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Comments on the artificiality of actual amount adsorbed Shivaji Sircar Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00852 • Publication Date (Web): 19 Apr 2018 Downloaded from http://pubs.acs.org on April 20, 2018

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Industrial & Engineering Chemistry Research

Comments on the artificiality of actual amount adsorbed Shivaji Sircar* Department of Chemical and Biomedical Engineering Lehigh University, Bethlehem, Pennsylvania 18015, United States

ABSTRACT

The artificiality of the actual amount adsorbed (AAA) estimated from the experimentally measured Gibbsian Surface Excess (GSE) of a gas is discussed. The use of AAA for calculation of isosteric heat of adsorption or mixed gas adsorption isotherms using Ideal Adsorbed Solution Theory can be misleading and incorrect. The GSE, on the other hand, can be unambiguously used to describe all practical aspects of adsorption science and technology.

INTRODUCTION

The true and only experimental variable to measure the extent of adsorption from a pure gas or a gas mixture is the Gibbsian surface excess (GSE). The specific GSE for adsorption of a pure gas ( , mole/kg) is defined by: 1 - 3 = ( − ) = ( − ); = = +

1 ACS Paragon Plus Environment

(1)

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

where (cm3/g) and (mole/cm3), are, respectively, the specific volume and the density of the of the adsorbed phase in equilibrium with a pure gas at pressure P and temperature T having a density of ρ (mole/cm3). The variable (mole/kg) is the specific actual amount adsorbed (AAA) at P and T. According to Equation 1, the GSE represents the amount adsorbed in excess of the amount that would be present in an adsorbed phase of same volume if that phase had the same density as that in the equilibrium gas phase (i.e. in absence of adsorption). Hence the name. The GSE framework of gas adsorption can be used to analytically describe all practical aspects of gas adsorption science and technology.3

- 9

These include multicomponent adsorption

thermodynamics, heats of adsorption, specific heat of adsorption, thermodynamic consistency tests for equilibrium adsorption isotherms, concept of ideal adsorbed solution for correlating pure and mixed gas GSE isotherms, adsorbate mass transfer into adsorbent particles, adsorption column dynamics, adsorption process design, adsorptive gas storage, etc. It may be important to point out that the mathematical formulations for all of these tasks using the GSE framework of adsorption are similar to those traditionally developed by using the AAA as the variable for describing the extent of adsorption. In fact, these two formulations are interchangeable by simply replacing AAA by GSE. Thus, the GSE framework of adsorption is inadvertently being used in the literature because the experimentally measured GSE is often inadvertently called the AAA.


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ESTIMATION OF ACTUAL AMOUNT ADSORBED (AAA) FROM GSE OF A PURE GAS Equation 1 indicates that the AAA of a pure gas [ (P, T)] can be estimated from the experimental GSE [ (P, T)] only if the volume of the adsorbed phase ( ) is known as a function of P and T. Since cannot be experimentally measured, the AAA is not an experimental variable. However, the AAA is approximately equal to the GSE of a pure gas ( ~ ) for the special case of adsorption of a strongly adsorbed pure gas at a very low pressure ( >> ρ). Otherwise, the difference between and can be substantial. Particularly for a weakly adsorbed gas at high pressures.4 Several authors have made various extra ordinary assumptions regarding the nature of the adsorbed phase in order to estimate the AAA of a pure gas from the experimentally measured GSE isotherm using Equation 1. The following two approaches are common: 10 - 15 (a) Pre- assume an analytical isotherm model in the AAA framework [ (P, T)], such as the Langmuir, the dual site Langmuir, the Sips, etc., and simultaneously, treat as an adjustable parameter, or assume that the adsorbed phase density ( = / ) is a constant. Then curve fit the pure gas experimental GSE isotherm [ (P, T)] to estimate the AAA model parameters and using Equation 1. (b) Assume that is equal to the entire specific pore volume of a porous adsorbent ( , cm3/g) under all conditions and calculate the AAA from the experimental GSE using Equation 1. A key issue with approach (a) is the pre-selection of an AAA isotherm. Different AAA isotherm models with a sufficient number of adjustable parameters and an adjustable can be used to curve fit a set of experimental GSE isotherms at different temperatures with reasonable accuracy. Consequently, the physical significance and reliability of the estimated AAA and the 3 ACS Paragon Plus Environment


