables=m+2
P
--pressure
Pc
--critical pressure
PR
--reduced pressure=P/Pc
I.E -Gas
constant
--total entropy
T
-temperature
Tc
--critical temperature
TR
--reduced temperature=T/Tc
U
--total internal energy
V
--total volume
V
--specific volume
Vc
--critical volume
V
-VPc
-
xa
--(letter subscript) mole fraction of component a
xz
--(number subscript) ith fundamental variable
S,
/RTc,
V or Nj)
--function of x, through x,=U
y
_
Ckk 1X
-Z
jz
(either
107
OTHER SYMBOLS
-- chemical potential
.
X(T) --purely temperature dependent part of the fugacity
W
-acentric
factor
--Legendre Transform of y from x, to £6 space=y-g,x,
(F) --pth Legendre Transform of y from x, through x, to CI
7
P
through 2p space=y-"C XL
L=i
--conjugate variable of xL= LyL
8
~L0o -- [ ]Lo
[1]
--the derivative operator [
CE
-- .427480
-b
--. 086640
,_
-- @a./( C ),•
-y~ •
•]
J, -- Nal/b
f
--Nb
OTHER SUBSCRIPTS
Subscripts
on U, A, G and H, and numerical subscripts
on y, Ifand T(P) indicate partial derivatives with
to
the
indicate
corresponding
variable.
Otherwise,
respect
subscripts
that the value is of the corresponding component.
of U with respect to S
US
-derivative
i,
--number of moles of component a
108
SUPEESCRIPTS
w
-evaluated
o-value
at the limit of infinite total volume
around which an expansion is being made
109
APPENDIX G
LITERATURE CITATIONS
[1] Gibbs, J. W.,
volume
of J. W. Gibbs,"
"The Collected Works
1, p. 56,
Press,
Yale University
New Haven
(1957).
[2] Ibid., pp. 110-112.
[3] Ibid., pp. 132-1353.
F.,F. L. Robinson
[4] Spear, R.
"Critical
Jr., and
Chao,
K. C.
States of Mixtures and Equations of State,"
Ind. Eng. Chem. Fund., 8, 2 (1969).
[5] Eberhart,
J. G.,
Eubble
"Homogeneous
Hydrocarbons
Kremsner,
Nucleation
and their Mixtures,"
"An International
its
W.
and
M. Blander,
in
Superheated
paper presented at
Workshop on Nucleation Theory
Applications," April 10-12,
and
1972, Clark College,
Atlanta, Georgia.
[6] Soave, G.,
Equilibrium
"1
Constants
of
Equation
Redlich-Kwong
from
a Modified
State,"
Chem. EnB. Sci., 27, 1197 (1972).
[7] Spear, R. R., R. L. Robinson Jr., and K. C. Chao, loc.
cit.
[8] Gibbs, J. W. loc.
[9] Prigogine,
cit.
I.,
and
pp. 45-135.
R.
Defay,
"Chemical
Thermodynamics," pp. 205-261, Longmans, Green and
Inc., New York (1954).
Co.
110
[10] Callen,
H1.B., "Thermodynamics," pp. 85-100, 131-145,
361-366, John Wiley & Sons, Inc., New York (1960).
[11] Munster, A.,
"Classical Thermodynamics,"
pp. 61-94,
159-192, Wiley Interscience, New York (1970).
[12] Modell,
M., and
R. C. Reid,
"Thermodynamics and its
Applications in Chemical Engineering," chapter 7,
(to
be published).
[13] Rowlinson,
J. S.,
"Liquids and
Liquid
Mixtures,"
Academic Press, New York (1959).
[14] Boberg,
T.
C.,
and
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