A Thermodynamic Theory of Adsorption - Advances in Chemistry (ACS

A Thermodynamic Theory of Adsorption L. E. COPELAND Research Department, Portland Cement Association, Skokie, Ill. T. F. YOUNG

Downloaded by UNIV OF MONTANA on December 13, 2014 | http://pubs.acs.org Publication Date: June 1, 1961 | doi: 10.1021/ba-1961-0033.ch037

George Herbert Jones Chemical Laboratory, The University of Chicage, Chicago, Ill.

The concept of a monodisperse liquid adsorbent has been used in the development of a nearly rigorous thermodynamic theory of adsorption. It is found to be instructive to discuss changes in integral (as well as partial molal) properties of the components of an adsorption system. Changes in the partial molal properties of the adsorbent (associated with changes in surface energies) are significant, and some are large. It is predicted that changes in the escaping tendencies of solid adsorbents will be several times as large as values calculated from adsorption isotherms without an allowance for failure of the solid adsorbent to attain equilibrium with itself.

A finely divided adsorbent is inherently unstable. Very small crystals in a powdered crystalline mass disappear as it ages. Eventually, especially, at high temperatures, a few relatively large crystals or even a single crystal remain. A spray of fine liquid droplets becomes a set of comparatively large drops or may coalesce to a single mass (13, 19). The thermodynamics of finely divided crystals is complicated by the fact that parts of even a single crystal are not, in general, in equilibrium with one another; the rigidity of a crystal may prevent the internal adjustments essential to the equalizing of the escaping tendencies of material in its surface and of material in the interior of the crystal. The thermodynamics of a drop of liquid is comparatively simple. Surface tension produces a pressure which increases the partial molar free energy at any point within the drop. The escaping tendencies of material within the drop and of material in its surface are thus equalized. Considerations of this equilibrium have been recognized as fundamental in classic discussions of the thermodynamics of surface tension (18) and in investigations of pressures within, and the shapes of, bubbles and drops (2, 22). 348

In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

COPELAND AND YOUNG

349

Thermodynamics of Adsorption

If the adsorbent is known to be a monodisperse liquid (9, 14), the thermody namics of adsorption is also comparatively simple. It is simplified by the fact that we can conceive of the existence of complete equilibrium of the adsorbent in all parts of an adsorption system and can indulge in thermodynamic reasoning with out mental reservations concerning slow exchange of adsorbate or adsorbent be tween particles of different sizes and shapes. Thermodynamic Processes


In general, the thermodynamic processes to be considered are of two general types, integral and differential. Two integral processes of frequent occurrence may be written as follows (4, 5, 12, 15) : n{Ki

+

—*• [system consisting of 7 2 1 X 1 and 7 2 2 X 2 ]

722X2

[System consisting of « 1 X 1 and

722X2]

+

(1)

ni ' X 2 —• [system consisting of « 1 X 1 and 7 Z 2

, /

(2)

X2]

The differential processes most frequently encountered in adsorption are the following: «2X2 722X2

—• 7 2 2 X 2 (adsorbed on X i , ni'/n\ = constant)

(gas)

(liquid)

—• 7 2 2 X 2 (adsorbed on X i ,

« 'Ai 2

(3)

= constant)

(4)

The values of n and n of Equations 3 and 4 are not related to the values of n and n of either Process I or Process 2, nor are they necessarily equal for Proc esses 3 and 4. Equation 3 represents the adsorption of n mole of gaseous X , the adsorbate, on an indefinitely large amount of X which already holds n mole of adsorbate for each η mole of adsorbent. The mole ratio n /n is, of course, unchanged—i.e., increased only infinitesimally. Equation 4 represents a similar differential process: the disappearance of n mole of liquid X and its reappearance in the infinitely large adsorption system. The free energy change for this differential process is important in the discussions that follow. It is readily calculated from the ratio of the fugacities of liquid X and of X in the equilibrated adsorption system. If gaseous X is a nearly perfect gas, the free energy change for the process is: 2

2

2

2

2

2

1

2

λ

2

±

2

2

2

2

2

AG (Process 4) = 72 A(J = Τ2 RT In ^ 2

(5)