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subsequent evaluation of the thermodynamic and physical properties using them will be ambiguous. Model dependent and anomalous behavior of isosteric heats of adsorption have been reported using this approach. 14, 15 Approach (b), on the other hand, allows estimation of the AAA without presetting its functional dependence on P and T. Thus, it is less restrictive. This assumption is popular for simulation of actual amounts adsorbed on porous adsorbents by molecular simulation techniques for gas adsorption.10, 11 A method called the Adsorbed Volume Mapping method (AVM) for estimating from the experimental GSE isotherm using the Density Functional Theory (DFT) has been proposed.16 However, the idea cannot be independently verified since cannot be measured experimentally. The main purpose of estimating AAA as functions of P and T for a pure gas has presumably been to utilize the conventional thermodynamic formulations, which are developed by using AAA as the primary variables to represent the extent of adsorption. Another reason is that the AAA is a monotonically increasing function of P, which permits unambiguous estimation of thermodynamic properties at high pressure. The thermodynamic applications of interest are: (a) estimation of the isosteric heat of adsorption ( , Kcal/mol) of a pure ideal gas [ ρ = P/RT] as a function of : 17 = RT2

(2)

where R is the gas constant. (b) estimation of mixed gas adsorption isotherms [ (P, T, )] from the corresponding pure gas AAA adsorption isotherms [ (P) at constant T] using the Ideal Adsorbed Solution Theory


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(IAST).18 The variable is the AAA of component i in equilibrium with a multicomponent gas phase (mole fraction of component i = ) at P and T.

ADSORPTION THERMODYNAMICS OF A PURE IDEAL GAS USING THE GSE FRAMEWORK

The GSE framework of adsorption can be used to estimate the iso-excess heat of adsorption ( , Kcal/mol) of a pure gas from the GSE isotherms at different temperatures by: 4,5, 7 = RT2

(3)

Equation (3) is identical to Equation 2 except that is replaced by . In other words, the iso- excess heat is equal to the traditional iso-steric heat when ~ . It should be noted that equation (3) can be practically applied only when increases monotonically with increasing P. Such is the case for most micro-meso-porous adsorbents of practical interest where the pure gas GSE isotherm at a supercritical temperature has a Type I isotherm shape by the Brunauer classification.17 It is characterized by a linear isotherm in the Henry’s law region [ = K(T) P as P → 0], and asymptotically approaching m as P→ ∞. The variable K(T) is the Henry’s law constant for the gas at T. The variable m is independent of T for an energetically homogeneous adsorbent.18 Similarly, the concept of an ideal adsorbed solution using the GSE framework created by mixing pure gases at constant surface potential (ϕ) and T, which is also the key postulation of the original IAST, yields identical correlations between pure gas and mixed gas isotherms as those in



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the original IAST.8, 19 The correlations are interchangeable by simply replacing the AAA with the GSE for the pure gas and the components of the gas mixture.8 The GSE ( ) of a pure gas is a function of P and T. Thus, a differential change in due to a differential change in P and T may be written using the chain rule of calculus as: d = dP + dT; =

; =

;

(4)

It follows from Equations (2), (3) and (4) that:

= − P [ ] ;

= − P [

( )

]

(5)

RELTIONSHIP BETWEEN and WHEN THE ADSORBED PHASE DENSITY IS A CONSTANT It is assumed that is not a function of at constant T. It, however, is a function of T. Equation 1 can then be rewritten for an ideal gas (ρ = P/RT) as: = (1 – βP); β =

(6)

where β is a function of T only. Equations (2), (3), (5) and (6) can be combined to get: [ − ]

!

$ = f(P) ["#

%& %

+

]; f(P) =

&

(7)

( & )

It should be noted that in the Henry’s law region (P→ 0) of the GSE isotherm, the quantity !

→ 1 and f(P) → 0, Thus, one gets: ' = '


(8)

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where ' and ' are, respectively the iso-excess and isosteric heats of adsorption in the Henry’s law region. They are identical. Furthermore, in the high- pressure region of a Type I isotherm, → 0, and f(P) → −1. Thus, Equations (6) and (7) yield: %& %

(

= −

;

% %

=+

( )

; %* = ( * − RT) > 0

(9)

where * and %* are, respectively, the iso -steric heat and the differential heat of a pure gas in the high- pressure region ( → m as P →∞). Equation (9) shows that increases with increasing T under the constraint of ≠ (P). Thus, decreases with increasing T at a faster rate than that for . Equations (7) and (9) can be reorganized to obtain: ,=[

) )

( ) )

]=

!. ! .

(10)

!.

! /( )0 .

where % [= ( - RT)] and % [= ( – RT)] are, respectively, the differential heats of adsorption of a pure gas under the GSE and the AAA frameworks. The variable %* is the limiting value of % at the limit of → m. The parameter , represents a dimensionless differential heat which is a function of the fractional GSE, 1 (=

). The limiting values of η

are, respectively, unity and zero at the limits of 1 → 0 and 1 → 1. RELATIONSGIP BETWEEN and WHEN 2 = 23 It is assumed that the total pore volume of the adsorbent ( , cm3/g) is not a function of P or T (structurally inert adsorbent). Then, Equation 1 can be written for an ideal gas as: = – αP; α =

4!

where α is a function of T only.


