Langmuir 1991, 7, 3065-3069
3065
Isosteric Heats of Multicomponent Gas Adsorption on Heterogeneous Adsorbents Shivaji Sircar Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195-1501 Received March 11, 1991. In Final Form: April 9, 1991 Isosteric heats of adsorption of the components of a gas mixture on a heterogeneous adsorbent can be estimated from multicomponentadsorption isothermsby using thermodynamicrelationships. It is shown by using the simplified Toth equation for heterogeneous binary-gas adsorption that the component isosteric heats of adsorption can be strong functions of specific loadings of the components. These functions are very different from the loading dependence of pure-gas isostericheats. It is also shown that the degree of adsorbent heterogeneity very strongly influences the component isosteric heats. Analytical mixed-gas heterogeneous adsorption isotherms are preferred for calculating the mixed-gas isosteric heats.
Introduction One of the most ignored thermodynamic variables in study of adsorption from gases is the isosteric heat of adsorption of the components of a gas mixture on heterogeneous adsorbents. Knowledge of these variables is critical in calculating the heat evolved (consumed)during the ad(de)sorption process from a multicomponent gas mixture in a dynamic adsorber for carrying out a gas separation process using a pressure swing adsorption (PSA) or thermal swing adsorption (TSA) scheme as well as during the measurement of nonisothermal adsorption kinetics of a multicomponent gas mixture. The differential change in the enthalpy (dH) associated with a differential change in the specific adsorbate loading (dni)of component i of a multicomponent gas mixture ( i = 1, 2, ...) is given by1
dH = x q i d n i
i = 1 , 2 , ...
(1)
where qi is the isosteric heat of adsorption of component i of the gas mixture at the specific adsorbate loading of ni for component i in the multicomponent adsorbed phase. The summation in eq 1 is over all components of the gas mixture. For adsorption of a pure gas or for adsorption of a single adsorbate from an inert gas, eq 1 reduces to
dH('= qpdnp
(2)
where qp is the isosteric heat of adsorption of pure gas i at the specific adsorbate loading of np. The superscript zero represents pure gas adsorption. The enthalpy change during a finite ad(de)sorption process can be calculated by integrating eqs 1 or 2 between the initial and final values of ni (or np) during the process. There lies the crux of the problem. If the adsorbent is energetically homogeneous (all adsorption sites characterized by the same energy of adsorption), and if the lateral adsorbate-adsorbate interactions are negligible, qi or qp are constants and they are independent of adsorbate loadings. This is often true for homogeneous microporous adsorbents. Consequently, the integration process is much simplified. Cn the other hand, if the adsorbent is energetically heterogeneous, qi or qp become functions of n, (or np) and the integration process becomes extremely difficult under real dynamic ad(de)sorption conditions.2 (1) Sircar, S. J. Chem. SOC.,Faraday Trans. 1 1985,81, 1527.
It is preferred that the functions qi(ni)and qp(np)be known analytically so that the integration process is facilitated. It has been a common practice in the published literature to assume that qi and qp are constants and equal (true for homogeneous adsorbents), or their variations with ni or np are identical, for design of adsorption columns and for studying nonisothermal adsorption kinetics even though the adsorbent is energetically heterogeneous (which is the case with most practical adsorbents). This can lead to severe misinterpretation of data or in their subsequent use in design of adsorbers. The purpose of this paper is to calculate qi and qp for a model heterogeneous adsorbent by using multicomponent adsorption thermodynamics and to demonstrate the role of adsorbent heterogeneity on these variables. Adsorption Thermodynamics Single-Gas Adsorption. The isosteric heat of adsorption of a pure gas (qp) at a given specific adsorbate loading (np) can be calculated from the adsorption isotherms of that gas at different temperatures by1 (3)
where P is the equilibrium adsorption pressure for the pure gas a t temperature T a t the specific adsorbateloading of np. Equation 3 can be used to calculate qp as a function of no. It is apparent that an analytic expression for qi0( n6i )can be obtained if the pure-gas specific adsorption isotherm np(P,T) is analytic. By using the chain rule of calculus, we can rewrite eq 3 as
(4) Equation (4) shows how an analytic expression for qp can be easily obtained if np(P,T) is analytically known. Multicomponent Gas Adsorption. The isosteric heat of adsorption of component i of a multicomponent gas mixture (qi) at a given specific adsorbate loading (ni, i = 1, 2 , ...) is given by1 (5)
where P is the total gas-phase pressure and yi is the gas(2) Sircar, S. PressureSwing Adsorption: Research Needs by Industry. Presented at the 3rd International Conference on Fundamentals of Adsorption, Sonthofen, Germany, 1989.