2

2

From a series of adsorption isotherms (vapor pressure vs. n /n^) the differential heat of adsorption may be determined for both of the differential processes. Equation 1 represents the adsorption of n moles of X on a finite amount, n moles, of adsorbent to produce the system containing n± moles of X and n moles of X . Equation 2 represents the addition of a specified finite number of moles of adsorbate, n , to a finite system containing η moles of X i and n moles of X . Since both n and the system are finite, the composition of the resulting system differs from that of the initial system. If it were possible to work with monodisperse liquid adsorbent, it would be important and interesting to compare values of thermodynamic properties de termined by measurements of different types. Agreement would indicate correct ness of the methods and their interpretation. It is more important to compare values of thermodynamic properties deter mined in different ways when solid adsorbents are used. Most adsorption experi ments are made with adsorbents consisting of small solid particles of varying sizes and shapes. Complete equilibrium in such systems is not possible, but for many 2

2

2

x

2

1

2

2

λ

2

2


2

350

ADVANCES IN CHEMISTRY SERIES

purposes it may be possible to disregard that fact. The most important available test of the possibility of such a departure from exactness is the comparison of re sults* of methods which differ from each other in the essential approximations in corporated within them. Probably the most significant comparisons which can be made are of values of properties determined from calorimetric measurements with values calculated from adsorption isotherms. Two general methods are available for the comparison of values of enthalpies determined from experiments of the two types. One involves two differentiations: The change in the partial molal enthalpy, ΔΗ , of X , for Process 4, is determined from the differentiation with respect to n /n of the inte gral heat of adsorption measured in a series of calorimetric experiments of the type represented by Equation 1. The values of the differential heats of adsorption (heats corresponding to the differential Process 4) are compared with values de termined from the temperature variation of AG /T for a series of values of n /n in Process 4. This type of comparison has been made successfully by several groups of authors (3, 5, 10). The other method for the comparison of enthalpies depends upon the integra tion of Equation 5 as indicated by Equation 6 for each of a set of adsorption iso2


2

2

x

2

I JO

AG =

2

r

(6)

Gidn

2

therms to determine a corresponding set of integral free energy changes for Proc ess 1. For a constant composition, AG/Τ is then differentiated with respect to temperature to yield an integral AH for Process 1. The calculated integral AH may then be compared with the integral AH measured directly in the calorimeter. This method, apparently, has been used very little, but is used by W u and Copeland (20). Changes in Integral Free Energy There is no general method for the determination of changes in integral free energy for a process such as Process 1. Incidental to a comparison of integral en thalpy changes, a series of integral free energy decreases has been calculated (20) for this process: 72iBaS0 + n H 0 4

2

2

—*• [system consisting of wiBaS0 and

rc H 0]

4

2

(7)

2

The plot of AG (Process 7) is shown in Figure 5 (20) for n /n varying from 0 to 0.03. The curve was calculated by Equation 6 from an adsorption isotherm from which the free energy change, n AG , of Process 4 had been derived. Shown also in Figure 5 is a plot of n AG . The large difference between the n AG and the integral free energy change is an indication of large changes in the partial molal properties of the adsorbent. 2

2

2

x

2

2

2

2

Partial Molal Properties of Adsorbent Little has been written about changes in the escaping tendency of adsorbents, and in some treatments of adsorption the effect has been ignored entirely. That changes in escaping tendency must occur during dispersal of a material and during adsorption of another substance on it is a consequence of the self-evident truth that substance X, cannot attract substances X unless X also attracts X]. This recognition of the existence of changes in escaping tendency does not imply that 2

2


COPELAND AND YOUNG

351


there are changes in free energy and other properties which are not part of, or are not associated with, changes in surface free energy. There can be no such changes. It seems that changes in various properties produced by adsorption can be rather large. For example, several authors have reported some surprising volume changes which occur in the adsorption of various gases on porous glass and other materials (1,17,21).