(11)


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Equations (20, (3), (5) and (11) can be combined to get:

! ) = [50 λ = ] ; α = )

!

4!

(12)

The parameter λ, defined by Equation (12), is another dimensionless differential heat which is a function of fractional GSE (1 ). The limiting value of λ in the Henry’s law region is given by 7' = [K/(α + K)], where K(T) is the Henry’s law constant of the gas at T. The parameter 78 is approximately equal to unity for most systems where K >> α, except for a very weakly adsorbed gas (see Table 1). On the other hand, λ → 0 in the high-pressure region, since → 0 as P→ ∞. Thus, %* = 0 [ * → RT] at that limit. This is a very interesting and unexpected result of assuming = . It may be seen from the definitions of η and λ that they are equivalent for the special case when %* = 0, which is satisfied only when is independent of both ρ and T. ARTIFICIALITY OF AAA DERIVED FROM GSE

Equations (10) and (12) show that there can be a substantial difference in the functional dependence of η and λ on 1 . In other words, there can be significant differences between the iso-excess heat ( ), which is evaluated using the raw GSE experimental isotherm data and Equation (3), and the isosteric heat ( ) which is evaluated by Equation (2) and the calculated AAA from the GSE isotherm by employing various model assumptions. The purpose of this article is to demonstrate the artificiality of AAA derived from a GSE isotherm of a pure gas. Consider an energetically homogeneous adsorbent where the heat of adsorption is not a function of the adsorbate loading. The classic multi-site Langmuir [MSL] model provides an analytical isotherm equation for describing a Type I GSE isotherm on such an adsorbent:19 8 ACS Paragon Plus Environment

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9

The variables b, a and [= - RT2(

%=

bP = [9] ; b = bo exp [ ] ; K = mb ; = %

:[ 9 ](;0

(19)

Thus, the maximum value of the extra gas storage capacity (UV ) offered by an adsorbentpacked vessel at temperature T is achieved at a gas density of V where the slope of the GSE isotherm at that temperature is positive and equal to [1/QR ]. In other words, V occurs at a density below the density where the GSE isotherm goes through a maximum [

= 0] and

where it is still a monotonically increasing function of ρ. The condition imposed by Equation (19) is valid for any shape of the GSE isotherm [Type I behavior or exhibiting a maximum]. Consequently, there is no problem in estimation of iso-excess heat using Equation (3) in the high- pressure region for the practical application of gas storage. It may be interesting to note that the skeleton density of many practical porous adsorbents (activated carbons, zeolites, silica and alumina gels) is ~ 2.1 g/cm3. Thus, the slope of the GSE isotherm at V ~ 0.5 cm3/g. An example of the density dependence of N described by Equation 18 is shown by Figure 8 for storage of CO2 in a silica-gel filled vessel.



Carbon Dioxide Storage on Silica Gel at 312 K 8 6

GSE

4

GSE , N, mmoles/g

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2 0 -2

0

5

10

15

20

25

-4

N

-6 -8 -10

Gas density, mmoles/cc

Figure 8. Plots of GSE and N vs ρ at 312 K for storage of CO2 using a vessel packed with Silica gel: Experimental GSE (circles), Smooth line through the data points (solid line); Estimated N vs ρ by Equation (18) (dashed line).

The Figure plots GSE and N as functions of ρ for storage of CO2 in a packed-vessel of silica gel (QR = 2.2 g/cm3) at 312 K. The GSE isotherm was reported in the literature.36 It may be seen from Figure 6 that the N vs ρ plot exhibits a maximum and then N monotonically decreases with increasing ρ. Eventually it becomes negative in the higher density region. Furthermore, the maximum occurs at a density which is below the density at which the GSE isotherm exhibits its maximum value. A key observation from Figure (8) is that N cannot be increased indefinitely by increasing storage gas pressure (or density). Furthermore, there is no advantage of using an adsorbent packed vessel for gas storage beyond ρ = V . The Figure also gives the pressure (density) when a compressed tank alone wins over the concept of adsorbed gas storage (N ≤ 0).


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The above-described discussions demonstrate that there is no problem for using Equation (3) in estimating from the GSE isotherms at different temperatures for all practical applications of high-pressure gas adsorption.

