Predicting the Integral Heat of Adsorption for Gas Physisorption on

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Predicting the Integral Heat of Adsorption for Gas Physisorption on Microporous and Mesoporous Adsorbents Peter B. Whittaker,† Xiaolin Wang, ‡ Wolfgang Zimmermann,§ Klaus Regenauer-Lieb,∥ and Hui Tong Chua*,† †

School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia ‡ School of Engineering, The University of Tasmania, Hobart TAS 7001, Australia § SPG-Steiner GmbH, Wittgensteiner Strasse 14, D-57072 Siegen, Germany ∥ School of Earth and Environment, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia S Supporting Information *

ABSTRACT: We have developed two predictive methods for the heats of adsorption that stem from isotherm models and benchmarked them against the Clausius−Clapeyron equation. These are the Tóth potential function model and the modified Clapeyron equation. Three adsorbate/adsorbent working pairs are used as examples: n-butane/BAX 1500 activated carbon, isobutane/BAX 1500 activated carbon, and ammonia/Fuji Davison type RD silica gel, all of which are examples of gas physisorption on adsorbents with both micro- and mesopores. Isotherms and corresponding integral heats of adsorption were measured in the range 298−348 K. For n-butane and isobutane, the pressures were up to 235 kPa, and for ammonia, the pressures were up to 835 kPa. Our two predictive methods consistently offer significant improvements over the Clausius−Clapeyron equation. Between the two predictive methods, the Tóth model is more robust across all three working pairs studied with predictions generally falling within 10−15% of the values of the measured heats.

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INTRODUCTION Most gas adsorption data in the literature are presented as experimentally obtained isotherm measurements without measured heats of adsorption. This is presumably because adsorption experiments that incorporate calorimetry are more expensive and more complicated to both set up and operate than experiments based upon volumetric or gravimetric measurements alone.1,2 However knowledge of the heat of adsorption can lend insight into surface phenomena.3 Additionally thermal management is an integral part of the design and operation of systems which make use of adsorption phenomena such as adsorption chillers,4,5 adsorbed gas storage tanks,6 and adsorption separation units,7 all of which require knowledge of the heat of adsorption. Conventionally experimentalists circumvent the dearth of calorimetric data by invoking the Clausius−Clapeyron equation, either through measuring many closely spaced isotherms and applying the equation directly to the data (to wit, the isostere method)8,9 or by fitting the isotherm data with an isotherm model and then applying the Clausius−Clapeyron equation to the model.10,11 The isostere method can lead to ambiguous results1,2,12 particularly if the number of isotherms obtained is low, if the isotherms are widely spaced, or where the uncertainty in the isotherm measurement is high. Moreover even at moderate pressure (p < 1 MPa) the ideal gas assumption inherent in Clausius−Clapeyron can lead to large errors in predicting the heat of adsorption.13 © 2014 American Chemical Society

Theoretical models (e.g., refs 14−17) have been developed for this problem by using numerous isotherm models consistent with statistical mechanics and specific to various different adsorbate/adsorbent modes of interaction. However it remains a challenge to know which model to apply to get a correct prediction of the surface interactions and heat of adsorption without significant prior knowledge of the behavior of the adsorbate/adsorbent system. Savara et al.18 developed a rigorous method for selecting between a number of 2D gas and Langmuir related models based on a van’t Hoff analysis of the expected entropy and enthalpy changes for the various models. However this method appears to be applicable only when the isotherm data can be described by a Langmuir isotherm, which is often not the case. In this paper we will offer two improved models to the use of the Clausius−Clapeyron equation so as to predict heat of adsorption based on isotherm measurements. The first method applies the Tóth potential function to the isotherm model; using literature data we had shown that this method was accurate to within ±10−15% of calorimetric values for predicting the isosteric heat of adsorption.19 The second Received: November 5, 2013 Revised: February 5, 2014 Published: March 11, 2014 8350