Thermodynamics of Small One-Component Particles The volume of a drop formed by coalescence of two small drops is not ex actly the sum of their volumes. The enthalpy, H, is not the sum of the enthalpies of the two small drops. In other words, a sufficiently sensitive measurement would reveal a AV and a AH associated with the coalescence. We can calculate the AH for the process with considerable precision if the drops are very small but large enough to ensure that surface tension is nearly the same as for the liquid in bulk. The calculation involves AH per unit area calculated from the temperature coefficient of surface tension (8) and the area which disappears in the process. (The coalescence of two drops of water at 2 5 ° C. each 0.01 cm. in diameter produces a temperature rise of 3 . 5 ° X 10~ ° C.) These facts mean that extensive thermodynamic properties are not truly "ho mogeneous" (11 ) in the sense in which that word is used in mathematics and ther modynamics. Experiments with relatively large particles show that thermody namic functions are homogeneous within the limits of precision ordinarily attained in the laboratory. The fact is simply that we usually do not detect deviations from homogeneity. It is not true that such differences do not exist. Most texts and treatises dismiss the problem so quickly that students are likely to conclude that even the mention of the word "homogeneity" is somewhat pedantic. The partial molal free energy of a spray of finely dispersed droplets is not so simply related to escaping tendency as it is in systems usually treated by thermo dynamic methods. Even the definition of the partial molal free energy is not en tirely devoid of difficulty. To discuss equilibrium we wish to think of the transfer of a small amount of adsorbent from each droplet in the spray to another phase— e.g., the gas phase (16). If a little material leaves each droplet, there must be a concomitant decrease in surface area, A, and an increase in specific area, (area per mole or per gram). The quantity related to escaping tendency of material is not the derivative of free energy with respect to n, the number of moles in the spray at either constant area or constant specific area. The condition is that the number of droplets, v, be constant. (This means that area divided by mass to the 2/3 power is nearly invariant.) The measure of escaping tendency for a monodisperse spray of one component droplet is 4

σ

G -

(ψ)

(8)

\Οη/ρ,τ,ν

This quantity is, of course, a partial molal free energy. The fact that the extensive free energy depends upon ν means that the free energy of each drop is not proportional to the amount of material in the drop. Hence, when the amount of material per drop is small, the extensive free energy, G, is not equal to nG. Constant Specific Surface Area Another definition of partial molal free energy avoids this particular diffi culty. Consider the addition of an increment of substance to the spray of finely In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

352


dispersed liquid adsorbent. To keep the drop size constant, the quantity of material in each addition must be equal to that in a whole drop, or a multiple of it. If an amount of material less than that in a single uniform drop were intro duced as a smaller drop, the system would be unstable and no single value of the molal or partial molal free energy of the substance could be stated. We shall define a new continuous function of η equal for all integral values of ν to the free energy of the system. A complete new set of thermodynamic functions may be defined in a corresponding manner. One such function is illustrated in Figure 1. The ordinate might be any extensive property of the system containing a varying number of moles of liquid. The volume has been chosen for illustration. The abscissa of point A is n , the number of moles of substance in a single drop of the uniformly dispersed adsorb ent; the ordinate is the volume of one drop. Point Β represents the number of moles of adsorbent in, and the volume of, two of the drops; point C, the moles in, and volume of, three drops, etc. Curve 1 represents the volume of a single drop of any size as a function of n, the number of moles of adsorbent. The curve represents the volume for both η less than n and η greater than n . Curve 2 represents the volume of two drops, the size of one being standard, the size of the other variable. Curve 3 represents three drops, two of standard size and the third variable. The straight line connects the series of discrete volumes contained in integral numbers of uniform drops. The volume V*, indicated by the straight


A

A

A

4

n, Number of Moles of Liquid

Figure 1.

Volume as a function of η for various integral numbers of drops

1. One drop, of varying size 2. One standard drop plus another of variable size 3. Two standard drops plus another of variable size A, B,C,D. Intersections Vt. Continuous function which equals actual to tal volume of any integral number of stand ard drops In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

COPELAND AND YOUNG


353

line, is a property which is a continuous and linear homogeneous function of n. The derivative of each of the curves 1, 2, 3, etc., where it crosses the Vt line is the derivative of the volume with respect to n, at constant v—i.e., it is the partial molal volume corresponding to G of Equation 8. The functions represented by curves 1, 2, and 3 are not homogeneous. The slope of V* is the derivative of the volume at constant specific area. V\ of course, was defined so that it is a homogeneous function of n. A qualitative representation of free energies of varying numbers of moles and varying numbers of drops of adsorbent is similar to Figure 1, but the curves corresponding to 1, 2, and 3 are convex toward the free energy axis. The slope of these curves where each crosses the G* line is the partial molal free energy of Equation 8. The slope of the G* line is also partial molal free energy, the derivative of free energy at constant specific area, σ.