0 1991 American Chemical Society
3066 Langmuir, Vol. 7,No. 12, 1991
Sircar Table I. Homogeneous Adsorption Isotherms Langmuir
Jovanovic
pure-gas isotherm temperature dependence of bi
= biP/(l + biP) d In bi/dT = - qr/RT
0; = 1- exp[biP] d In bi/dT = - q l / R T
pure-gas isosteric heat of adsorption, q; mixed-gas isotherm mixed-gas isosteric heat of adsorption, qi
q; 0; = biPyi/(l
4 = (biPyi/EbiPyi)[l-
4
phase mole fraction of component i in equilibrium with the adsorbent with specific adsorbate loadings of ni at temperature T. Again, by use of the chain rule of calculus or the Jacobian transformation, eq 5 can be rewritten to express qi in terms of (dni/dP)T,,, (dni/dyi)p,T,and (dni/dT)p,, as in eq 4. Consequently, if the specific adsorption isotherms ni(P,T,yi) of each component i from a multicomponent gas mixture were known, qi could be calculated as functions of P, T, and yi or ni. For a binary gas mixture (i = 1, 2), eq 5 can be transformed to give1
Q1
RP
Equations 6 and 7 show that analytical functions for ql(n1,nz) and ~ 2 ( n l , n 2can ) be obtained if ni(P,T,yi) are analytically known. The need for having analytical expressions for ni(P,T,yi)must be emphasized here. Equations 6 and 7 show that even if one has such analytical component isotherm equations, the calculations of qi may not be trivial. In the absence of such isotherm equations, calculation of qi from experimental ni(P,T,yi)data may be impossible due to the enormous volume of data needed to cover the range of interest. Direct calorimetric measurement of qi at a given ni or direct measurement of (Pyi)for a given ni at different temperatures may provide easier solutions of this problem if experiments can be devised to generate those kinds of data. In the latter case, eq 5 can be directly used to estimate qi(ni). Even then, a large volume of data will be needed to satisfy the design needs of practical adsorbers. I am aware of only one case of such measurem e n t ~ .Clearly, ~ the complexity of the problem will be increased many-fold when the number of components in the gas mixture exceeds two.
Model Isosteric Heats of Adsorption of Pure Gases and Binary Mixtures The effects of adsorbent energetic heterogeneity on the isosteric heats of adsorption of pure gases and on the component isosteric heats for adsorption of binary gas (3) Bulow, M.; Lorezn, P. In Proceedings of Second International ConferenceonFundamentals ofAdsorption;Liapis, A. I., Ed.; Engineering Foundation: New York, 1987; pp 119-128.
+ ZbiPyi)
q; exp(-XbiPyi)]
qc
mixtures can be evaluated by using thermodynamic equations (3-7) in conjunction with model heterogeneous adsorption isotherms for pure and multicomponent gas mixtures. A few examples follow. Homogeneous Adsorbent. Single-GasAdsorption. Two commonly used homogeneous adsorption isotherms are the Langmuir and the Jovanovic model^.^ They are given in Table I. The isotherms are expressed in terms of fractional coverages (e: = .:/mi) of the adsorbate i as functions of gas-phase pressure P at any given T. mi is the temperature-independent saturation adsorption capacity of the adsorbate i. bi is the Henry’s law constant [(&$/dP)T-,bi as P 01 for pure adsorbate i, which has an exponential temperature dependence. Applications of eqs 3 or 4 to these isotherms show that the isosteric heat of adsorption of the pure gase qf (= qi*) is independent of$. (Hence,theadsorbentisenergeticallyhomogeneous.) Multicomponent Gas Adsorption. Table I alsoshows the multicomponent adsorption isotherms by the Langmuir and the Jovanovic models for the special case where all adsorbates have equal saturation capacities (mi = m). The fractional coverages of component i (ei = ni/m) are given as functions of P, T, yi and pure-gas adsorption parameters bi. These mixed-gas isotherms can also be derived by applying ideal adsorbed solution theory4 to the pure-gas isotherms, and hence, they are thermodynamically consistent. It can be shown by applying eqs 6 and 7 to these homogeneous multicomponent adsorption isotherms in their binary forms (i = 1, 2), as well as by applying eq 5 directly, that qi for component i adsorption is independent of Bi, and it is equal to q: (= qi*) for pure-gas adsorption. Thus qi and qp for a homogeneous adsorbent are equal and constant (independent of adsorbate loading). qi, however, may not be equal to qp when mi # mj. HeterogeneousAdsorbent. Single-Gas Adsorption. The Toth m0de1~7~ provides a relatively simple but flexible framework for describing single-gas adsorption on heterogeneous adsorbents:
