Impact of the adsorbate compressibility on the calculation of the micropore volume

Carbon, Printed

Vol. 31, No. 7. pp. in Great Britain.

1015-1019.

1993

WJ8-6223/93 8 1993 Pergamon

Copyright

$6.00 + .I0 Press Ltd.

IMPACT OF THE ADSORBATE COMPRESSIBILITY ON THE CALCULATION OF THE MICROPORE VOLUME

Institute

M. M. of Physical

DUBININ, A. V. NEIMARK,” Chemistry of the Russian Academy

(Received

28 August

1991; uccepted

and V. V. of Sciences,

SERPINSKY 117915, Moscow,

in revised form 5 March

Russia

1993)

Abstract-The theory of micropore volume filling has been modified in taking into account the adsorbate compressibility. A new equation has been obtained for calculating the micropore volume from the experimental value of the limit of adsorption. The estimations, with the example of benzene adsorption on activated carbons, show that because of the adsorbate compressibility, the real micropore volume can exceed the values calculated from the conventional Dubinin-Radushkevich theory by about 10% at room temperature, and this difference increases with an increase in temperature. Kev Words-Adsorption, Rahushkevich theory.

micropores,

activated

1. INTRODUCTION

= a,u, ;

pore

the bulk

Physical adsorption in micropores whose size is comparable with the molecular sizes differs from physical adsorption on an open surface. The restricted geometry causes the overlap of adsorption potentials, which leads to the volume filling of micropores by molecules adsorbed. The theory of volume filling of micropores (TVFM) had been developed by Dubinin in a number of publications. The TVFM is a conventional basis for the characterization of active carbons[ 1,2]. In this paper we discuss the problem of micropore volume calculations. In the framework of the TVFM, it is supposed that limit micropore volume, W,, is related to the limit of adsorption, a,, at the saturation conditions when the adsorbate pressure, P, approaches the saturation pressure, P,, and the density of liquid-like adsorbate in micropore equals the density of bulk liquid adsorbate at given conditions, pS. In this way the micropore volume is calculated simply as a limit adsorption at saturation pressure, P = P,, multiplied by molar volume of liquid adsorbate, u,? = Ilp,:

W” = u,lp,

carbon,

(1)

that is, the limit of adsorption is equal to the ratio of the limit adsorption volume in micropores W, to the molar volume of liquid adsorbate at given temperature T and saturation pressure P,. In this paper we consider the alteration of the adsorbate density in micropores owing to the adsorbate compression in the field of adsorption forces. The compressibility leads to an increase of the average adsorbate density in micropores from *Author to whom correspondence should be addressed. Current address in 1991193: Inst. Anorg. & Anal. Chem., Johannes Gutenberg University, D-6500, Mainz, Germany.

volume,

value

pJ.

compressibility,

Therefore,

the

Dubinin-

definition

of the

due to eqn (I) demands this correction. In general, the value of molar volume of adsorbate in eqn (1) depends on the temperature T, and also on the characteristic energy E of adsorbateadsorbent interaction. In order to make the micropore volume definition more precise, it is necessary to introduce the correction factor Q-in eqn (1): micropore

volume

w,, = uou,,7. The correction factor 7 can be calculated from the general thermodynamical equation librium in an external field.

(2) starting of equi-

2. THEORY

The molecules adsorbed in micropores are in the variable external field of the adsorption forces[3]. As in the frameworks of TVFM, WC would suppose that at saturated conditions, P = P,, the adsorbate in the adsorption field is a liquid-like phase, but therein we would reject the assumption concerning its homogeneity. The spatial inhomogeneity of external field at the equilibrium conditions inevitably causes the spatial inhomogeneity of the adsorbate density p(Z) and of the pressure field P(S) (2 = radius-vector of the points in pore space). According to the main principles of thermodynamics, the chemical potential of the adsorbate, pn, answers the following equation of equilibrium in an external field[41: PO(Z) - E(k) = PS(T).

(3)

Here ~(2) is the potential of the adsorption forces equal to the potential of the adsorption field taken with the opposite sign. The potential ~(2.) does not depend on the external thermodynamical variable quantities, P and T, as well as on the quantity ad-

1016

M. M.

DUBININ

sorbed, a. In this way, an adsorbent is considered an inert solid with the constant volume of micropores Wa; pS(T) is the chemical potential of the adsorbate at P = P,. The chemical potential of the adsorbate, &.G), is a function of the local value of pressure P,(x) and temperature T. For the dependence p,(P,, T), the following thermodynamic relationship takes place: p.

P*,(P,,T) = /h(T) +

1u,(P,,WP,.

et al.

