Langmuir 1987, 3,121-124
121
Heat of Adsorption in Liquid Mixtures and Their Vapors A. L. Myers* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
S. Sircar Air Products and Chemicals, Inc., Allentown, Pennsylvania 18105
Received December 31,1985. I n Final Form: August 12, 1986 Enthalpic effects accompanying adsorption are needed for energy balances in the design of adsorbers. Heats of adsorption are also useful for predicting (1) the selectivity of the adsorbent for different substances and (2) the temperature coefficient of adsorption. The heat of immersion of an adsorbent in a liquid mixture (AH) is connected by thermodynamics to the isosteric heat of adsorption (q,J of its vapor.
Introduction The relationships between adsorption from liquids and adsorption from vapors have been derived in a series of papers.'-5 For vapor mixtures the adsorption isotherm of the ith component is the individual amount adsorbed ni, which is a function of pressure and the composition of the vapor phase yi. For liquid mixtures the adsorption isotherm of the ith component is the surface excess n;, which is a function of the composition of the bulk liquid phase xi. The results are summarized in three key equations; the first is for the surface excess2 n; = lim (ni - mi) (1)
pors. The purpose here is to derive a companion set of equations for the relationship between the heat of immersion of an adsorbent in a liquid (AH) and the isosteric heat of adsorption of its vapor qat. The usefulness of the enthalpy extends beyond the calculation of heat effects; for example, for an ideal liquid solution; the heat of immersion determines the temperature coefficient of adsorption from liquids through the equation4 (4)
Theory The first step is to devise a reversible process for calculating the free energy of immersion AG in a liquid mixture containing N components. Let P and yi be the bubble-point pressure and equilibrium vapor composition, respectively, for a liquid mixture of composition xi. Then the free energy of immersion AG is given by eq 2 with the integration at constant T and yi. Also, let AH be the isothermal enthalpy of immersion of clean adsorbent in liquid of composition xi. It has been shown that4
P-P
where n = Cni. The limit in eq 1is evaluated along a locus of constant vapor composition yi that is in equilibrium with the bulk liquid phase at its bubble-point composition xp Thus xi in eq 1 is a constant but ni and n increase with pressure. This equation may be used to calculate liquid adsorption from experimental data on vapor adsorption isotherm^^.^ or to derive analytical equations2i5for n;. The second equation is for the free energy of immersion (AG) of a solid adsorbent in a pure liquid?
AG = -RT
1:;
df
(2)
The gas-phase fugacity (f) may be replaced by pressure (P) when the gas is ideal. Integration is from the limit of zero pressure up to the vapor pressure of saturated liquid (P). AG is a negative quantity because isothermal adsorption is spontaneous. The free energy of immersion determines the selectivity of the adsorbent through the third key equation:'
When (AG1l >> lAG21,the adsorbent has a high selectivity for compnent no. 1 relative to no. 2, and nle is large and positive. y1 is the activity coefficient of component no. 1 in the bulk liquid. Equation 3 also provides a test of thermodynamic consistency1 of a surface excess isotherm (rile vs. xl) with adsorption from its constituent vapors AG2).
The above equations summarize the connection between the extent of adsorption from liquids and from their va(1) Myers, A. L.;Sircar, S.J. Phys. Chem. 1972, 76, 3412. (2) Myers, A. L.;Sircar, S. J.Phys. Chem. 1972, 76, 3415. (3) Sircar, S.;Myers, A. L. MChE J. 1971, 17, 186. (4) Sircar, S.;Novosad, J.; Myers, A. L. Ind. Eng. Chem. Fund. 1972, 11, 249. (5) Sircar, S.;Myers, A. L. AIChE J. 1973, 19, 159.
0743-7463/87/2403-0121$01.50/0
where hi is the partial molar enthalpy of ith component in the bulk liquid phase. AH* is the actual heat of immersion ( A l l ) corrected for heat effects in the bulk liquid solution caused by its change in composition during adsorption
m* = AH - (n*)o[Ahm(xi) - Ahm(x?)]