dx.doi.org/10.1021/jp410873v | J. Phys. Chem. C 2014, 118, 8350−8358

The Journal of Physical Chemistry C

Article

Because of the availability of data, by far the most frequently used reference state is the normal boiling point. Various methods exist of dealing with the thermal coefficient of limiting adsorption α. Common methods include the following: a. using an arbitrary constant value (often 0.0025 K−1)22 b. making α equal to the inverse of the absolute temperature23 c. making α a function of other physical properties such as the critical temperature and van der Waals volume11 d. if isotherm data at multiple temperatures are available, α can be a fourth fitting parameter10,24 Choosing a value for α prior to data fitting amounts to making an assumption about the behavior of the adsorbed phase volume; that there are several methods in common usage suggests that none of the methods are consistently useful or particularly physically meaningful. Letting α be a fitting parameter may produce a better isotherm fit than the methods that predetermine the value of α. However if the adsorbed phase volume is only weakly a function of temperature across the isotherms being fitted, then using a form of the D−A equation with α as a fourth fitting parameter risks the value of the fitting parameters being statistically indeterminate. As the three methods of predicting the heat of adsorption in this paper are dependent on obtaining a good isotherm fit and getting the correct fitting constants out of that isotherm fit, we follow Saha et al.25 in lumping the adsorbed phase volume vs together with the fitting constant W0, to form a new fitting constant n* and writing the D−A equation as

method is a modified Clapeyron equation, whereby we explicitly incorporate the full equation of state of adsorbing gas. To benchmark our two proposed methods of predicting the heat of adsorption against the Clausius−Clapeyron equation we have measured isotherms for three adsorbate/adsorbent working pairs: n-butane/BAX 1500, isobutane/BAX 1500, and ammonia/ silica gel. Additionally we have simultaneously and directly measured their corresponding integral heat of adsorption so that the predictions can be compared to calorimetric values. In this paper, all three predictive methods (namely, Tóth potential function model, Clapeyron equation, and Clausius− Clapeyron equation) involve deriving a model for the heat of adsorption from an isotherm model. The coefficients for each heat of adsorption model are then found from fitting the isotherm model to the isotherm data. Once the value of the coefficients has been determined from the isotherm fitting, the coefficients are then used in the corresponding heat of adsorption model to predict the heat. We emphasize that no fitting has been done with respect to the heat data and that the heat data are solely used as a benchmark against which the merits of all three methods can be compared. For n-butane/BAX 1500, isobutane/BAX 1500, and ammonia/silica gel, we have found that the Dubinin−Astakhov (D−A) isotherm equation fit the isotherms very well. Dubinin and Astakhov (D−A) developed the equation based on the theory of volume filling of micropores.20 While activated carbon and silica gel may contain mesopores as well as micropores, for energetic reasons the micropores are expected to fill first, and in fact the theory of volume filling of micropores was experimentally validated with data from adsorption onto activated carbons. It should be noted that the D−A isotherm has the following limitations: it cannot be used to describe low uptake (below 10 and 20% of the adsorbent micropore capacity), it should only be used to describe a family of isotherms for which the Gibbs free energy of adsorption can be considered temperature independent, and it is meant for adsorbate/adsorbent working pairs where the heat of adsorption is expected to decrease with increasing uptake.20,21

⎧ ⎡ RT ⎛ psat ⎞⎤m⎫ ⎪ ⎪ n = n* exp⎨ −⎢ ln⎜ ⎟⎥ ⎬ ⎪ ⎪ ⎩ ⎢⎣ ΔE ⎝ p ⎠⎥⎦ ⎭ s

This avoids both making assumptions about the behavior of the adsorbed phase volume and the statistical indeterminacy trap. As W0 was formulated to be the specific volume of the available micropores of the adsorbent, n* can be regarded as a maximum uptake capacity within that micropore volume. Previously, we have shown that it is possible to predict the heats of gas adsorption to within 10−15% by finding the Tóth potential function corresponding to an isotherm equation.19 The method can be summarized as

