Treatment of Water by Granular Activated Carbon - ACS Publications

4 Theory of Correspondence for Adsorption from Dilute Solutions on Heterogeneous Adsorbents

Downloaded by CORNELL UNIV on May 18, 2017 | http://pubs.acs.org Publication Date: March 15, 1983 | doi: 10.1021/ba-1983-0202.ch004

A. L. MYERS University of Pennsylvania, Chemical Engineering Department, Philadelphia, PA 19104 S. SIRCAR Air Products and Chemicals, Inc., Allentown, PA 18105

The thermodynamics of adsorption from dilute solutions can be formulated analogous to that for adsorption from pure vapors. A principle of correspondence for adsorption of dilute solutes is derived. It is found that K functions [K = (mRT/ΔG) ln a] of different solutes coincide. Adsorp tion isotherms can be predicted from the K function of a reference solute. The constants of the theory are the saturation capacity of the adsorbent for the solute and the free energy of immersion of the adsorbent in the pure solute.

E

play an important role in the performance of adsorbers and chromatographs. Experimental deter mination of equilibrium data is tedious and costly. Design engineers are interested in the development of theories that can be used to predict equilibrium adsorptive data from a minimum of information. Several theories have been proposed for the calculation of multicomponent vapor-solid (1) and liquid-solid (2-4) adsorptive behavior. The starting point for all of these methods is either pure vapor adsorption isotherms or adsorption from binary liquid mixtures. Thus, the minimum information referred to is the adsorption of each component of a vapor mixture or, in the case of liquid adsorbates, the selectivity of the solid surface for each solute-solvent pair. QUILIBRIUM PROPERTIES O F ADSORBENTS

0065-2393/83/0202-0063$06.00/0 © 1983 American Chemical Society

McGuire and Suffet; Treatment of Water by Granular Activated Carbon Advances in Chemistry; American Chemical Society: Washington, DC, 1983.


64

TREATMENT OF WATER BY GRANULAR ACTIVATED CARBON

In a search for a simpler approach, several workers (5-9) tried to generate "characteristic adsorption isotherms" for the vapor-solid interface, so that isotherms of different vapors can be coalesced into a single curve for a particular adsorbent The idea is to search for certain constants called affinity coefficients that can be used to define new, reduced adsorptive properties. If such a characteristic curve could be defined and if the affinity coefficients could be correlated with the properties of the adsorbate, then it would be possible to predict adsorption isotherms using a single measured isotherm of a reference substance. This process would provide an enormous simplification of the described design problem. The procedure of finding affinity coefficients that will produce a characteristic isotherm for different adsorbates is empirical. The underlying Polanyi potential theory (10) is not actually a theory because its derivation is based upon so-called equipotential surfaces whose existence can be neither proved nor disproved. In this work, the objective of correlating experimental data is the same but the theory is based upon specific assumptions about the interactions of adsorbate molecules with the surface. Instead of adjustable parameters like affinity coefficients, the reducing variable is a thermodynamic quantity: the free energy of immersion. If our result is called a characteristic adsorption isotherm, it will be classified as another in a long line of extensions of the Polanyi theory (11). The Polanyi theory has indeed been useful, but after nearly 70 years of study its applications probably have been exhausted. To avoid the empirical connotation associated with Polanyi plots and characteristic isotherms, the approach discussed here is called a principle of correspondence because of its similarity with the principle of corresponding states for pure fluids. In both cases, the intention is to cast the volumetric properties of a fluid (adsorbate) in terms of dimensionless functions that are the same for all fluids (adsorbates). Recently (8), we proposed a principle of correspondence for adsorption of pure vapors. Extension to the case of adsorption from liquids is complicated because of competition of the solvent for available surface area. Neglect of this competition is justifiable when the solvent is sterically excluded from the surface of the adsorbent, e.g., adsorption of water on 3A molecular sieve from a dilute solution of water in cyclohexane (12). In general, however, solvent molecules compete effectively for the surface even when the solute is much more strongly adsorbed than the solvent, because the solvent is present in relatively high concentration. Consider, for example, the mole fractions of solute and solvent in the adsorbed and bulk liquid phases in Table I. Here, the solute is much more strongly adsorbed than the solvent as indicated by a selectivity of 4000: 0.8/0.001 ^2/^20

0.2/0.999

= 4000


4.

