Unified Separation Science

J Calvin Giddings
Chapter 2
Equilibrium the driving f
orce for separative displacement
All isolated systems move, rapidly or slowly, by one path or another, towards equilibrium. In fact
essentially all motion stems from the u
niversal drift of eventual equilibrium. Therefore, if we wish to
obtain a certain displacement of a component through some medium, we must generally establish
equilibrium conditions that favor the desired displacement. Clearly,
knowledge
of that equilibriu
m
state is indispensable to the study of the displacements leading to
separation.
In many separation processes (chromatography, countercurrent distribution, field

flow fractionation,
extraction, etc.), the transport of components, in one dimension at least
, occurs almost to the point of
reaching equilibrium. The equilibrium concentrations often constitute a good approximation to the
actual distribution of components bound within such systems. Equilibrium concepts are especially
crucial in these cases in pre
dicting separation behavior and efficacy.
2.1 MECHANICAL VERSUS MOLECUALR EQUILIBRIUM
We can identify two important classes of equilibria:
(a)
Mechanical
–
defines the resting place of macroscopic bodies.
(b)
Molecular
–
defines the spatial distribution of molecu
les and colloids at equilibrium.
Of the two, (a) is more simple. With macroscopic bodies, it is unnecessary to worry about thermal
(Brownian) motion,
which greatly complicates equilibrium in molecular systems. This is equivalent
to stating that entropy is
unimportant. This is not to say that entropy terms are diminished for large
bodies, but only that energy changes for displacements in macroscopic systems are enormous
compared to those for molecules, and the swollen energy terms completely dominate the sma
ll
entropy terms, which do not inherently depend on particle size.
Without entropy consideration, equilibrium along any given coordinate
x
is found very simply as
that location where the body assumes a minimum potential energy
P
; the body
will eventually c
ome
to rest at that exact point
. Thus, the mechanical equilibrium is subject to the simple criterion
which is
d
P/
d
x
= 0 or d
P
= 0
(2.1
)
equivalent to saying that there are no unbalanced forces on the body.
Systems out of equilibrium
–
generally in the process of moving toward equilibrium
–
are
characterized by (d
P/
d
x
≠ 0).
A rock tumbling down a mountainside and a positive test charge
moving toward the region of lowest electrical potential are both manifestations of the tendency
toward a simple mechanical equilibrium.
Molecular equilibrium, by contrast, is complicated
by entropy. Entropy, being a measure of
randomness, reflects the tendency of molecules to scatter, to diffuse, to assume different energy
states, to occupy different phases and positions. It becomes impossible to follow individual
molecules through all the
se conditions, so we resort to describing statistical distribution of
molecules
, which for our purposes simply become concentration profiles.
The molecular statistics
are described in detail by the science of statistical mechanics. However, if we need only
to describe
the concentration profiles at equilibrium, we can invoke the science of thermodynamics.
We discuss below some of the arguments of thermodynamics that bear on common separation
systems. We are particularly interested in the thermodynamics of
equilibrium between phases and
equilibrium in external fields, for these two forms of equilibrium underlie the primary driving forces
in most separations systems. A basic working knowledge of thermodynamics
is assumed. Many
excellent books and generally mo
nographs on this subject are available for review purposes (1

4). In
the treatment below, we seek the simplest and most direct route to the relevant thermodynamics of
separation systems, leaving rigor and completeness to the monographs on thermodynamics.
2.2
MOLECULAR EQUILIBRIUM IN CLOSED SYSTEMS
A
closed system
is one with boundaries across which no matter may pass, either in or out, but one in
which other changes may occur, including expansion
, contraction
,
internal diffusion, chemical
reaction, heating
, and cooling. First law of thermodynamics gives the following
expression for the
internal energy increment
dE
for a closed
system undergoing such a change
dE
=
q + w
(2.2)
where
q
is the increment of added heat (if any) and
w
is the increment of work done on the system. If
we assume for the moment that only pressure

volume work is involved, then
w
=

p dV
, the negative
sig
n arising because positive work is done on the system only when there is contraction, that is,
when
dV
is negative. For
q
we write the second law statement for entropy
S
as the inequality:
dS ≥ q/T,
or
T dS ≥
q
. With
w
and
q
written in the above forms, E
q. 2.2 becomes
dE ≤ T dS
–
p dV
(2.3)
an equation which contains the restraints of both the first and the second law of thermodyna
mics.
We hold this equation briefly for reference.
By definition, the
Gibbs free energy
relates to enthalpy
H
and entropy
S
by
G = H
–
TS = pV
–
TS
(2.4)
from which direct differentiation yields
dG = dE + p dV + V dp
–
T dS
–
S dT
(2.5)
The substitution of Eq. 2.3 for the
dE
in Eq. 2.5 yields
dG ≤

S dT + V dp
(2.6)
Therefore, all natural processes occurring at constant
T
and
p
must have
dG ≤
0
(2.7)
while for any change at equilibrium
dG
= 0
(2.8)
In other words, the equilibriu
m at constant
T
and
p
is characterized by minimum in
G
. This is
analogous to mechanical equilibrium, Eq. 2.
1, except that
G
is
the master parameter governing
equilibrium ins
tead of
P
.
For example, if a small volume of ice is melted in a closed container at 0
0
C and 1 atm pressure, we
find by thermodynamic calculations that
dG =
0, representing ice

water equilibrium, which is
reversible. At 10
0
C, we have
dG
< 0, representing t
he spontaneous, irreversible melting of ice above
0
0
C, its melting (equilibrium) point. Spontaneous processes such as diffusion, of course, are likewise
accompanied by
dG <
0
.
2.3 EQULIBRIUM IN OPEN SYSTEMS
An
open system
is one which can undergo all the c
hanges allowed for a closed system and in
addition it can lose and gain matter across its boundaries. An open system might be one phase in an
extraction system, or it might be a small

volume element in an electrophoretic channel, such
systems, which allow
for the transport of matter both in and out, are key elements in the description
of separation process.
In open systems, we must modify the expression describing
dG
at equilibrium in closed systems,
namely
dG =

S dT + V dp
(2.9)
to account for small amounts of free energy
G
taken in and out of the system by the matter crossing
its boundaries. For example, if
dn
i
moles of component
i
enter the system, and there are no changes
in
T
and
p
and no other components
j
crossing in or out,
G
will change by a small increment
proportional to
dn
i
dG =
(∂
G/
∂
n
i
)
T
,
p, n j
dn
i
(2.10)
The magnitude of the increment depends, as the above equation shows, on the rate of change of
G
with respect to
n
i
, providing the other factors are held constant. This magnitude is of such
importance in equil
ibrium studies that the rate of change, or partial derivative, is given a
special
symbol
μ
i
=
(∂
G/
∂
n
i
)
T
,
p, nj
(2.11)
Quantity
μ
i
is called
chemical potential
. It is, essentially, the amount of
“
G
”
brought into a system
per mole of added constituent
i
at
constant
T
and
p.
Dimension
ally, it is simply energy per mole.
If we substitute
μ
i
all
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