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THEORY The Dubinin−Astakhov isotherm equation is20 ⎧ ⎫ ⎡ RT ⎛ psat ⎞⎤m⎪ ⎪ W0 ⎢ ⎥ ⎬ n = − exp⎨ ln ⎜ ⎟ ⎪ ⎪ vs ⎩ ⎢⎣ ΔE ⎝ p ⎠⎥⎦ ⎭

qst = Δλ + λp + ZRT

s

(4)

where qst is the isosteric heat of adsorption and Δλ = RT ln(ψp /p) is the Tóth potential function, in which ψ = (ns/p)(∂p/∂ns)|T − 1 is Tóth’s correction factor to the Polanyi potential. The latent heat of condensation at the isotherm equilibrium pressure is symbolized by λp, and the gas compressibility factor is Z. Inserting the pressure explicit form of eq 3, sat

(1)

s

where n is the specific adsorbate uptake on the adsorbent, vs the specific volume of the adsorbed phase, R the universal gas constant, T the isotherm temperature, p the equilibrium pressure, and psat the saturation pressure of the adsorbate at the isotherm temperature. The fitting parameters are W0, ΔE, and m. According to the theory of volume filling of micropores W0 is the specific volume of the adsorbent micropores, ΔE the characteristic energy of the adsorption process, and m a coefficient related to the distribution of adsorption sites. For the adsorbed phase specific volume, which cannot be directly measured, the most commonly used model is the one first proposed by Nikolayev and Dubinin:21 vs = v0 exp[α(T − T0)]

(3)

1/ m ⎧ ⎪ −ΔE ⎡ ⎪ ⎛ n* ⎞⎤ ⎫ ⎜ ⎟⎥ ⎬ p = psat exp⎨ ln ⎢ s ⎪ ⎪ ⎩ RT ⎣ ⎝ n ⎠⎦ ⎭

into eq 4, taking the derivative with respect to uptake, and canceling terms leaves ⎤ ⎡ n* (1/ m) − 1 ⎥ ⎢ ΔE ln ns ⎛ n* ⎞1/ m qst = ΔE ln⎜ s ⎟ + RT ln⎢ − 1⎥ ⎝n ⎠ mRT ⎥ ⎢ ⎦ ⎣

( )

(2)

in which α is the thermal coefficient of limiting adsorption and v0 and T0 are the specific volume and temperature of the adsorbate as a liquid at a known reference state, respectively.

+ λp + ZRT 8351

(5)



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as the model for the isosteric heat. The compressibility factor Z is thermodynamically a function of uptake and temperature. Depending on the functional form of Z used it may be possible to integrate eq 11 analytically; if not it can always be integrated numerically. In the Supporting Information we show a complete derivation of eq 11.

as the Tóth potential function model for the isosteric heat of adsorption. The differential heat of adsorption can be found from the isosteric heat by subtracting out ZRT26 or simply RT if the gas phase is considered to behave ideally. An analytical expression for the integral heat corresponding to eq 5 could not be found; therefore numerical integration is necessary in making integral heat predictions with this model. For adsorption at pressures below the triple point we have found that for λp the heat of vapor−liquid phase change at the triple point gives good results and that heats of sublimation should not be used.19 The Clapeyron equation is a fundamental thermodynamic relationship between volume change, the pressure−temperature gradient, and heat released during phase change. For gas adsorption it is written as qst = T Δv

∂p ∂T

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EXPERIMENTAL SECTION Using the sensor gas calorimeter (SGC) constructed at the University of Siegen Germany, we have measured adsorption isotherms and integral heats of adsorption for pure (99.998%) n-butane and isobutane on the activated carbon BAX 1500, a wood based activated carbon manufactured by the Westvaco Corporation. Measurements were made at temperatures of 298, 323, and 348 K and over a range of pressures up to 235 kPa. Additionally we have measured adsorption isotherms and heats of adsorption for pure (99.998%) ammonia on Fuji Davison type RD silica gel at temperatures of 298 and 348 K and over a range of pressures up to 835 kPa. The apparent density of BAX 1500 is between 0.27 and 0.35 g·cm−3. The total pore volume and micropore volume are 1.29 cm3·g−1 and 0.50 cm3·g−1 respectively. The mean pore diameter is 1.32 nm, and the BET surface area is 1350 m2·g−1. The total pore volume and pore volume distribution are consistent with those reported by Wilhelm et al.;27 however, they reported a significantly greater BET surface area of 2173 m2·g−1. The physical properties of Fuji Davison type RD silica gel were studied extensively by Chua et al.28 They found the BET surface area to be 838 ± 3.8 m2·g−1 and the total pore volume and micropore volume to be 0.37 cm3·g−1 and 0.18 cm3·g−1 respectively. A sketch of the sensor gas calorimeter, for simultaneous measurements of gas uptake and heats of gas adsorption, is shown in Figure 1. It had previously been described by