MYERS AND SIRCAR

Adsorption from Dilute Solutions

65

Table I. Typical Concentrations in Bulk Liquid and Adsorbed Phases for Preferential Adsorption of a Dilute Solute from a Liquid Solvent Adsorbed Phase, Xj


Solute 1 Solvent 2

0.8 0.2

Liquid Phase, x

i0

0.001 0.999

but comparable amounts of solute and solvent are present in the adsorbed phase. Three assumptions will be utilized in the theoretical development of the conditions necessary for the existence of a principle of correspondence or a "universal" adsorption isotherm for different vapors adsorbed on a particular adsorbent: 1. The solute is only sparingly soluble in the solvent, e.g., no more than a few mole percent. 2. The solute is strongly adsorbed relative to the solvent so that *1,2 »

I-

3. The surface of the adsorbent is heterogeneous. These conditions are usually satisfied in practical separation problems for the separation of trace impurities such as dehydration of hydrocarbons and purification of water containing organic substances. It will be shown that these systems can be treated as the adsorption of a single component (the solute) by subtracting the properties of the solvent from the adsorption isotherm. This procedure permits the adsorption of single solutes to be treated in a manner analogous to the adsorption of single vapors. Therefore, procedures developed for the vapor-solid interface (8) maybe, applied to adsorption from liquids. First, it is necessary to consider the equations of thermodynamic equilibrium for adsorption from dilute solutions.

Thermodynamics of Adsorption: Excess Properties The Gibbsian model of the liquid-solid interface is adopted (13). The surface excess for adsorption of Solute 1 or Solvent 2 is: nf = n(x - x ) t

(1)

i0

where n is the total amount adsorbed per unit mass of adsorbent, x is the mole fraction of solute or solvent in the adsorbed phase, and x is the (

i0


66


corresponding mole fraction in the bulk liquid phase at equilibrium. It follows from the definition that: nf + n| = 0

(2)

so that a positive surface excess of solute (nf) implies an equal in magnitude but negative surface excess of solvent (n|). Excess Gibbs free energy for adsorption from solution is defined as:


A G * = n(g - go) -

Znf/tf

(3)

where g is the molar Gibbs free energy of the adsorbed phase and g is the value for the equilibrium liquid solution. Excess properties defined this way are consistent with the definition of surface excess in Equation 1. The terms g and go refer to different compositions, and the purpose of the summation term is to cancel the standard-state chemical potentials (/$ which arise due to this difference in composition. AG is always negative for the spontaneous process of adsorption. Similar excess functions can be defined for enthalpy and entropy. The Gibbs free energy of the adsorbed phase is defined by (10): 0

e

G = ng = yA +

+ n fi 2

2

() 4

Thus, the free energy of the adsorbed phase can be considered as the sum of contributions from the solid surface (yA), the solute ( n ^ ) and the solvent (n /ia). In Equation 4, y is the surface tension of the liquid-solid interface, and A is the surface area per unit mass of adsorbent. In the case of microporous adsorbents, the meaning of A loses its significance but the product (yA) is always a measurable quantity (14). The molar Gibbs free energy of the bulk liquid is: 2

go = *ioMio +

Z20M20

(5)

Because of equilibrium: /*! = / i

1 0

and /x = 2

M20

( ) 6

Substitution of Equations 4-6 into Equation 3 yields: A G * = yA + fiK/ii - Mi) + n|(/i2 - /4)

(7)

For adsorption of pure solvent, nf = n| = 0 and: A G = yoA


(8)

4.


