Solid Adsorption Isotherm Equations

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

185, 228–235 (1997)

CS964562

Some Consequences of the Application of Incorrect Gas/Solid Adsorption Isotherm Equations JO´ZSEF TO´TH Research Laboratory for Mining Chemistry, Hungarian Academy of Sciences, H-3515 Miskolc-Egyetemva´ros, Hungary Received May 8, 1996; accepted August 18, 1996

Adsorption Equilibrium Data Handbook (D. P. Valenzuela and A. L. Myers, Prentice–Hall, Englewood Cliffs, NJ, 1989) includes many data on single gas isotherms excellently described by the To´th and UNILAN equations. This paper proves that in spite of the good mathematical applicability, the forms of the two mentioned equations—from a thermodynamic standpoint—are incorrect. As a consequence, the correct specific surface areas of the adsorbents cannot be calculated. The To´th equation can be transformed to a correct relationship and from this modified form correct values of the specific surface areas and isosteric heats of adsorption can be calculated. This correct form does not violate the old principle of the dynamic equilibrium of physical adsorption.

The well-known and thermodynamically correct Gibbs equation can be written in a special form,

* c( U )d U, 1

pr ( U ) Å

[1]

U

where U is the coverage and the function c( U ) has an implicit form defined as c( U ) Å ( U /p)(dp/d U ),

[2]

q 1997 Academic Press

Key Words: adsorption; thermodynamics; isotherm equations; surface area; isosteric heat.

where p is the equilibrium pressure, i.e., dp/d U is the differential quotient of the isotherm;

INTRODUCTION

pr ( U ) Å

D. P. Valenzuela and A. L. Myers published a handbook in 1989 (1) including data on 200 single-gas isotherms, binary gas isotherms, and binary liquid isotherms measured in laboratories over the past 30 years. In this respect this handbook is a great value indeed for both scientists and engineers interested in problems of physical adsorption. The authors of this handbook have described single gas isotherms by the UNILAN equation (2) and by the To´th equation (3). This latter equation has been applied to 80% of the single gas isotherms, and therefore, the author of this paper has received many questions of the same content: How can the fact that the specific surface areas calculable from both equations are much greater than those mentioned in the handbook and given by the producers of the adsorbents be explained? The answer is very simple: The forms of both equations applied in the handbook are, from a thermodynamic standpoint, incorrect. This statement is discussed in this paper. CONDITIONS OF THERMODYNAMIC CORRECTNESS

The two conditions of thermodynamic correctness of gas/ solid isotherm equations have already been defined (4); therefore, only a short summary is presented here.

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[3]

where p( U ) is equal to the function of the free energy of the adsorbent surface; and pid Å RT/ fm ,

[4]

where fm is the surface of the adsorbent covered by one mole of the adsorptive at U Å 1. One of the conditions of thermodynamic correctness is lim c( U ) Å 1,

[5]

Ur0

which is equivalent to the requirement that the isotherm equation should have a linear (Henry) section when p r 0 and U r 0. Myers was the first who declared this condition (5), and, therefore, it is evident that both equations applied in his handbook meet Eq. [5]. However, it is also evident that the relative free energy of the surface ( pr ) is limited from ‘‘above’’; that is, pr must have a finite value at U Å 1. This condition can be defined by Eq. [1], namely

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0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

p( U ) , pid

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INCORRECT ISOTHERM EQUATIONS

d pr Å c( U ). dU

[6]

Substituting this equation into Eq. [1] we get the original Gibbs equation in integrated form:

* Up dp. 1

Therefore, another condition of correctness is

pr ( U ) Å

[13]

U

lim c( U ) Å finite value.