(6)

ns

where Δv is the change in volume between the gas phase and the adsorbed phase. This is commonly simplified to the Clausius−Clapeyron equation by assuming ideal gas behavior for the gas phase adsorbate and negligible volume for the adsorbed phase, thus RT 2 ∂p p ∂T

qst =

(7)

ns

Taking the pressure explicit form of eq 3 and substituting it into eq 7 results in ⎛ n* ⎞1/ m qst = λT + ΔE ln⎜ s ⎟ ⎝n ⎠

(8)

as the Clausius−Clapeyron expression for the isosteric heat. Here λT is the latent heat of condensation at the isotherm temperature, which arises from the saturated pressure term in eq 3. The expression for the integral heat of adsorption corresponding to eq 8 is ⎡ ⎛ n* ⎞⎤ 1 qint = ns(λT − RT ) + ΔEn*Γu⎢1 + , ln⎜ s ⎟⎥ ⎣ m ⎝ n ⎠⎦

(9)

where Γ is the upper incomplete gamma function, or Γ (a,x) = −t a−t ∫∞ dt. In developing this model for the integral heat, RT x e t was subtracted from eq 8 rather than ZRT before integration, in order to be consistent with the ideal gas assumptions of Clausius−Clapeyron. Because eq 3 has no adsorbed phase volume model associated with it, this form of the D−A equation cannot be used with the classic form of Clapeyron’s equation (eq 6). We have accordingly modified Clapeyron’s equation by explicitly treating the gas phase adsorbate as a real gas, while neglecting the adsorbed phase volume. We will demonstrate that this results in a substantial improvement over Clausius−Clapeyron. Starting from the modified Clapeyron’s equation u

u

qst = Tvg

∂p ∂T

ns

(10)

where vg is the gas phase specific volume and applying eq 10 to eq 3 yields qst = λT

⎛ n* ⎞1/ m p vg s ⎜ ⎟ Z [ p ( n , T ), T ] E ln + Δ ⎝ ns ⎠ psat ΔvT

Figure 1. Schematic of a sensor-gas calorimeter. Adapted with permission from ref 2. Copyright 2003 Elsevier Science B.V.