To carry out specific calculations by means of eqn (8), one has to employ, first, the physical-chemical data concerning the dependence of molar volume of adsorbate on pressure P, and temperature T, u,(P,,T), and, second, a particular distribution function of the adsorption potential (O(E). A preliminary estimation of the compressibility contribution can be done in an open form by using the linear approximation of the dependence u,(P,, T) for liquid adsorbate:

(4) u,(f’,,T)

PS Here u,(P,,T) is the adsorbate pressure P, and temperature T.

molar volume

at

Here & is the isothermal adsorbate compressibility.

- P,).

(10)

coefficient of the liquid Thus, we obtain

PO

P”W

I

= u,(T) - /3,u,UW’,

u,(P,,T)dP,

= E(Z).

(5)

PS

Thus, it follows that the local value of pressure P,(Z) and correspondingly the local value of the adsorbate density p,(Z) are stipulated by the local value of potential E(Z). By using a particular dependence u,. (P,,T), one can find the dependencies sought, P,(Z) = P,(E(?)) and u,(Z) = u,(e(Z)), by means of the relationship (5). The limit of adsorption u0 depends on the average density of adsorbate in the micropore volume, name1 y a, =

I

W,

d3x QF)’

.r

u,dP, = u,(TH(P,- PJ - ,&@'a- P,)*W p, = (US- u:m,4> (11) and in accordance u, = v, Vl

with eqn (5)

- 2&&h, = u,(l - P&u,).

Here in approximate equality, we have left out the parts of more than the first order of a bit. The corresponding local values of pressure and density are as follows: P, = P, + (II, - u,)Ip$l,

(6)

Here the integration is to be done over the micropore volume W, that could be replaced by the integration over the potential distribution. By introducing, as it has been doing in the potential theory of adsorption (PTA)[S], the differential distribution function of the potential of adsorption forces, +(a), the limit adsorption a,, can be calculated from:

(12)

p. = l/u, = P,U

Putting the (6), we have adsorption, compression

a0 = :

j

= P, + E/U,

+ PTEIU,).

(13)

(14)

relations obtained for U,(E) in the eqn got a new equation for the limit value of which takes into account the adsorbate in a linear approximation:

Cp(E)[l

+

ds = :

&&J,]

[l + /?Ti?/U,].

0

a0 = wo]&k$. 0 a

(15) (7)

Here, Z is the value of the adsorption potential averaged over the volume of pores

To obtain more convenient formulae for calculating, it is reasonable to transform the variable quantity E in the integral (7) to the new one P, using the relationship (5) and the following differential relationship ds = u,dP,.

(8)

In this way we obtain the equation for a0 in the form of

(9)

m

EC

I EL&) d&.

(16)

Thus, in linear approximation, the correction factor in eqn (2) depends on the average adsorption potential:

7-l = 1 + priYu,jr==

1 -&E/u,

(17)

More accurate calculations can be carried out using instead of linear approximation (10) one of the known empirical equations for the liquid adsorbate

Impact of the adsorbate compressibility molar volume un(P,, T), which are reliable for a wide range of pressure and temperature. For such adsorbates as, for example, benzene, the Tate equation seems to be applicable[6]:

on micropore volume

1017

which formally leads starting from the PTA to the main equation of the TVFM-Dubinin-Radushkevich (DR) equation for the adsorption isotherm, (22)

u,(P,,,T)

= u,(T)(I

- Cln(1

+ (P,, - P,,)iB(T))). (18)

Here, C is a constant that does not depend on temperature, and B(T) is a function of temperature with dimension of pressure. Usually, another form of the Tate equation is accepted, where the value of the liquid molar volume extrapolated to zero pressure is taken as the level of counting out[6]. However, for the purposes of given research, while considering the effect of compression at high pressures up to thousands of atmospheres, the Tate equation in the form of (18) does not lead to any appreciable error. For benzene C = 0.09377, and B(T) in the range of temperature from 10°C to 65°C varies from 1070 atm to 692 atm. From the Tate equation (18) and the equation of equilibrium (5), the algebraic equation follows for the determination of the overpressure, AP = P,, - P,. in the adsorbate phase and correspondingly for the difference between molar volumes, Au = u,, - u,, as a function of E:

The choice of distribution function in the form of eqn (21) is motivated by an apt correspondence of the DR equation to numerous experiments with microporous adsorbents, including activated carbons. Therein, the adsorbate-adsorbent interaction is stipulated quantitatively by the characteristic energy of adsorption E, and eqn (9) transforms to

exp[-(jo,dP,)1/E’]

_ exp( - E~/E*) ds = $

E,

(24)

0

I u,, dP, = p\

u,[( I + C) AP - CB( 1 + APIB) In(l + APlB)] = u,B](l

(23)

In the case of the distribution function (20), the average adsorption potential E is proportional to the characteristic energy of adsorption E,

P‘, &=

dP,.

