Unified Separation Science- JC alvin Giddings by STB7n70v

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									                                 Unified Separation Science

                                                                                   - J Calvin Giddings

                                              Chapter 2

                    Equilibrium the driving force 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 universal 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 equilibrium

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 predicting 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 molecules 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 small

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 come

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 these 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 monographs 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

sign 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, Eq. 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 thermodynamics.

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 equilibrium 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 instead of P.

For example, if a small volume of ice is melted in a closed container at 00C and 1 atm pressure, we

find by thermodynamic calculations that dG = 0, representing ice-water equilibrium, which is

reversible. At 100C, we have dG < 0, representing the spontaneous, irreversible melting of ice above

00C, 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 changes 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 dni 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 dni

                                      dG = (∂ G/ ∂ ni )T, p, n j dni                            (2.10)
The magnitude of the increment depends, as the above equation shows, on the rate of change of G

with respect to ni , providing the other factors are held constant. This magnitude is of such

importance in equilibrium studies that the rate of change, or partial derivative, is given a special

symbol

                                         μi = (∂ G/ ∂ ni )T, p, nj                                 (2.11)

 Quantity μi 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. Dimensionally, it is simply energy per mole.

If we substitute μiall

								
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