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Zimmermann and Keller,2 and its features are merely summarized here. The sensor gas calorimeter setup consists of a standard volumetric measurement system for isothermal measurements and an enclosing sensor gas jacket for the measurement of heat transferred during adsorption processes. For the isothermal measurement of gas uptake, its construction and working principles are similar to the static volumetric measurement which has been described by Wang et al.24 The volumetric part of the sensor gas calorimeter consists of an adsorption vessel and a connected gas reservoir vessel. The pressures and temperatures in these two vessels are monitored by calibrated pressure transducers and temperature sensors respectively. The temperature of the gas reservoir is maintained at 25 °C by using a thermoelectric cooler/heater while the temperature of the adsorption vessel is held constant by means of a thermostatic bath. The adsorbent sample under investigation is loaded into the adsorption vessel and is heated prior to uptake measurements for possible outgassing and surface cleaning at a temperature of 423 K for around 3 h at a residual pressure of 10−3 Pa. The dead volume after loading the sample into the calibrated adsorption vessel is determined by expansion of helium gas, which is assumed to adsorb to no more than a negligible degree on the BAX 1500.29 The adsorbate is first charged into the gas reservoir from a gas cylinder, and the amount of gas is quantified by the measured pressure and temperature when the calibrated gas reservoir vessel reaches thermal equilibrium. Thereafter the valve labeled 5 in Figure 1 is opened such that a certain amount of adsorbate is introduced into the adsorption vessel and closed again after approximately 30 s. After the two vessels return to thermal equilibrium, the pressure and temperature of both vessels are recorded so that the adsorbent uptake on the BAX 1500 sample can be calculated as the difference between the reduction of adsorbate in the gas reservoir and the increase of adsorbate in the dead volume of the sample vessel. The corresponding adsorption pressure is monitored by the pressure sensor inside the adsorption vessel. During this measurement process, it is important that the velocity of gas flow is small enough to allow thermal equilibration of the gas temperature to thermostat temperature. The heat released during the adsorption process is measured via the sensor gas jacket which encloses the adsorption vessel. The sensor gas jacket is connected with a reference gas pressure vessel of identical volume. All the heat generated during the adsorption process must be rejected to the constant temperature bath via the sensor gas jacket. In consequence the sensor gas in the thermal jacket is heated and its pressure increases transiently. This temporary pressure rise is measured by a highly sensitive differential pressure manometer against the constant pressure in the reference vessel leading to an asymmetric peak signal. By calibrating the differential pressure response against a known energy input from a resistance heater in the adsorption vessel prior to adsorption measurements, the energy released from the adsorbate during the adsorption process can be accurately calculated from the area beyond the differential pressure signal peak. By stepwise increase of adsorbate pressure at constant temperature, isotherms can be obtained along with the simultaneous measurement of the differential heat released during the adsorption process. In the same way, the heats of desorption can also be evaluated by stepwise decrease of adsorbate pressure. See Zimmermann and Keller2 for a further explanation of how the SGC was calibrated.

We have performed a rigorous error propagation analysis, the methodology of which is presented in the Supporting Information. For the adsorption isotherm measurements at very low uptake the measurement uncertainty is about ±10% of the measured values, and as the pressure and number of measurement steps increase, the uncertainty reduces to ±5%. Similarly for the heat of adsorption measurements the uncertainty is ±5−10% of the measured value.

Figure 2. Adsorption isotherm data and D−A equation fits to that data. Fitting constants are given in Table 1. 8353



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Figure 3. Left column: measured heats of adsorption (symbols) and predicted heats (lines) for n-butane on BAX 1500. Right column: error in predictions of the integral heat for n-butane/BAX 1500. Error was calculated as error = 100 × (fitting value − measured value)/measured value.

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RESULTS

parameter model was deemed insignificant and therefore rejected. The results of the fitting are shown in Figure 2, and the fitting parameters are given in Table 1. Original data recorded for the pressure and uptake, as well as the integral heat data, are furnished in the Supporting Information. The corresponding integral heats of adsorption are presented in Figures 3−5. Overlaying the data are predictions made using the fitting parameter values from Table 1 and the three methods of predicting the heats of adsorption described in the Theory section (Tóth potential function heat model, modified

The isotherm data were fitted with eq 1 with each of the adsorbed phase volume models (eq 2 with the various methods of determining α) and with eq 3. Of the various D−A models eq 3 gave a better fit to the data (as determined by least-squares fitting and visual inspection) than eqs 1 and 2 with the various methods of determining α. Letting α be a fourth fitting parameter does lead to a very slight improvement in fit, but as eq 3 gave a fit that was generally inside of our estimated uncertainty of about ±5%, the improvement offered by the four 8354



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Figure 4. Left column: measured heats of adsorption (symbols) and predicted heats (lines) for isobutane on BAX 1500. Right column: error in predictions of the integral heat for isobutane/BAX 1500. Error was calculated as error = 100 × (fitting value − measured value)/measured value.

the dip in the integral heat. At 323 and 348 K the modified Clapeyron heat model best predicts the data. For isobutane/BAX 1500 (Figure 4) the Tóth heat model overall best predicts the data at all temperatures. For the heats measured at 323 K the prediction of the Tóth heat model agrees best with the data at low uptake (

Predicting the Integral Heat of Adsorption for Gas Physisorption on

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