-
n; = mb,P/[l+ (biP)k]”k
(8)
m is the temperature-independent saturation adsorption capacity. bi is the Henry’s law constant for component i. k (I1.0) is the dimensionless adsorbent heterogeneity parameter, which is a function of T (dk/dT > 0). According to this model, the adsorbent is homogeneous when 12 = 1. In that case, eq 8 reduces to the Langmuir model. The adsorbent heterogeneity increases as k decreases. It has been shown that the Toth model can be derived by assuming that the adsorbent is “patchwise homogeneous”, consisting of a collection of Langmuirian sites with a skewed Gaussian-like distribution of site adsorption energies.5 However, it is not necessary to assume patchwise homogeneity, if lateral interactions between the adsorbate molecules are neglected. ~~
~
~~~~~
(4) Sircar, S.;Myers, A. L. Surf. Sci. 1988, 205, 353. (5) Jaroniec, M.; Toth, J. Colloid Polym. Sci. 1976,254, 643. (6) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibria Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989.
Langmuir, Vol. 7, No. 12,1991 3067
Zsosteric Heats of Multicomponent Gas Adsorption
Equations 3 and 4 can be applied to eq 8 to obtain qp as a function of 0: for the Toth model as follows:
d In k
293.1 K 313.1 K 333.1 K 293.1 K 313.1 K 333.1 K
(9) u) Q)
0
d In bi
qi*
dT
RP
-=--
0.01 1
-
qr* is the isosteric heat of adsorption of pure gas i at the
limit of zero coverage (0; 0). It also defines the temperature coefficient of the Henry’s Law constant for the pure gas (eq 11). The function F,(dp), defined by eq 10, has a negative value for (0 5 0: 5 1). Fr(O;)is equal to zero 0 and its absolute value increases as at the limit of 0; 0; increases. Thus qp(8;) is less than qr* since ( d k / d n >. 0 and qp decreases as 0: increases. In other words, the isosteric heat of adsorption of the pure gas decreases with increasing coverage, which is a direct proof of adsorbent heterogeneity. It should be mentioned here that the Toth model corresponds to an asymmetric distribution of energy sites, which broadens towards the lower energy region. Thus, this isotherm model and the corresponding heats (eqs 9 and 10) only apply to such cases. Multicomponent Gas Adsorption. The Toth model has been extended to describe multicomponent gas adsorption on heterogeneous adsorbents for the special case of (m, = m ) and for the same values of the heterogeneity parameter k for each component of the gas m i ~ t u r e : ~
-
ni =
mbiPyi [ l + (Cb;Py;)k]l’k
i = 1,2, ...
(12)
Thus, eq 12 represents multicomponent adsorption on a heterogeneous adsorbent where all adsorbates have the same saturation capacities and the same degrees of heterogeneity but their Henry’slaw constants are different. It was assumed in the derivation of eq 12 that the differences between the site adsorption energies of any two components of the gas mixture on any Langmuirian site were ~ o n s t a n t .This ~ means that the selectivity of adsorption of component i over component j is a constant (= bi/bj) on all sites. Equation 12 can also be derived by applying ideal adsorbed solution theory to eq 8 under the restrictions of interest indicating that they are thermodynamically consistent. Equations 6 and 7 can be applied to eq 12 to obtain qi as a function of 01 and 82 for adsorption of a binary gas mixture as
The function F(0) in eq 13 has the same form as that in eq 10 except that 0; is replaced by total fractional coverage 0 (= 81 + 02). Thus q; can be calculated for a given 01 and 02 or the set (P,T,yi). Equation 13 shows that qi decreases as 8 increases and q;* is the limiting value of qi when ’h 0 ( P 0 for any yi).