(6)
where Ah" is the molar enthalpy of mixing of the bulk liquid, evaluated before contact with the adsorbent [Ah"(x?)] and after equilibration [Ahm(xi)]. ( d ) O is the total amount of liquid per unit mass of adsorbent before contact of the solid with the liquid phase. The heat of mixing Ah" and partial molar enthalpies hi are measured independently of the adsorption experiment. For the special case of an ideal liquid phase, Ah" and hi are zero, AH = AH*, and eq 5 simplifies to AH = -P-(-) a AG (7)
aT
Also, for a pure liquid:
AH
T
x,
(-)
= -P d AG
dT T Returning to eq 5 , it is desired to translate the partial derivative of AG into a heat of adsorption. Let I be the 0 1987 American Chemical Society
122 Langmuir, Vol. 3, No. 1, 1987
Myers and Sircar
definite integral of eq 2, so that
Substitution of eq 13, 16, and 17 into eq 15 gives
Since P is a function of temperature as well as liquid composition, Leibniz’ rule yields
The isosteric heat of adsorption is defined by7
(10) where ne is the number of moles adsorbed in the limit as P P. For the case of multilayer adsorption, ns approaches infinity a t condensation and the right-hand-side of eq 10 has the form (a - a). However, the limit exists because both AG and AH are finite quantities. It should be mentioned in this connection that the largest contributions to AG and AH occur a t low surface coverage. Experimental uncertainties in the value of n near saturation are not critical because their effect upon the integral of eq 2 is relatively small. The partial derivative in the first integral of eq 10 may be written
-
Finally, if eq 10, 12, 18, and 19 are combined and substituted into eq 5, a general equation for the relationship between the heat of immersion AH* and the isosteric heat qst is obtained:
AH* = C(n;hi) -
L O
P [qst -
Cyi(Xi - hi)]dn (20)
Summations are over the N components of the liquid mixture. Equation 20 is the key equation relating the enthalpies of adsorption of liquids and their vapors. There are several important special cases: for example, if the bulk solution is ideal, then hi is zero, AH* = AH, and
Furthermore, for a pure liquid for constant composition yi. If eq 11 is substituted into the first integral of eq 10 and the independent variable is changed from pressure to moles adsorbed then
where all derivatives and integrals are a t constant vapor composition yi. In order to simplify the equations and focus upon the properties of the condensed phases, a perfect gas is assumed for the vapor phase. At saturation, vapor a t pressure P is in equilibrium with the bulk liquid phase6 where ai is the activity of the ith component in the liquid phase and Pi” is its vapor pressure. By summation over the components of the liquid mixture N
P = CPtai
(14)
i=l
Taking the partial derivative of both sides of eq 14 with respect to temperature,
(g)x,
+
= zNa i P t (
(%)x,)
(15)
The Clapeyron equation for vapor a t low pressure6 gives the relation between the slope of the vapor pressure curve and the heat of vaporization of the pure liquid ( X i ) : d In Pi” Xi =-
dT RT2 The change of activity with temperature is governed by the excess partial molar enthalpy (hi)in the liquid phase:6
xi
hi RT2
(6) Smith, J. M.; Van Ness, H. C. Introduction to Chemical Engineering Thermodynamics, McGraw-Hill: New York, 1975;pp 185,250, 614.
Equations 20-22 connect liquid-phase calorimetric measurements (AH)with isosteric heats of adsorption (qsJ for the vapor phase. qst is used in energy balances for flow systems such as adsorption in fixed beds and is simply related to the differential heat of adsorption ( q d ) from calorimetry7 by (23) qst = qd + R T Alternatively, if adsorption isotherms have been measured at a series of temperatures, qst may be obtained from eq 19: qst = R P ( -jjT), a In P Throughout this paper it is assumed that the vapor phase obeys the perfect gas law, but no other assumptions about either the liquid or the adsorbed phase are made. In most applications, the vapor pressures are near or below atmospheric so that the perfect-gas assumption is justified; the error in eq 20-22 is no more than 1% for vapor pressures less than 500 kPa.