MYERS AND SIRCAR

67

where y refers to the surface tension of the solvent-solid interface. To subtract the properties of the solvent from Equation 7, A G * is redefined relative to the adsorption of pure solvent: 0

A G * = (y - y )A + nf(/ii " Mi) + nS(/*2 " ^

( ) 9

0

so that A G * = 0 for adsorption of solvent alone. According to the assumption that the solute is present at low concentration, the mole fraction of the solvent is close to unity and JLL — ju.?- Equation 9 reduces to: 2

AG" = (y — y ) A + nftMi - ML)

(10)


0

Since the solute concentration is assumed to be less than a few mole percent, its activity is given by Henry's law: jd! = fii + RTlnfl!

(11)

where:

and y", the activity coefficient at the limit of infinite dilution, is assumed to be constant up to the mole fraction at saturation, xf\ Equation 10 becomes: A G * = (y - y )A + n^RTln^

(13)

0

By using the fact that x

20

— 1, it follows from Equations 1 and 2 that:

n* = -nf, = ~n(x

~ x)

2

20

= nx = n x

Y

(14)

Thus, the surface excess of solute, nf, is the actual amount of solute in the adsorbed phase, n so Equation 13 may be written as: h

A G * = (y - y )A + n ^ T l n ^ 0

(15)

The surface tension of the interface may be calculated from the Gibbs adsorption isotherm (10): —Ady — riidfii + n d[i 2

2

(16)

Since the selectivity of the adsorbent for the solute relative to the solvent, Si , is large, the magnitude of the ratio: 2


68


n dfi 2

_

2

x d\na 2

riidfii

2

1 x dlnx

_

20

s x dlnx

Xidlnai

10

1 dx

_

20

1_

20

s

10

dx

s

i0

is small and n dn^ may be neglected in comparison with n^/ii. Therefore, Equation 16 reduces to: 2

(17)

—Ady = n d\ii = rtiRTdlnai x


Integration of Equation 17 from immersion in pure solvent to a particular loading of solute gives: -A(y

— y ) = RT \ Jni

(18)

n^Xna,

0

=0

Substituting Equation 18 into Equation 15 gives: AG /RT= e

n^na! -

/

'

n^ln^

(19)

Integration by parts of the integral in Equation 19 gives the simple result: AGVRT

=

f

lniiidm

ni

Jni

=

(20)

o

In spite of the discontinuity of the integrand at the lower limit, this improper integral exists and is a well-defined thermodynamic quantity. For example, at very low solute loading, Henry's law for adsorption (15) is obeyed: Hi =

Kdi

and in this range the integral is: AGVRT

= K /

1

J x=0

Inxdx = K(a \na 1

l

- a ) = n ^ l n a ! - 1) x

From Equation 15, it is found that at saturation of the solute (a = 1): Y

lim AG = ( y e

sat

- y )A = A G 0

(21)

so that A G * at saturation is the free energy of immersion of the adsorbent in pure liquid solute, relative to the free energy of immersion in pure solvent Both free energies are negative and since the solute is more strongly adsorbed, A G is also a negative quantity. We propose that, at saturation of the solute, the last of the solvent is displaced from the


4.

MYERS AND SIRCAR


69

adsorbent surface due to the immiscibility of the solute-solvent pair and the preference of the surface for the solute. Therefore, a fractional filling of the micropores of the adsorbent may be defined: (22)

6 = n /m l

l

where m is the saturation capacity of the adsorbent for the solute. The fractional filling of the micropores by solute varies from zero to unity, while the portion of the surface not filled by solute contains solvent molecules. The solvent molecules are displaced by solute during the loading process. Downloaded by CORNELL UNIV on May 18, 2017 | http://pubs.acs.org Publication Date: March 15, 1983 | doi: 10.1021/ba-1983-0202.ch004

x

Surface Heterogeneity The final and most important of our assumptions is that the adsorbent is heterogeneous (16). If the surface has a distribution f(E) of energies such that adsorption of a solute molecule on a portion of the surface with energy E is: (23)

0 = 0(a T,E) l9

then £ is a positive quantity corresponding to the decrease in energy for the transfer of one mole of solute from the liquid phase to the adsorbed phase. Thus, 9(a T, E), the local adsorption\isotherm, is distinguished from 9(a ,T), the overall adsorption isotherm for the entire surface. During adsorption of solute, solvent molecules must be displaced so E refers to the energy exchange process of replacing solvent molecules by solute molecules. The total adsorption isotherm is given by an integration of Equation 23: u