-
____
.-
c
-
LANGMUIR MODEL EXPERIMENT (SZEPESY & ILLES)
I
1
10
100
1000
P(KPa) --+ Figure 1. Adsorption of pure ethylene and methane on Nuxit charcoal. Table 11. Parameters of Pure-Gas Adsorption Models
m, mmol/g bl, (kPa)-l ql, kcal/mol k (293.1K) k (313.1K) k (333.1K) dk/dT (KIF
ethylene Toth Model“ 11.0
7.802X 9.122 0.34 0.37 0.40 0.0015
methane 11.0 7.602X lo3 7.560 0.34 0.37 0.40 0.0015
Langmuir Model m, mmol/g bi at 293.1 K (kPa)-l
4.0
0.0263
4.0 0.00275
Temperature coefficient of Toth parameter b, = b: exp (qrlR1T3.
The quantity (RP/k)(d In k/dT) in eqs 9 and 13 are very weak functions of T because dk/dT is very small for most systems.6 Thus, qp or qi are weak functions of T. Example of Binary Isosteric Heat of Adsorption on Heterogeneous Adsorbent The experimentally measured7 isotherms for adsorption of pure ethylene and methane on Nuxit-al charcoal can be adequately described by the Toth model. Figure 1show the best fit (solid lines) of the experimental data (circles) at three different temperatures by eq 8. The average error in describing the ethylene isotherms is within A3 7% and that for the methane isotherms is within f7 7% The model parameters are given in Table 11. The same values of m and k(T) were used to describe both systems so that the mixed gas Toth model (eq 12) can be used to calculate the binary adsorption isotherms. The 12 values for both gases were substantially less than unity indicating that the adsorbent was sufficiently heterogeneous for their adsorption. The temperature coefficient of k was positive but small (dk/dT = 0.0015 K-l). The dashed lines in Figure 1 show the best fit of the isotherm data at 293.1 K by the Langmuir model, using constant m. The values of Langmuir parameters are given in Table 11. Obviously, the Langmuir model is not adequate due to the heterogeneous nature of the adsorbent. Figure 2 shows the calculated (solid lines) isosteric heats of adsorption of pure ethylene and methane on the charcoal as functions of coverage according to eqs 9 and 10. It also shows isosteric heats for pure gases obtained by applying eq 3 directly to the experimental isotherms (circles). The isotherms at 293.1,313.1, and 333.1 K were used in a pair-
.
(7) Szepesy, L.; Illes, V. Acta Chim. Hung. 1963, 35,37.
Sircar
3068 Langmuir, Vol. 7, No. 12, 1991 .n
"I
-
t S
I
O0o
I
BINARY
GAS AT SAME ei
----PURE
I
0
FROM ISOTHERM
0.1
0.2
r
-
T O T H MODEL
EXPERIMENT (SZEPESY & ILLES)
0
.Ir
-
T = 293.1
TOTH MODEL
0
P = 99.2 KPa, T = 293.1 K
t
t
P = 99.2 (KPa)
-
0Figure 2. Isostericheats of pure ethyleneand methane on Nuxit charcoal.
4
I
10
I
0.2
0.4
0.6
0.8
1 .o
Yl-
Figure 3. Adsorption of binary mixture of ethylene (1) and methane (2) on Nuxit charcoal at 293.1 K. wise manner to obtain qP(0:) by using eq 3. The circles represent the average values of qp from such calculations. The error bars show the variation in calculated qp by this method. It can be seen from Figure 2 that the qp values for adsorption of ethylene by these two methods compare very well, but those for adsorption of methane differ by 0.5-1.0 kcal/mol. Both systems show a decreasing qp with increasing 0;. The limiting values of qi* (qp at 0:- 0) are, respectively, 9.122 and 7.560 kcal/mol for adsorption of ethylene and methane on the charcoal. This indicates that ethylene is more strongly adsorbed than methane on the charcoal, and it will be more selectively adsorbed from a mixture with methane. The mixed-gas Toth model (eq 12) was used to calculate the isobaric-isothermal binary-gas adsorption equilibria for ethylene (1)+ methane (2) mixtures at 293.1 K and at a total gas pressure of 99.2 kPa. Figure 3 shows the isotherms (solid lines). The experimental binary isotherms for this case8 are also shown (circles) in Figure 3. The model predicts the ethylene isotherm from the mixture very well (within h2%). The methane isotherm is underpredicted by 6 % in the lower gas-phase composition of ethylene where methane adsorption is moderate. It is underpredicted by 15% at higher compositions of ethylene where methane adsorption is very small. The calculated selectivity of adsorption for ethylene over methane on the charcoal at 293.1 K is 15. The experimental selectivity randomly varies between 11.5-14.8 over the composition range.698
-
-
(8) Szepesy, L.; Illes, V. Acta Chim. Hung. 1963, 35,245.