Values of Immersion Functions Values of free energy (AG) and enthalpy (AH) of immersion have been calculated for a variety of systems by using eq 2 and 22, respectively.8 The entropy of immersion ( A S ) is then obtained from AS =
A H - AG T
For example, for immersion of BPL activated carbon in liquid carbon dioxide at 260 K, the values calculated from vapor adsorption isothermsg are AG = -56.3 J f g, AH = -65.8 J/g, and TAS = -9.5 J f g. Negative values for AG and AH are expected because adsorption from liquids is spontaneous and exothermic. Less obvious is the negative (7)Young, D.M.; Crowell, A. D. Physical Adsorption of Gases; Butterworths: London, 1962;pp 71,147,163. (8) Valenzuela, D.; Myers, A. L. S e p . Pur$ Methods 1984, 13, 153. (9)Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 336.
Langtnuir, Vol. 3, No. 1 , 1987 123
Heat of Adsorption in Liquids and Their Vapors sign for the entropy of immersion, which is typically 10-20% of AH.
Heat of Immersion from Adsorption Isotherm Equations Equation 22 may be used to calculate the heat of immersion from vapor adsorption isotherms. An example is the BET theory7 of multilayer adsorption, for which the potential energy of a molecule in the first adsorbed layer is -El, and the potential energy in the second and higher layers is equal to its potential energy in the bulk liquid (-EL). E, and ELare positive constants and El> EL.In this case the heat of immersion in the bulk liquid is obviously AH = -m(E, - E L ) , where m is the amount adsorbed in the first layer. The calculation of AH by eq 22 is not trivial and is therefore outlined below. First eq 22 is written in a more convenient form by using eq 16 and 24:
Table I. Adsomtion of Benzene on Silica Gel' 30
"C
50 "C
P. kPA
nu.mollke
P, kPa
nu,mol/kg
0.0125 0.109 0.397 0.640 1.573 2.893 4.106 5.093 6.799 8.466 9.906 13.332
0.207 0.842 1.638O 2.059 2.689 3.284 3.629" 3.754 3.835 3.880a 3.907 3.959
0.0644 0.249 0.519 0.800 1.586 2.853 4.493 6.373 8.626 10.946 14.745 19.665 22.971 26.678
0.268 0.650 1.001" 1.199 1.569 2.096 2.580 2.928" 3.296 3.518 3.694 3.769 3.807" 3.857
Desorption point.
where x =P/P
Equation 26 applies when the adsorption isotherm is explicit in n. For pressure-explicit isotherms like the BET equation, a transformation of variables in eq 26 from n to x gives
The BET equation is' n=
f
mcx [l - x][l + x ( c - l ) ]
2
and j, and jLare partition functions for internal degrees of freedom of an adsorbate molecule in the first adsorbed layer and in the liquid, respectively. From eq 28, dc 1 (29) = mx [ l x(c - 1)]2
(&),
do +
By substitution of eq 29 into eq 27,
AH
= -mcR-
d In c
1
dx (30)
The value of the definite integral is l/c, so d In c AH=-mRd(l/T)
Finally, from eq 28 with j, and jLequal AH = -m(E, - E L )
(32)
Comparison of Calculated and Experimental Heats of Immersion In the first example, isosteric heats of adsorption are derived from vapor adsorption isotherms. The adsorbent is silica gel (Davison grade 37, mesh size 12-42, surface area 350 m2/g), degassed under vacuum for 12-15 h at 280-300 "C. Gravimetric measurements of the adsorption of benzene vapor a t 30 and 50 "C are given in Table I and Figure 1. These data were first differentiated according to eq 19 for qat, which was substituted into eq 22. The
4
6
s
10
12
14
PRESSURE, kPa
Figure 1. Adsorption isotherms of benzene on silica geL5 (0) 30 "c;(A)50 "c.
