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 criterionwhich is
dP/ dx = 0 or dP = 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 (dP/ dx ≠ 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 constantT and p. Dimensionally, it is simply energy per mole.
If we substitute μiall