0
0.2
0.4
0.6
0.8
1.0
Y1-
Figure 4. Binary isosteric heats of adsorption of ethylene and methane on Nuxit charcoal. Mixture Isosteric Heats. It may be concluded that eq 12 describes ethylene-methane binary adsorption data on the Nuxit charcoal fairly well at the P and Tof interest. Equation 13 was then used to calculate qi for this system as a function of yi at the same P and T. 0i was first estimated as a function of yi by using eq 12, and then qi was calculated by using eq 13. Figure 4 shows the results (solid lines). The isosteric heat of adsorption of ethylene from the mixture decreases with its increasing composition in the gas phase, while that of methane increases with its increasing gas-phase composition. It may be seen from Figure 4 that Qi values for both components of the mixture vary significantly over the gasphase composition range (0 5 y 1 5 1)which is a reflection of the energetic heterogeneity of the adsorbent. Both q1 and q 2 decrease with increasing y1 because 0 increases as y1 increases. The difference (q1- q 2 ) is constant over the whole composition range because this difference should be equal to (q1* - q2*) according to eq 13. Figure 4 also shows the pure-gas isosteric heats of adsorption (qp) of each component (dashed lines) of the mixture at 0; values that are equal to t9i from the mixture at any given yi. The differences between qP(0: = 0i) and qi(0i) are striking. This is because si(&) depends on total fractional coverage, 0 (= 01 + 02), according to the Toth model. For this system, 0 is closer to 01 at any given y1 because of.high selectivity of ethylene over methane except when y1 is low. Thus ql(Ol) and qy(0: = 01) are close at higher values of y1 but they differ when y1 is low. 0 is much larger than 02 over the entire composition range and thus, ~ ( 0 2 is) much lower than = 02). This example demonstrates that the common assumption of constant isosteric heats of adsorption from mixtures (qi(0i) = qi*) or the assumption of equal isosteric heats of adsorption for components from mixtures and pure gases a t equal coverages [qi(&) = qp(0p = Oi)] can be extremely misleading. qi(0i) is a complex function of 0i, which is determined by the gas-phase conditions (P, T , and yi) of the adsorption system. This reemphasizes the need for having analytical multicomponent isotherm equations so that qi(0i) can be calculated from thermodynamics by using eqs 6 and 7 or their multicomponent extensions. Effect of Adsorbent Heterogeneity on qi. A parametric study of the effect of the degree of adsorbent heterogeneity on si(&) was carried out with the Toth model (eqs 12 and 13) for binary gas adsorption by varying the heterogeneity parameter k . It was assumed that the Henry's Law constants for the components ( b J and qi*
Langmuir, Vol. 7, No. 12, 1991 3069
Zsosteric Heats of Multicomponent Gas Adsorption
0.9
T = 293.1 K
0.8
0.7
1.-
a
0.6 0.5
Y
0.4
u-
0.3 0.2
0
0.1 0 0
0.2
0.4
0.6
0.8
1.0
YlFigure 5. Effect of Toth heterogeneity parameterk on fractional amounts adsorbed from a binary gas mixture: (a) k = 1.00; (b) k = 0.67; (c) k = 0.34. Parameters: bl = 0.0495 (kPa)-l; bz = 0.0033 (kPa)-l; m = 11.0 mmol/g.
values were same as those for ethylene (1)and methane (2) adsorption on the charcoal. A total gas pressure of 99.2 kPa and a system temperature of 293.1 K were also assumed. Figure 5 shows the calculated 8i(yJ isotherms for the binary mixture at three values of k (1.0,0.67, and 0.34). The specific amounts adsorbed of both components at any yi were the largest when the adsorbent was homogeneous (it = 1.0). Both 81 and 192 decreased at any yi as the adsorbent became more heterogeneous. Figure 6 shows the corresponding qi(yi) plots for the three values of k. The isosteric heats of adsorption of both components were constant over the entire composition range when the adsorbent was energetically homogeneous. The qi(ui) became stronger functions of yi as k decreased and the difference between the absolute value of qi and qi* at any yi became larger when the adsorbent was more heterogeneous. Thus, adsorbent heterogeneity played a very significant role on the component isosteric heats of adsorption from a mixture even though the relative selectivity of adsorption for the components remained the same. It should be noted here that the mixed-gas Toth model for heterogeneous adsorption used in this study is a rather simplified model. The assumptions of constant m and it
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(1)
2 -
(2)
1
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