POTENTIAL ERROR IN MEASUREMENT OF GSE ISOTHERM AT HIGH PRESSURE

Helium at low pressure and high temperature is often used as a nonadsorbing gas for calibration of the void volume inside the adsorbent cell of an isotherm measuring device or for calibration of the adsorbent density. It has been shown that the adsorption of helium, albeit very small, introduces a significant error in calculation of the GSE at very high pressures.2, 37 The following correction must be applied: XYS (T*) ρ = GG +W

(20)

where is the true GSE of a pure gas at P and T after correction for finite adsorption of helium; GG is the apparent GSE of a pure gas at P and T before the correction and ρ is the

XYS (T*) is the Henry’s law constant for equilibrium pure gas density at P and T. The variable W XYS YS ] at temperature T* at which the void volume or the Helium GSE isotherm [YS =W

adsorbent density is measured. The correction term of Equation (20) can be fairly large compared to GG when the gas density is very high. In particular, the correction can be very large for a weakly adsorbed pure gas.37 Thus, is always larger than GG . Surprisingly, the

maximum observed in many reported apparent GSE isotherm data vanishes when the correction XYS (T*) from the available described by Equation 15 is applied.2, 37 A protocol for estimation of W helium isotherms at other temperatures is available.37



The helium GSE isotherm at 298 K on the DAY zeolite of Figure 7 is shown by Figure 9.31, 32 The same temperature was used to measure the adsorbent density. The Figure shows that the XYS (T*) was helium GSE isotherm is linear in the range of the data. The slope of the isotherm [W estimated to be 0.3934 cc/g.

GSE of Helium on DAY Zeolite at 298 K GSE, Helium, mg/g

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8 6 4 2 0 0

5

10

15

20

Gas Phase Helium Density, mg/cc

Figure 9. Adsorption of He on DAY zeolite at 298 K. Experimental data points (circles); Smooth line connecting the data points (solid line)

It may be seen that the apparent maximum and the negative GSE exhibited by the isotherm of Figure 7 vanishes (Dashed line) when the helium correction is applied. The corrected GSE isotherm for N2 has essentially a Type I shape. It should be mentioned here that the GSE isotherm of Figure 8 was not subjected to this correction. It should be emphasized that all published adsorption isotherm data in the literature are GSE. They are inadvertently called AAA and directly used to estimate properties like heats of adsorption, mixed gas adsorption characteristics by IAST, etc. In other words, the thermodynamic formulations in the Gibbsian excess frame work, which is analogous to those derived using AAA as primary variables, is already being used. Since the GSE framework of adsorption is fully capable of describing all practical aspects of adsorption science and


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technology, there is no need for estimation of AAA which can lead to misleading interpretation of adsorptive properties.

CONCLUSIONS

1. The actual amount adsorbed (AAA) cannot be experimentally measured. Estimation of AAA from the experimentally measured GSE isotherm requires extra-ordinary assumptions regarding the properties of the adsorbed phase which cannot be independently verified. Thus, the AAA is an artificial variable. 2. Calculation of the isosteric heats of adsorption and binary adsorption isotherms from pure gas adsorption isotherms using AAA can be misleading and severely incorrect. 3. Estimation of AAA from an experimentally measured GSE isotherm is dependent on the model used to describe the nature of the adsorbed phase. Hence all subsequent adsorptive properties estimated by using AAA are also model dependent. 4. The Gibbsian Surface excess (GSE) and the Iso-excess heat of adsorption are the only true and experimental variables for describing adsorption of gases. They can be measured unambiguously using macroscopic measurement techniques and they can be used for describing all practical thermodynamic, kinetic, column dynamic and process modeling issues of an adsorptive separation process. 5. The Iso-excess heat can be unambiguously estimated from the GSE isotherms at different temperatures in the region where the GSE is a monotonically increasing function of pressure (or density). Calorimetry must be used to measure the iso-excess heat at higher pressures where that condition does not hold.



6. The optimum highest- pressure levels for operation of most practical gas separation processes and gas storage in an adsorbent-packed vessel are generally in the region where the GSE is a monotonically increasing function of gas density, and thus, there is no problem in estimating iso-excess heats from the GSE isotherms for these applications. 7. The GSE isotherm of a pure gas ( vs ρ) can exhibit, in principle, unusual behaviors like local maximum and negative values in the high- pressure region. They, however, can be caused by ignoring helium adsorption during the measurement of the system void volume or the adsorbent density required for measurement of GSE.

AUTHOR INFORMATION

Corresponding Author *[email protected] Funding Sources: None Orcid: [email protected]/Sircar1958

TOC Graphic:


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Dimensionless Differntial Heats η and λ vs Dimentionless GSE for CO2 on Silicalite at 300 K 1 0.8

η, λ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


λ

0.6

η

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Dimensionless GSE

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Comments on the Artificiality of Actual Amount Adsorbed - American

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