[7]

UÅ1

Equation [7] is the condition which is not met by the forms of equations applied in the handbook mentioned above. DERIVATION OF CORRECT ISOTHERM EQUATIONS

The explicit form of the implicit differential relationship ( U /p)(dp/d U ) in Eq. [2] can easily be calculated if the isotherm equation, both in the form U Å f ( p) and in the form p Å f( U ), can be expressed explicitly. It is evident that in these cases the explicit function c(p) Å ( U /p)(dp/d U )

Quite similarly, it can be proved that Eq. [11] is also a direct consequence of Eq. [1]. Equations [10] and [11] possess all thermodynamic requirements defined by Gibbs’ thermodynamics, which has very important consequences as discussed below. However, here it is worth mentioning that if condition [5] is primarily fulfilled, then the isotherm equations derived from Eqs. [10] or [11] are thermodynamically entirely correct. CORRECTNESS AND INCORRECTNESS OF THE TO´TH AND UNILAN EQUATIONS

The forms of To´th and UNILAN equations applied in Valenzuela and Myers’s handbook are

[8] UÅ

can also be calculated. It follows from Eqs. [2] and [8] that for conjugated pairs ( U, p) of the isotherm equation,

n p Å , m (b / p t ) 1 / t

[14]

and c(p) Å c( U )

[9] UÅ

is valid. This fact makes it possible to express from Eqs. [2] and [8] two integral equations having the forms p Å exp

F*

U

1

G

[10]

,

[11]

c( U ) dU U

and U Å exp

F*

p

pm

dp c(p)p

G

where pm is the equilibrium pressure when U Å 1. Equations [10] and [11] are isotherm equations with the following properties. If the relationships c( U )/ U or [ c(p)p] 01 are analytically integrable functions then the explicit isotherm equation can be calculated. From Eqs. [1], [2], [8], and [9] it follows that Eqs. [10] and [11] are exact and rigorous consequences of the Gibbs equation [1]. To prove this statement let us write Eq. [10] in the following form:

F

n 1 c / pe s Å ln m 2s c / pe 0s

G

,

[15]

where n is the adsorbed amount (mmol g 01 ), p is the equilibrium pressure (kPa), and m is the monolayer adsorption capacity (mmol g 01 ) at p r ` . Both equations reduce to a Langmuir isotherm, the To´th for t Å 1 and the UNILAN for s Å 0. s, c, and b are constants; the latter two parameters characterize the average adsorptive potential of the surface and the parameter t depends on the heterogeneity of the adsorbents. According to Eq. [8], the function c(p) for the To´th equation is cT (p) Å

1 t p /1 b

[16]

and that for the UNILAN equation is cU (p) Å

F G

f ( p) f(p) f ( p) ln cap f(p)

,

[17]

where U c( U ) Å d ln p. dU

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a Å e s 0 e 0s

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[18]

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230 and

lim cT ( UT ) Å lim f ( p) Å c / pe s;

UTÅ1

f(p) Å c / pe 0s .

and lim cT ( UT ) Å 1.

1 1 0 Ut

[27]

UTÅ0

[20]

In order to better understand the thermodynamical correctness and the mathematical applicability of Eq. [23], some remarks should be made:

and cU ( U ) Å

2saU s

e /e

0s

0 [e s ( 2U01 ) / e 0s ( 2U01 ) ]

.

[21]

Based on Eqs. [20] and [21], the thermodynamic properties of the two equations can easily be defined. Since lim cT ( U ) Å lim cU ( U ) Å infinity, UÅ1

[22]

UÅ1

both equations are incorrect although the conditions

(a) Since we have proved that Eqs. [10] and [11] meet the Gibbs requirements, the derivation of any correct isotherm equation can therefore be carried out from these relationships; that is, adequate c( U ) or c(p) functions can only be substituted into Eq. [10] or [11] and not inversely. This rigorous mathematical rule is prescribed by Gibbs’ thermodynamics. (b) From point (a) it follows that a correct isotherm equation has real physical (thermodynamic) meaning only in the range p £ pm . This fact is mathematically emphasized by the limiting value of Eq. [23]:

lim cT ( U ) Å lim cU ( U ) Å 1 UÅ0

UÅ0

lim UT Å lim are met. In spite of these facts there is an important difference between the two equations. The function [ cT (p)p] 01 belonging to the To´th equation is an analytically integrable relationship; therefore, this equation can be transformed into a correct relationship but the UNILAN equation cannot. This transformation of the To´th equation by Eq. [11] has been shown in several papers (4, 6–9); the result of the integration in Eq. [11] is UT Å

