Adsorption from Liquid Mixtures on Solids: Thermodynamics of Excess Properties and Their Temperature Coefficients Shivaji Sircar, Jaromir Novosad, and Alan 1. Myers* School o j Chemical Engineering, Cniversity o j Pennsylvania, Philadelphia. Pa. 19104
The Gibbsian thermodynamics of adsorption from liquid mixtures on solids is formulated in terms of excess properties. The excess properties are calculated from experimental phase equilibrium and calorimetric data. The equations for the excess functions are used to derive a general expression for the temperature coefficient of the surface excess in terms of the heat of immersion of the adsorbent. For the special case of an ideal adsorbed phase, the heat of immersion cf the adsorbent in liquid mixtures may b e calculated from heats of immersion in the pure liquids. Experimental data on nonisothermal adsorption and heats of immersion are reported for the system benzene f cyclohexane adsorbed on silica gel.
C o n v e n t i o n a l adsorptive methods of separation such as fixed-bed adsorption are based upon the establishment of two regions of different coiiceiitration: the bulk phase and the adsorbed phase. X neiv class of pulsed separation processes has been developed for which the effectiveness of the separation depends upon the temperature coefficient of the amouiit adsorbed. These processes include pulsed adsorption (Jenczemki and Myers, 1970), cycling zone adsorption (Pigford, et ai.,1969), and parametric pumping by direct thermal mode (Sweed and Vilhelm, 1969). Prospective mixtures for separat’ion by these new processes must possess a high temperature coefficient of adsorption. T h e direct measurement of adsorption a t different temperatures is a time-consuming experiment which might be avoided by relating the ternperat’ure coefficient to some other more readily measurable quantity. The purpose of our work is to derive a therniodyiiamic relationship between the temperature coefficient of the amount adsorbed and the heat of inimersioii of the adsorbent in the liquid mixture.
the properties of the bulk and adsorbed phase-, respectively, a t equilibrium in term> of the Gibbs model (see Figure lb). ,Ifo (-110 = X I ) is the same propert1 of the solution prior to contact with the adsorbent. Unlike V and -If’, A M is independent of the value of b. For the particular case of mass, A M 15 zero and eq 1 reduces to
AJf
=
nm f n’m’ - na?no
I n the Gibbs model of an interface, the real system consisting of a hornogeiieous bulk liquid phase and a n iuhoniogeneous adsorbed phase is replaced by two homogeneous phases (bulk liquid and adsorbed) extending up to a surface of division. The properties of the homogeneous adsorbed phase in the Gibbs model are functions of the location of the dividing surface. Figures l a and l b are schematic represenbations of the real system and the Gibbs model, respectively. The dividing surface in the Gibbs model is located a t a n arbitrarily chosen distance b from an origin located within the adsorbent. h balance of any extensive thermodynaniic property ( M ) of the system, based upon unit mass of adsorbent, yields
(3)
where m 1s any molar extensive thermodynamic property defined by
vz
=
;VI’n
(4)
may be molar enthalpy, molar Gibbs free energy, mole fraction, etc., but not a partial niolar quantity. We define next a n excess property ( X e ) for adsorption
112
=
Gibbs’ Model of Interface
(2)
no=.+.’
Equation 1 may be written in terms of molar properties
no(mo-
?TL)
+ ALII
(5)
Since all of the quantities on the right-hand side of eq 5 are experiniental variables, X e 13 measurable. In other nords, the value of J f e is iiirariant to the location of the Gibbs dividing surface. From eq 2 , 3, and 5 M e
= n’(m’ - 112)
(6)
Thus X e , in addition to being an experimental variable, is given in terms of the hypothetical but useful parameters (m’and n ’ ) of the Gibbs model. Equation 6 shows that M e is the difference between the ~ a l u eof -If i n the adsorbed phase and the value of ,M in a hypothetical adborbed phase a t the rame state ( L e . )the same \ alues of intensive properties) as that of the bulk liquid. Surface Excess
-11 may be enthalpy, Gibbs free energy, number of moles of the i t h compound, etc. The change in the value of d l due to adsorption ( A X ) is measured by its value before (MI) and after (M,) contact with the solid adsorbent. and N ’ are
Let V be the number of moles of the i t h component (nL). Since mass is conserved, AJ4 = An, = 0. From eq 5 nie =
nyx,o - x,)
(i
=
1, 2,
’
’ ’
~
S,)
Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 2, 1 9 7 2
(7) 249
where AG is the change in free energy of the system due to formation of the solid-liquid interface. The free energy of the adsorbed phase is given by (Bering, et al., 1970)
t
G’
Figure 1. Gibbsian model of solid-liquid interface
zi (zi = ni/n) is the mole fraction of the i t h component; N , is the total number of components of the liquid mixture. np is called the surface excess of the i t h component (Sircar and Myers, 1971). Since the mole fractions must sum to unity, i t is seen from eq 7 that nie = 0
G
ni* = 0
p