+ C - AuIu,~)exp(AulCu.,)

and the limit of adsorption can be calculated approximation, using eqn (15):

in linear

- (1 - C)l. (19)

Equation (19) cannot be transformed into a linear form despite the expected small quantity of ratio Au/v, because of a small value for constant C. Substituting eqn (19) in eqn (9) leads to the correction factor in the form of

u,B

I 0

co(u,B](l

+ C)y - C(1 + y)ln(l

+ y)l)dy. (20)

While deriving eqn (20) for the convenience of calculations, the variable quantity has been made dimensionless:

The limit adsorption n,, at P = P, corresponds to the complete filling of the adsorption volume W, with the liquid adsorbate, but the consideration of its compression in the adsorption field leads to the increase of u0 compared with the value corresponding to the D-R theory (I). In this way, from the experimental limit of adsorption a,, one can calculate the volume of micropores W,, which, due to the effect of compressibility, tends to be less than the value corresponding to relationship (1). This shift is proportional to the product of the isothermal compressibility coefficient by the characteristic energy of adsorption. The correction factor T equals

y = APIB. 3. MODIFICATION

OF THE DUBININRADUSHKEVICH THEORY

To provide quantitative estimations of the adsorbate contribution, we would use as a distribution of potential a normal distribution,

(P(E) = g

exp( - &*/E*),

(21)

Let us estimate the correction obtained for the case of benzene adsorption on activated carbons at 20°C. Using a typical value of characteristic energy of adsorption E 2 20 kJ . mole-‘, molar volume of

M. M. DUBININ et al.

1018

benzene u = 0.9 . 10e4 m3 . mole-‘, and isothermal coefficient of compressibility /3r = 0.7 . 10e4 atm-’ (the last value corresponds to 20°C in the range of G pressures from 296 to 494 atm[6], we obtain 2 P&Iv, = 0.1, It means that at the benzene adsorption on activated carbon, the consideration of compressibility can lead to the corrections of order 10%. Therein, the average overpressure in the liquid-like adsorbate filling the micropores turns out to be G equal, p0 - P, = F/U, = ~Elv, = 2. lo5 kJ . rnw3 = 2000 atm. This estimation shows that the compressibility of adsorbate has to be taken into account in the analysis of adsorption in microporous adsorbents. This effect has a particular value at the adsorption conditions close to the critical ones, when the isothermal coefficient of compressibility, &., increases without any limit. Moreover, &depends on temperature rather strongly, and this fact can lead to the essential corrections of the thermal characteristics of adsorption. The Tate equation (18) for the relationship u, (P,, T) leads to the following equation for the correlation factor 7: m r-i=

1 +2&C

ln(1 +y)

I

0

[U

+

exp[-~~((1

C)Y - CO + Y)MI + Y)I * + C)y - C(1 + y) ln(1 + Y))~]dy. (27)

Equation (27) has been derived from the general equation (20) by using the distribution function (21). Here, we have introduced a dimensionless parameter, u(T)

=

h(T)B(T). E

(28)

For benzene adsorption on a microporous adsorbent with characteristic energy E = 20 kJ . mole-i, the values of correcting factor r,, have been calcu-

lated by means of eqn (27) at 10 different temperatures from 10°C to 65°C. Numerical values of B(T) and u,(T) were taken from the reference book[7]. With increase in temperature, the influence of compressibility on the calculated micropore volume increases: The correlated factor T decreases from r. = 0.9 at T = 10°C up to r = 0.87 at T = 65°C. It corresponds to the increase of the calculated limit of adsorption ao, comparing with the result of the D-R theory (l), in the first case of 11% and in the second case of 14%. 4. CONCLUSION For determining the micropore volume IV,,taking into account the adsorbate compressibility, it is necessary to use the following procedure: first, to determine the characteristic energy of adsorption E and limit value of adsorption in micropores a, via the standard method of treating adsorption isotherms in coordinates of the D-R equation; second, for the obtained value of E and given temperature T, to calculate the correction factor r via eqn (27); and finally, to calculate the volume of micropores via relationship (2). The results of numerical estimations, mentioned above, show that ignoring the adsorbate compressibility leads to overstating the micropore volume W,, calculated by means of D-R theory (for benzene adsorption on activated carbon at room temperature by 10% and more). REFERENCES 1. M. M. Dubinin, In Progress in Surface and Membrane Science (Edited by D. A. Cadenhead), Vol. 9, pp.

l-70. Academic Press, New York (1975). 2. M. M. Dubinin, Carbon 27, 457 (1989). 3. S. Brunauer, The Adsorption of Gases and Vapours. Oxford University Press (1945). 4. L. D. Landau and E. M. Lifshits, Statistical Physics. GITTL, Moscow (1951). 5. M. Polanyi, Verb. Dent. Phys. Ges. 16, 1012; 18, 55 (1914). 6. Reference Book of Chemist (Edited by B. P. Nikolsky), Vol. 1, p. 636. GITChL, Moscow (1951). 7. Phvsical-Chemical Prooerties oflndividual Hvdrocarbans (Edited by M. D. Tilicheev), Vol. 2,. p. 242. Gostoptechizdat, Moscow (1947).