Page 54 - Pressure Swing Adsorption

1,, , ·;1 i

28 PRESSURE SWING ADSORPTION FUNDAMENTALS OF ADSORPTION 29

Equation 2.6 contams three constants (b, qs, and n), but it should be stressed vapor-phase system, the different1at1on would ha:ve to be performed ar
that this form is purely empincal and has no sound theoretical basts. constant total pressure, mamtainmg the same number of moles of s and
changing the number of moles of a by adding or removrng an men compo-
nent to mamtam the total moles (and totai oressuve) constant. This would
2.2.6 BET Isotherm
v1eld a measure of the partial molar interaction energy between components
Both the Langmmr and Freundlich isotherms are of type I form (in Brunauer's a and s, which would be verv small. For an adsorbed phase <J.) can be
dassificatton). This 1s the most commonly observed form of isotherm, particu- regarded as the change m mternal energy, per umt of adsorbent, due to the
larly for nucroporous adsorbents. However, materials such as activated alu- spreading of sorbate over the surface. This change m energy may be regarded
mina and silica gel commonly show type 11 behavior. This form 1s commonly as a work term-the product of a force and a displacement. Thus, depending
11 on whether one chooses to regard the adsorbed ohase as a two-dimensional
represented by the BET eouation :
fluid (area A per mole) or a three-dimensional fluid contained within the
b( p/pJ
(2.7) oore voiume (V oer moie):
(I - v/p,)(1 - P/P, + bp/p,)
<J>dn, = 7TdA = <f,dV (2.10)
where /Js 1s the saturatmn vapor pressure, although the physical model from
where 7T 1s the "spreading pressure" and <f> the thrne-dimens1onal analog. It
which this expression was ongrnally denved is probably not realistic, particu-
1s evident that <P (or -rr) fulfills the roic of the pressure ma bulk svstem and
larly for mtcroporous solids. The BET model ts most commonly encountered
the relevant free energy quantity for an adsorbed phase ( fJ 1s given bv:
in connection with the experimental measurement of surface area by nitrogen
adsorption at cryogenic temperatures, but it has also been used to represent (2.11)
the isotherms for moisture on activated aiumrna, where the isotherms are of (smce Gs )::: As). The sm1ilantv with the defimtiOn of Gibbs free energy, for a
the well-defined type !I form. 12
bulk phase (G = A + PV) 1s obvious.
Followmg essent1allv the same logic as m the derivation of the
2.2, 7 Spreading Pressure and the Gihbs Adsorption Isotherm Gibbs-Duhem eauat1on leads directly to the Gibbs adsorption isotherm:
To understand the Gibbs adsorption isotherm reqmres a short digression into . ,Jrr )' _ RT n,
or l tlo r - PA (2.12)
the formal thermodynamics of adsorptlOn and an rntroductmn to the concept
of "spreading pressure.'' It is convenient to adopt the Gibbstan formulation
By msertmg different equations of state for the adsorbed phase Irr( 11\, A, T)1,
and consider the adsorbent simply as an inert framework that orovides a
corresponding forms for the e(1uilibnum adsorptmn isotherm a( p) may
force field that alters the free cncq,.ry (and other t11ermodynam1c properl1es)
therefore be found.
of the sorbate-sorbent system. The changes in the thermodvnam1c prooerttes
are ascribed ent1reiy to the sort>ate. Since the adsorbed layer 1s a condensed
phase, its thermodynamic orooerties are relatively msens1t1ve to the ambient 2.2.8 Binary and Multicomponent Sorption
pressure.
The Langmuir modei (Ea. 2.3) yields a simple extension to binarv ( and
If we consider na moles of adsorbent and n moles of sorbate, the
5 multicomponent) systems, reflecting the comoet1tion between species for the
chemical ootenUal of the adsorbed phase 1s given by:
adsorption sites:
(2.8) ( 2.13)
JUSt as for a binary bulk system contaming n,. moles of component s and n" It 1s clear that at a given temperature (which determmes the value of b) and
moles of component a. We may also define a soecific energy <I> by the partial at given partial pressures the quantity of component 1 adsorbed will be lower
derivative: than for a single-component system at the same oarual oressure. Like the
smgle-comoonent Langmmr ec1uat1on, Eq. 2.13 provides a useful approxuna-
(2.9) t10n to the behavior of m;_my svstems, but 1t 1s quant1tat1vely accurate only for
a few systems. It 1s however widely used m the modeling of PSA systems
This quant1tv has no direct analog for H hulk phase. For example, for a largely because of its simplicity h11t also hce,111sc many PSA wstcm<. oncrnte