A ;IfODIFICATION O F THE FREUNDLICH ADSORPTIOK ISOTHERM BY WILLIAM ROGERS, JR. AND RlARTIN SCLAR
Of the numerous theoretical and empirical adsorption equations which have been proposed,' that of Freundlich has been found to describe best' the experimental facts, and hence has been used almost, exclusively. The Freundlich isotherm may be used to calculate: (a) the mass of adsorbent necessary to cause a desired change in concentration or pressure, except where t h e equilibrium concentration or pressure is zero, and (b) the equilibrium concentrabion or pressure, when the initial concentration or pressure, the volume of solution or dead space, and t'he mass of adsorbent are known. But the latter calculat'ion is very laborious even when carried out in the simplest manner.2 The purpose of this paper is to modify Freundlich's equation so that it may be used for all the above purposes with the same efficiency, but with much great,er ease. The modified equation also has the advantage that it may be used to calculate the mass of adsorbent just necessary for tot,al adsorption. Freundlich's isotherm for adsorption from solution is x/m = k C"p where x is the amount adsorbed, m is the mass of adsorbent, CZis the equilibrium concentration of the solute, and k and n are empirical constants. For an infinitesimal adsorption the isotherm becomes dxjdni = k C" But dx = V dC, where V is the volume of the solution, so that VdGjdm = k C" and VdC/C" = k dm. This equation may be integrated as follows:
where C1is the initial con where r = (I-n). Then
T'/m (Crl - Cr2) = k r Let k r = K since both k and r are constants.
V/m (er,- Cr2) = K ' (1). Solving equation (I) for Cp: (Cri - K m/P)"' = C2.. . . . , . . . . . . . . . . . . . . . ( 2 ) . When CZis taken as zero: Crl V/m, = K and Crl V/K = m,. . , . , . , , , . . . . , ' ( 3 ) . 1
Swan and Urquhart: J. Phys. Chem., 31, 2j1 (1927). Rogers and Sclar: J. Phys. Chem., 35, 2758 (1931).
MODIFICATION O F FREUNDLICH ADSORPTION ISOTHERM
2285
Equation ( I ) is to be used to calculate the values of r and K if the method of trial and error is to be used. Equation (2) is used for a calculation of the equilibrium concentration when C1, m, and V are known. Equation (3) is used for a calculation of the mass of adsorbent, mor just required for total adsorption of the solute. It is interesting to note that equation (3) is similar to one previously derived3 in a different manner. The latter equation, however, is restricted to constant volume. The method used to derive equations (I), ( 2 ) , and (3) may be used to derive a similar set of equations for the adsorption of gases. The Freundlich isotherm for the adsorption of gases is x/m = k PnZ where Pzis the equilibrium pressure, and the other terms retain their previous meanings. If x is expressed in mols, then x = ni - n2, where ni is the number of mols admitted to the adsorption chamber, and nz is the number of mols not adsorbed. Corresponding to nl and n2 there would be an initial pressure P, and an equilibrium pressure PZrespecti0ely. The volume of the adsorption chamber less the volume of the adsorbent is V. Then assuming a perfect gas, PIV = nl R T PzV = n2 R T n2 = P2V/RT nl = PlV/RT dx = V/RT d P x = V/RT (Pi - Pz) For an infinitesimal adsorption the Freundlich adsorption isotherm would become, dx/dm = k P". Substituting in this equation the value of dx from above, V/RT . dP/dm = k P" and
J,$P/Pn
=
kRT/V
&Lo dm
Proceeding exactly as in the derivation of the equation for adsorption from solution, the results are: V (P.1 - P.2) = K.. .................... (4).
mRT
pZ= (Yl
m,
=
-
K RT
V Pri KRT.
m)'Ir.. . . . . . . . . . . . . . . . . , . ( 5 ) .
......................
(6).
Equations (4), (s), and (6) are to be used similarly to equations (I), (2), and (3) respectively. Since PiV = nlRT Pl/ni = RT/V and these may be used interchangeably in any equation if to do so is more convenient. 8
Freundlich, translated by Hatfield: "Colloid and Capillary Chemistry," 3rd Ed., 178.
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WILLIAM ROGERS, J R . , AND MARTIN SCLAR
The validity of equations (2), (3), and (j) will be seen in the following data. There are no data presented (because they are not available) in supgort of equation (6). But its validity may be assumed without much apprehension because of its similarity to equation (3), and its following directly from equat,ion (s), both of which equations are supported by the data presented. In the data listed, experiment,al and calculated values of equilibrium concentrations and pressures are compared. A brief examination of the data, especially that for adsorption of gases, will show that were t,he experimental and calculated values of x/m compared, as is the custom in demonstrating the validity of adsorption isotherms, the agreement would be better. TABLE14 Adsorption of Benzene in Et,hyl Alcohol on Pure Carbon C, mol frac. o ,00653 ,01286 ,02604 ,05273 ,1090
r
H/m millimols/g. 372.2 333.6 356.5 366.8 353.7
C2 obs. mol frac. 0.00513 .or050 ,02289 ,04854 '1033 I/r = 1 . 8 5 2
= 0.54
C2calc.
mol frac. 0
,00504 .OIOjj ,02298
,0486 I
I
