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Combining the last two equations with the Gibbs adsorption isotherm gives:
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where x and yi represent adsorbate and gas-phase mole fractions respectively i
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Pure-Gas Adsorption
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Basic to the experimental study of pure-gas adsorption are measurements at constant temperature of n , the moles of gas adsorbed, as a function of P, the pressure in the gas phase Each set of data represents an adsorption isotherm for the pure gas on a particular solid adsorbent Available data are summarized by Valenzuela and ~ y e r s " The correlation of such data
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2 0 ~ P Valenzuela and
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A L Myers, Adsorption Equilibrium Data Handbook, Prentice Hall, Englewood Cliffs, NJ,
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CHAPTER 14 T o ~ i c in Phase Eauilibria s
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requires an analytical relation between n and P , and such a relation should be consistent with Eq (14102) Written for a pure chemical species, this equation becomes:
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The compressibility-factor analog for an adsorbate is defined by the equation:
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Differentiation at constant T yields:
Replace the last term by Eq (14103) and eliminate n / R T in favor of z / a in accord with Eq (14104) to yield:
Substituting a = A / n and da = - A d n / n 2 gives:
-dlnP =
-z- - d z
dn n
Adding d n l n to both sides of this equation and rearranging,
Integration from P = 0 (where n = 0 and z = 1) to P = P and n = n yields:
The limiting value of n / P as n + 0 and P + 0 must be found by extrapolation of experimental data Applying l'H6pital's rule to this limit gives:
Thus k is defined as the limiting slope of an isotherm as P + 0, and is known as Henry's constant for adsorption It is a function of temperature only for a given adsorbent and adsorbate, and is characteristic of the specific interaction between a particular adsorbent and a particular adsorbate The preceding equation may therefore be written:
148 Equilibrium Adsorption of Gases on Solids
This general relation between n, the moles adsorbed, and P, the gas-phase pressure, includes z, the adsorbate compressibility factor, which may be represented by an equation of state for the adsorbate The simplest such equation is the ideal-gas analog, z = 1, and in this case Eq (14105) yields n = k P , which is Henry's law for adsorption An equation of state known as the ideal-lattice-gas equation2' has been developed specifically for an adsorbate:
where rn is a constant This equation is based on the presumptions that the surface of the adsorbate is a two-dimensional lattice of energetically equivalent sites, each of which may bind an adsorbate molecule, and that the bound molecules do not interact with each other The validity of this model is therefore limited to no more than monolayer coverage Substitution of this equation into Eq (14105) and integration leads to the Langmuir
Solution for n yields:
Alternatively, where b = m/ k , and k is Henry's constant Note that when P + 0, n / P properly approaches k At the other extreme, where P -+ oo, n approaches m , the saturation value of the specific amount absorbed, representing full monolayer coverage Based on the same assumptions as for the ideal-lattice-gas equation, Langmuir in 1918 derived Eq (14106) by noting that at equilibrium the rate of adsorption and the rate of desorption of gas molecules must be the same23For monolayer adsorption, the number of sites may be divided into the fraction occupied 0 and the fraction vacant 1 - 0 By definition,
n rn- n and 1-0=rn rn where m is the value of n for full monolayer coverage For the assumed conditions, the rate of adsorption is proportional to the rate at which molecules strike the surface, which in turn is proportional to both the pressure and the fraction 1 - 0 of unoccupied surface sites The rate of desorption is proportional to the occupied fraction 0 of sites Equating the two rates gives:
1 3 -
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