1

Though G+ is not an exact measure of escaping tendency, the homogeneity of G? confers upon it a distinct advantage. We offer the postulate that in practice it will prove to be an adequate measure of escaping tendency; we believe that it will also lead to interesting predictions and experiments. To discuss the addition of adsorbate to a monodisperse adsorbent we shall use a similar device, a function Gt for the system of two components that strictly is equal to the free energy only when the number of moles of adsorbent, n is a multiple of the number of moles of adsorbent in a single standard droplet. For each value of n however, G* = G varies continuously as a function of n . G may be defined in the usual manner: l9

2

l9

/

-(Pi

'ηι,Ρ,Τ,σ

\

ΟΠ2 /

2

io

ηι,Ρ,Τ,σ

The subscript, σ, is redundant in both of these derivatives but is included as a reminder that each droplet is to contain an unvarying amount of X . The specific surface area, σ, is defined therefore as the area per unit mass that would be ob served if the adsorbate were removed. G^ is defined as in Equation 9, except that it is now necessary to include the restriction that both n and σ are to be constant. x

2

\

ΟΠι/ηΐ,Ρ,Τ,σ

At constant σ and constant n^/rx^ the free energy, G^, is proportional to n± and hence proportional to n . Gt is therefore a homogeneous function of n and n . Consequently, an equation similar to the Gibbs-Duhem equation is valid. 2

x

M G i t + n dG = 0 2

2

2

(12)

The change in may be evaluated by integration of Equation 12. It bears an interesting relation to the difference between the integral free energy change of This defined extensive property bears a relation to the discrete values of the vol umes of uniform drops which is somewhat analogous to the relation of the gamma func tion of ζ to the factorials of integral values of z. Chemical thermodynamics, in general, involves an approximation of a similar kind. We think in terms of mathematical opera tions which are dependent upon the transfer of infinitesimal quantities of material, al though we are not quite aware that transfers cannot involve—even in thought—fractions of molecules. We have become so accustomed to this discrepancy between mathematics and realizable processes that we seldom mention it in thermodynamic arguments. lr


354


a process such as that represented by Equation 1 or 7 and the free energy change of the differential process represented by Equation 4. Γη I JO

2

ηιΔ&τ =

AÔ2dn

2

-

n AQ 2

2

(13)

This equation follows from the integration of Equation 12 and the "formula for integration by parts" (7). An illustration of its use with a solid adsorbent is given by Wu and Copeland (20, Figure 5). There is also an interesting relation between AGjt and the surface free energy, γ, defined by Gibbs [cf. Equations 514 and 675 of (6) ].


A G i f

=

(14)

AAy

Here Δγ is the change in surface free energy for a process such as Process 1 or 7. Changes in the extent of surface, usually small, are ignored. Adsorption on Solids Most adsorbents studied in the laboratory are solids. The thermodynamic treatment of solid particles is more difficult than the treatment of uniformly dis persed liquid droplets. Some interesting conclusions concerning them can be drawn, however. As an approximation divide the solid material into two classes: interior matter and surface matter. In general they are not in equilibrium; the escaping tendency of material in the surface exceeds that of material in the interior. There may be a continuous gradation of escaping tendency over a wide range, but in keeping with our approximation we shall treat the escaping tendency of material in the interior as if it had a single value (the average interior value) and escaping tendency of surface matter as if it had another single (usually larger) value. Moreover, we shall regard the escaping tendency of material in the surface as conditioned merely by the fact that it is in the surface—i.e., differences arising from the shapes of solid particles will be ignored in this approximation. A discussion of the escaping tendency of material in the interior infers the existence of a (conceptual) process by which material could be added to the interior without a change in amount of adsorbent in the surface. We can imagine removal of part of a surface, addition of substance to the interior, replacement of the surface, and distortion of the particles to restore the extent of surface to its former value. The derivative of the free energy in this thought process with respect to the number of moles in the interior is G (interior). Similarly, a sub stance may be added to the surface with an accompanying distortion to keep con stant the quantity of material in the interior. The derivative of free energy with respect to number of moles in the surface is G (exterior). These definitions are in harmony with Equation 8—i.e., each of these partial molal free energies is a measure of escaping tendency. x

1

The extensive free energy of a system cannot be observed directly. A n attempt to measure partial molal free energies would, therefore, be a problem of a different type. We may conceive of the measurement of vapor pressure, solu bility, or the e.m.f. of a cell in which the adsorbent functions as an electrode. A successful measurement of such an intensive property would yield information concerning the escaping tendency or partial molal free energy of material of one class, presumably the exterior kind, of material. A partial molal free energy so obtained would not be an average for both interior and exterior material. Even a measurement only moderately successful could yield a change in partial molal In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.