result of this calculation is AH = -67.8 f 6.7 J/g. The uncertainty arises primarily from the differentiation step. The calorimetric heat of immersion4 of this adsorbent in liquid benzene at 30 "C is AH = -65.3 J/g. Thus the agreement lies well within the accuracy of our experiments. In the second example, isosteric heats of adsorption are measured by calorimetry. Values of qat for nitrogen vapor on magnesia "2640" (surface area 112.7 m2/g) were measured a t 74.9 K and are reported in Figure 3 of ref 10. Integration of these isosteric heats of adsorpton by eq 22, with X = 5690 J/mol for the heat of vaporization of liquid nitrogen at 74.9 K, gave the calculated value AH = -11.3 f 0.6 J/g. For comparision, the calorimetric valuelo for immersing the same adsorbent in liquid nitrogen at 78.15 K is -11.8 f 0.4 J/g. Again, the heat of immersion calculated from isosteric heats of adsorption of the vapor agrees with the direct measurement.
Estimating Selectivity from Heats of Immersion In the process of selecting an optimum adsorbent for the separation of a particular liquid mixture, the heats of immersion in pure liquids provide valuable information on selectivity in the following way. Assuming that the (10) Chessick, J.. J.; Young,G. J.; Zettlemoyer, A. C. Trans. Furuday SOC.1954, 50, 587.
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124 Langmuir,-Vol.3, No. 1, 1987
monolayer capacity (m) is the same for both components and ignoring bulk and adsorbed phase nonidealities, the surface excess of component no. 1 is2 (33)
where
K = exp( AG2 - bG1 mRT
)
(34)
As discussed previously, A S of immersion is small relative to AG and AH, so that a useful approximation is (35) The advantage of eq 35 is that heats of immersion are relatively easy to measure, compared to adsorption isotherms. m can be estimated from the surface area of the adsorbent. Thus the selectivity of an adsorbent for a separation can be characterized approximately in terms of its heats of immersion in the components of the liquid mixture.
Summary (1)Equation 20 connects the heat of immersion of an adsorbent in a liquid mixture (AH-)with the isosteric heat of adsorption of its vapor (qst). (2) The integral heat of adsorption in tlie saturated vapor is the sum of the heat of immersion in the liquid (AH) and the heat of condensation of the vapor (-A). (3) The temperature coefficient of adsorption in a liquid mixture (dnIe/d7')xl is proportional to the derivative of the heat of immersion with respect to the composition of the liquid (a CLH/dxl)T. (4) The heat of immersion in a pure liquid may be calculated from vapor-phase adsorption isotherms from eq 26 or 27. (5) Heats of immersion in pure liquids are useful for estimating the selectivity of an adsorbent by eq 35.
Notation activity constant, eq 28 potential energy in first adsorbed layer, J/mol E, potential energy in second and higher layers, J/mol EL free energy of immersion, J / g AG AH enthalpy of immersion, J / g corrected enthalpy of immersion, eq 6, J / g AH* partial molar enthalpy of ith component in bulk hi liquid phase, J/mol molar excess enthalpy (heat of mixing) in bulk Ahm liquid phase, J/mol definite integral of eq 2 I partition function for internal degrees of freedom j K constant, eq 33 m monolayer capacity, mol/g n amount adsorbed, mol/g surface excess of ith component, mol/g n? na amount adsorbed at dew point of vapor, mol/g P pressure, Pa dew-point pressure of vapor mixture, Pa P vapor pressure of ith component, Pa Pi" differential heat of adsorption, J/mol qd isosteric heat of adsorption, J/mol gas constant entropy of immersion, J/(g K) AS T absolute temperature, K mole fraction of ith component in liquid phase Xi mole fraction of ith component in vapor phase Yi Greek Letters activity coefficient in liquid phase Y enthalpy of vaporization of pure liquid, J/mol x a C
%t
Subscripts i refers to ith component Superscripts value at dew point of vapor S refers to bulk liquid phase 1 refers to value before contacting adsorbent with 0 liquid