S

xT b / pt

D

1/t

p,

[23]

where xT Å

b /1 p tm

and

UT Å

n . mc

[24]

Comparing Eq. [23] and Eq. [14] we have mc Å

m ; ( x ) 1/t

[25]

that is, the total monolayer capacity in the correct form (mc ) is always less than that in the incorrect form (m) because x 1T/ t ú 1. The entire correctness of Eq. [23] is evident because

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[26]

[19]

The functions c( U ) for the two equations are cT ( U ) Å

UTÅ1

xT Å finite value xT 0 U tT

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prpm

S

xT b / pt

D

1/t

p Å 1.

[28]

Equation [28] proves also the thermodynamic reality of Eq. [23] because in the range p £ pm the functions p( UT ) and c( UT ) are convergent (see Eq. [26]). This convergency is the most important demand of Gibbs’ thermodynamics. Evidently, Eq. [28] does not mean the mathematical inapplicability of Eq. [23]. On the contrary, in most cases Eq. [23] can be fitted excellently to the experimental data in the range p ú pm , too, but this application is only a mathematical formalism without any thermodynamic and physical meaning. (c) Beginning from Eq. [27], UT and UL give the correct value of the coverage when they are compared with U in the incorrect To´th and Langmuir equations. CONSEQUENCES OF APPLICATION OF AN INCORRECT AND A CORRECT ISOTHERM EQUATION

The only but most important problem of the application of Eq. [23] is the determination or calculation of the value of pm ( xT ), i.e., the equilibrium pressure where the total monolayer capacity is completed. When the specific surface areas of the adsorbents (a s ) are known from the producer’s data determined by the BET nitrogen method and checked by using surface area reference samples, the calculation of pm is very simple. The parameters m, b, and t in Eq. [23] can be calculated by a fitting procedure; the latter two param-

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INCORRECT ISOTHERM EQUATIONS

TABLE 1 Parameters of the Correct and Incorrect To´th Equation Applied to Isotherms of Carbon Dioxide and Ethane Measured on BPL Activated Carbon at Different Temperatures

Temperature K (7C)

Saturation pressure (k Pa)

xT

Parameter of the incorrect Eq. [14] m (mmol g01)

am (nm2)

a Calc. by the incorrect value of m (m2 g01)

as Given by the producer (m2 g01)

1.2163 1.2474 1.2382

12.9362 11.6389 10.0393

0.174 0.198 0.222

1356 1388 1342

988 988 988

1.0852 1.1050 1.0985

8.7611 7.8229 6.5130

0.235 0.269 0.300

1250 1267 1176

988 988 988

Parameters of the correct Eq. [23] pm (k Pa)

b (k Pat)

t

mc (mmol g01)

s

Carbon dioxide 212.7 (060.5) 260.2 (013.0) 301.4 (28.2)

437 2430 6930

172.4 1139 3653

0.6189 0.6503 0.7074

5.2386 24.0558 78.9266

9.4275 8.2847 7.4225 Ethane

212.7 (060.5) 260.2 (013.0 301.4 (28.2)

373.4 1720 4484

230.9 996.5 2667

0.3599 0.4008 0.5369

0.6040 1.6710 6.8055

6.9803 6.0981 5.4679

eters are independent of the value of the surface area. If a s is given (m2 g 01 ), then mc (mmol g 01 ) Å

as , 602.3am

[29]

where am is the molecular cross-sectional area in nm2 occupied by the adsorbate molecule in the complete monolayer. In calculating the value of am , it is assumed that the monolayer of molecules is a close-packed array and the molecules have a spherical form. In this sense we have q

am Å 0.02 3

S D M r q 4 2L

2/3

,

[30]

where L is Avogadro’s number, M is the molecular mass, and r is the specific volume of the liquid adsorptive at the temperature of the isotherm expressed in g and cm3 g 01 , respectively. From Eqs. [24] and [25] it follows that xT Å

S D m mc

t

[31]

and pm Å

S

b xT 0 1

D

1/t

.