COPELAND AND YOUNG

355


free energy much closer to the value for the exterior material than to the value for interior material. It might in such a case be several times as large as the "average value." This interesting complication is a consequence of the lack of equilibrium within solid adsorbents. It would not occur if a monodisperse liquid adsorbent could be used. If the quantity of adsorbent should be increased by the addition of more material in the same state of subdivision, we could measure changes in volume, in heat capacity, and in certain other extensive properties which can be directly observed. We could differentiate any one of these properties with respect to the number of moles of adsorbent (whether the system contains one, two, or more components) to obtain a partial molal property. The partial molal property so obtained would be the weighted average for interior and exterior adsorbent and is in harmony with Equations 9 and 11. A change in partial molal free energy of the adsorbent calculated from adsorption isotherms is an average for the exterior and interior adsorbent because we have taken the total number of moles of adsorbent as n This average, multiplied by the total number of moles of adsorbent, is an extensive free energy function. Its temperature variation yields a heat of adsorption which is also extensive. This heat divided by the number of moles of adsorbent is the average heat of adsorption per mole. Fortunately, it is this quantity which is most directly determined in a calorimeter. v

In the following paper average heats of adsorption for one system (BaS0 and H 0 ) are compared with values calculated from adsorption isotherms at three temperatures (20). The satisfactory agreement obtained shows that the prevailing concepts and approximations are adequate for an understanding of these phenomena for this particular system involving a powdered adsorbent. Undoubtedly there are other systems and other types of properties for which it will not prove adequate to ignore the complications inherent in solid adsorbents. 4

2

Acknowledgment We are indebted to Stephen Brunauer, Rulon E . Johnson, Paul Seligmann, and Y. C . W u for interesting and instructive discussions, suggestions, and penetrat ing arguments. We do not imply that all of our arguments are in complete agreement with the opinions of each of these scholars.

Literature Cited (1) Amberg, C. H., McIntosh, R., Can. J. Chem. 30, 1012 (1952). (2) Bashforth, F., Adams, J. C., "An Attempt to Test the Theories of Capillary Action," Cambridge University Press, Cambridge, 1883. (3) Brunauer, S., "Adsorption of Gases and Vapors. Vol. I. Physical Adsorption," p. 218, Princeton University Press, Princeton, N. J., 1945. (4) Dole, M., McLaren, A. D., J. Am. Chem. Soc. 69, 651 (1947). (5) Dunford, H. B., Morrison, J. L., Can. J. Chem. 33, 904 (1954). (6) Gibbs, J. W., "Collected Works," Vol. 1, Yale University Press, New Haven, Conn., 1944. (7) Granville, W. Α., Smith, P. F., Langley, W. R., "Elements of Calculus," p. 283, Ginn, Boston, 1946. (8) Harkins, W. D., "Physical Chemistry of Surface Films," pp. 80-3, Reinhold, New York, 1952. (9) Johnson, I., LaMer, V. K., J. Am. Chem. Soc. 69, 1184 (1947). (10) Kington, G. L., Aston, J. G., Ibid., 73, 1929 (1951). (11) Klotz, I. M., "Chemical Thermodynamics," p. 11, Prentice-Hall, New York, 1950. (12) Ibid., p. 211. In SOLID SURFACES; Copeland, L., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1961.

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ADVANCES I N CHEMISTRY SERIES

(13) Kolthoff, I. M., Sandell, Ε. B., "Textbook of Quantitative Analysis," 3rd ed., p. 116, Macmillan, New York, 1952. (14) LaMer, V. K., Gruen, R., Trans. Faraday Soc. 48, 410 (1952). (15) Lewis, G. N., Randall, M., "Thermodynamics, rev. by K. S. Pitzer, L. Brewer, p. 388, McGraw-Hill, New York, 1961. (16) Ibid., p. 482. (17) Powers, T. C., Proc. Am. Concrete Inst. 43, 584 (1947). (18) Thomson, W., Phil. Mag. (4) 42, 488 (1871). (19) Treadwell, F. P., Hall, W. T., "Analytical Chemistry," Vol. II, 9th ed., p. 26, Wiley, New York, 1942. (20) Wu, Y. C., Copeland, L. E., ADVANCES IN CHEM. SER. No. 33, 357 (1961). (21) Yates, D. J. C., Advances in Catalysis 9, 481 (1957). (22) Young, Thomas, Phil. Trans. Roy. Soc. 65, Part I (1805). September 25, 1961.


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A Thermodynamic Theory of Adsorption - Advances in Chemistry (ACS

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