Y

P

0(a T)= u

J

£

m

"

6( ,mf(E)dE ai

(24)

min

where / (E) is the energy density function and is normalized:

f

Emdx

J

£

f(E)dE = 1

min

It has been shown (8) that when the dimensionless dispersion of the energy distribution: O" =

RT

is large, as is the case for adsorbents such as active carbon and silica gel, the adsorption isotherm has the form:


(25)

70


-RT\na

{

= F~ (l

- 6)

l

(26)

where F~ is the inverse function of the cumulative energy distribution function: l

F(z) = £fa)dz

(27)

and:


z = E-

£

(28)

m i n

Equation 26 may be written in the form: ln =/(7V ic d

CD

•£•5

16

o U

o

B

CO 00 l> i—i

O

o o u

i—i

O

o od d 00

»—i

co io oo oq co

—i

Tf

CO o

I

5

o CO

O J8

cq o H

l>

O

6

co ^

O

I

IO ^ CO

io O

CO

3 o o u

£

I g I


MYERS AND SIRCAR



4.


75

76


which is the ratio of the average molar free energy of adsorption to the thermal energy, is proportional to the strength of adsorption of the solute.


Literature Cited 1. Sircar, S.; Myers, A. L. Chem.Eng.Sci. 1973, 28, 489. 2. Jossens, L.; Prausnitz, J. M.; Fritz, W.; Schlünder, E. U.; Myers, A. L. Chem. Eng. Sci. 1978, 33, 1097. 3. Minka, C.; Myers, A. L. AIChE J. 1973, 19, 453. 4. Sircar, S.; Myers, A. L. AIChE J. 1973, 19, 159. 5. Grant, R. J.; Manes, M.; Smith, S. B. AIChE J., 1962, 8, 403. 6. Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad, Nauk SSSR 1947, 55, 331. 7. Lewis, W. K.; Gilliland, E. R.; Chertow, B.; Cadogan, W. P. Ind. Eng. Chem. 1950, 42, 1362. 8. Myers, A. L.; Sircar, S. AIChE J. 1981, in press. 9. Potter, C.; Sussman, M. V. Ind. Eng. Chem. 1957, 49, 1763. 10. Aveyard, R.; Haydon, D. A. "An Introduction to the Principles of Surface Chemistry"; Cambridge University Press: London, 1973; pp 12, 15, and 166. 11. Polanyi, M. Verh. Dtsch. Phys. Ges., 1914, 16, 1012. 12. Sircar, S.; Myers, A. L.; Molstad, M. C. Trans. Faraday Soc., 1970, 66, 2354. 13. Sircar, S.; Novosad, J.; Myers, A. L. Ind. Eng. Chem. Fundamentals 1972, 11, 249. 14. Bering, B. P.; Myers, A. L.; Serpinsky, V. V. Dokl. Akad. Nauk SSSR, 1970, 193, 119. 15. Young, D. M.; Crowell, A. D. "Physical Adsorption of Gases"; Butterworths: London, 1962; pp. 64-70, 137, and 167. 16. Zolandz, R. R.; Myers, A. L. In "Progress in Filtration and Separation" Wakeman, R. J., Ed.; Elsevier: New York, 1979; pp. 1-29. 17. Langmuir, I. J. Amer. Chem. Soc., 1918, 40, 1361. 18. Brunauer, S.; Emmett, P. H.; Teller, E. J. Amer. Chem. Soc., 1938, 60, 309. 19. Toth, J. Acta Chim. Acad.Sci.Hung., 1971, 69, 311. 20. Myers, A. L.; Sircar, S. J. Phys. Chem., 1972, 76, 3415. 21. Gurvitsch, L.J.Phys. Chem. Soc. Russ. 1915, 47, 805. 22. Wohleber, D. Α.; Manes, M. J. Phys. Chem., 1971, 75, 61. RECEIVED for review August 3, 1981. ACCEPTED for publication June 29, 1982.


Treatment of Water by Granular Activated Carbon - ACS Publications

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