[32]

The question connected with this method of calculation of the values pm and xT is an old problem: is the BET nitrogen method reliable or not? An IUPAC Committee Re-

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port (10) verified in 1985 that the nitrogen method checked by reference samples—in spite of the oversimplifications of the BET model—can be used as a reliable measurement. The author of this paper completes this IUPAC’s opinion with the statement that the BET nitrogen method is also reliable because the monolayer component isotherm of the BET equation is thermodynamically entirely correct (9). Taking the above-mentioned facts and statements into account, in Table 1 the parameters of the incorrect and correct To´th equations are applied to carbon dioxide and ethane isotherms measured on BPL activated carbon at different temperatures (5). Quite similar results are obtained when the correct Eq. [23] is applied to the data in the Valenzuela– Myers’ handbook (5). From the data in Table I the following conclusions can be drawn: (a) The correctness of Eq. [23] is expressed not only by the fact that the thermodynamic requirements defined by Eqs. [26] and [27] are met and the values of a s are identical to those given by the producers, but it means also that the values of pm calculated by Eq. [32] are less than the saturation pressures at any temperatures. According to the incorrect equations [14] and [15], the total adsorption capacities are completed at pm r ` , which is physical nonsense at lower temperatures than the critical ones. (b) The thermodynamic correctness and incorrectness are represented by the functions c( U ) in Fig. 1. The incorrect equations do not have finite limits at U Å 1. (c) Very important differences can be observed between the isotherms, plotted as coverage–equilibrium pressure functions, when both a correct and an incorrect equation are applied to this calculation. The reason for these differences is the different total monolayer capacities (m and mc ) in Table 1. These phenomena are represented in Fig. 2.

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FIG. 1. The functions c( U ) of correct and incorrect isotherm equations applied to carbon dioxide isotherms measured on BPL activated carbon at 28.27C. h, function cT ( U ) calculated by Eq. [20] (incorrect); s, function cU ( U ) calculated by Eq. [21] (incorrect); l, function cT ( UT ) calculated by Eq. [26] (correct).

The differences shown in Fig. 2 have great importance when these isotherms are applied to calculate the isosteric heats of adsorption (q st ) based on the Clausius–Clapeyron type equation having the correct thermodynamical form

S

Ì ln p ÌT

D

Å0 G

q st , RT 2

[33]

where G is the surface concentration, defined as

GÅ

nMa n Å s. A a

[34]

In Eq. [34] A is the real surface area (m2 ) of the adsorbent and Ma is its mass; therefore, n can only be applied in Eq. [33] instead of G if a s is a real physical value. Another requirement is that according to

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mc U, as

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GÅ

the ratio mc /a s must include the corresponding pair of values mc –a s ; i.e., it is a great mistake to calculate G by the m/a s ratio, where a s is a correct specific surface area but the value of m in Eq. [14] is incorrect. (d) It is evident that the value of mc , which is present in all equations in this paper, is not obtainable independently without the aid of the controlled BET method. Connected with this statement should be emphasized that the importance of Eq. [23] is not the independently obtainable value mc , but its advantages can be drafted as follows: (i) Equation [23] has proved that the coverage UT calculated with the aid of the BET surface area leads to convergent functions pr ( UT ) and c( UT ); that is, the widely used and well-known BET method is compatible with Gibbs’ thermodynamics. This statement can be made in other words: From isotherms equations containing divergent functions c( U ) or pr ( U ), surface areas approximately compatible to the BET areas (see Table I) cannot be calculated. (ii) Equation [23] is the first among the known equations which perfectly meet the requirements of Gibbs’ thermodynamics. (e) The greatest mistakes in the calculation of q st based on Eq. [33] will be made if condition [7] of thermodynamic correctness is not taken into account. This statement, in rela-

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FIG. 2. Coverage–equilibrium pressure isotherms of carbon dioxide measured on BPL activated carbon at different temperatures (5). s a , coverage b , coverage calculated by the incorrect value of m in Eq. [14]; h, 212.7 K ( 060.57C); ,, 260.2 calculated by the correct value of mc in Eq. [23]; s K ( 0137C); s, 301.4 K (28.27C); — , calculated by the To´th equation.

