Adsorption Kinetics - The Rate of Adsorption

 

The rate of adsorption, Rads, of a molecule onto a surface can be expressed in the same manner as any kinetic process. For example, when it is expressed in terms of the partial pressure of the molecule in the gas phase above the surface:

Rads = k' P x

where:

x - kinetic order

k' - rate constant

P - partial pressure

 

If the rate constant is then expressed in an Arrhenius form, then we obtain a kinetic equation of the form :

Rads = A exp ( -Ea / RT ). P x

 

 

where Ea is the activation energy for adsorption, and A the pre-exponential (frequency) factor.

It is much more informative, however, to consider the factors controlling this process at the molecular level....

The rate of adsorption is governed by:

 

1. The rate of arrival of molecules at the surface.

2. The proportion of incident molecules which undergo adsorption.

 

i.e. we can express the rate of adsorption (per unit area of surface) as a

product of the incident molecular flux, F , and the sticking probability , S .

Rads = S . F
[molecules m-2 s-1 ]

 

The flux of incident molecules is given by the Hertz-Knudsen equation-

Flux , F = P / (2pmkT)1/2
[ molecules m-2 s-1 ]

 

where

P - gas pressure [ N m-2 ]

m - mass of one molecule [ kg ]

T - temperature [ K ]

The sticking probability is clearly a property of the adsorbate / substrate system under consideration but

must lie in the range 0 < S < 1; it may depend upon various factors - foremost amongst these being the existing

coverage ofadsorbed species (q) and the presence of any activation barrier to adsorption. In general , therefore ,

S = f (q) . exp ( -Ea / RT )

 

where, once again, Ea is the activation energy for adsorption and f(q) is some, as

yet undetermined, function of the existing surface coverage of adsorbed species.

 

Combining the equations for S and F yields the following expression for the rate of adsorption :

 

Notes :

 

1.The above equation indicates that the rate of adsorption is expected to be first order with regard to the partial pressure of the molecule in the gas phase above the surface.
2.It should be recognised that the activation energy for adsorption may itself be dependent upon the surface coverage , i.e. Ea = E(q).
3.If it is further assumed that the sticking probability is directly proportional to the concentration of vacant surface sites (which would be a reasonable first approximation for non-dissociative adsorption) then f(q) is proportional
to (1-q) ; where, in this instance, q is the fraction of sites which are occupied (i.e. the Langmuir definition of surface coverage).

For a discussion of some of the factors which determine the magnitude of the activation energy of adsorption you should see Section 2.4 which looks at the typical PE curve associated with various types of adsorption process.

Estimating Surface Coverages arising as a result of Gas Exposure.

If a surface is initially clean and it is then exposed to a gas pressure under conditions where the rate of desorption is very slow, then the coverage of adsorbed molecules may initially be estimated simply by consideration of the kinetics of
adsorption.

 

As noted above, the rate of adsorption is given by : Rads = S . F ,,, i.e. ;

 

 

 

Where : Nads is the number of adsorbed species per unit area of surface.

In general, this equation must be integrated to obtain an expression for Nads,

since the sticking probability is coverage (and hence also time) dependent.

However, if it is assumed that the sticking probability is essentially constant

(which may be a reasonable approximation for relatively low coverages), then this integration simply yields:

 

Thus any water-bonded glycerin groups and fatty acids are amenable

to selective adsorption by the "X" factor in kinetic order of coarse.

 

These substances can share both oil and water bonds so it is relatively easy to exploit

these forces of hydroscopic attraction as stated by and in the Hertz-Knudsen equation.

 

Acusorb beads and other selective hydroscopic agents within them can cleverly be employed to rid waste oils of the

worst types of water-bonded glycerin and fatty acid groups down to a satisfactory and acceptable level for fuel use.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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