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FIG. 3. Isosteric heats of adsorption calculated from carbon dioxide isotherms measured on BPL activated carbon at 060.57C, 0137C, and 28.27C. The calculations are based on Eq. [33] but these are performed by Eq. [36] (incorrect, s ) and by Eq. [37] (correct, l ).

tion to Eqs. [14] and [23], means that the incorrect calculation of q st is based on the equation ln p Å ln U 0

F

1 1 ln (1 0 U t ) t b

G

;

[36]

however, the correct calculation can be performed by the equation ln p Å ln UT 0

F

1 1 ln ( xT 0 U tT ) t b

G

.

UÅ

[37]

The essence of the difference between the two equations is that by Eq. [37]

n p Å m bL / p

lim cL ( U ) Å lim

UTÅ1

UÅ1

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[38]

is also an incorrect relationship because

lim (ln p) Å ln pm Å finite value, but by Eq. [36] this limiting value does not exist. The consequence of this incorrect calculation of q st can be seen in Fig. 3. The correct calculation leads to a finite value of q st at UT Å 1; therefore, there are increasing differences between the correct and incorrect values of q st at coverages greater than 0.7.

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(f ) The last but very important problem can be stated as the following question: How does the incorrectness of adsorption isotherms influence the principle of dynamic equilibrium of physical adsorption? It is very easy to answer this question from the precedent of the old Langmuir (L) equation, because the concept applied here is of general validity. We have proved (4) that the original L equation

UÅ1

1 Å` 10U

[39]

The correct (modified) form of the L equation (4) derived from Eq. [11] is

UL Å

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n xL p Å , mc bL / p

[40]

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INCORRECT ISOTHERM EQUATIONS

where

where bL xL Å 1 / . pm

UT Å U( xT ) 1 / t .

[41]

The dynamic equilibrium equations relating to Eqs. [38] and [40] are ka (1 0 U )p Å kd U

[42]

ka (1 0 UL )p Å kd UL ,

[43]

[47]

Equations [42] and [43] and Eqs. [45] and [46] prove that both the incorrect and the correct forms of the isotherm equations discussed here do not violate the old principle of dynamic equilibrium but different specific surface areas (coverages) belong to the dynamic equations. We have proved in this paper that the correct values of a s correspond to the thermodynamically correct isotherm equations.

and ACKNOWLEDGMENT This work has been supported by the Hungarian Scientific Research Foundation (No. T 014924).

where, evidently, REFERENCES

UL Å UxL .

[44]

We get quite similar dynamic equilibrium equations relating to Eqs. [14] and Eq. [23], ka (1 0 U t )p t Å kd U t

[45]

ka (1 0 U tT )p t Å kd U tT ,

[46]

and

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1. Valenzuela, D. P., and Myers, A. L., ‘‘Adsorption Equilibrium Data Handbook.’’ Prentice–Hall, Englewood Cliffs, NJ, 1989. 2. Honig, J. M., and Reyerson, L. H., J. Phys. Chem. 56, 140 (1952). 3. To´th, J., Acta Chim. Acad. Sci. Hung. 69, 311 (1971). 4. To´th, J., J. Colloid Interface Sci. 163, 299 (1994). 5. Myers, A. L., ‘‘Proceedings of the Third International Conference, Sonthofen, Germany, 1989.’’ pp. 128–132. 6. To´th, J., Colloids Surf. 49, 57 (1990). 7. To´th, J., Acta Chim. Hung. 129, 30 (1992). 8. To´th, J., Colloids Surf. 71, 233 (1993). 9. To´th, J., Adv. Colloid Interface Sci. 55, 1–239 (1995). 10. Sing, K. S. W., et al., Pure Appl. Chem. 57, 603 (1985).

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