Page 39 - Adsorption Technology & Design, Elsevier (1998)

36 Fundamentals of adsorption equilibria

pressure above a plane surface, p~ is that above a curved surface and Tis the
absolute temperature at which the comparison is made. Now if an annular
ring of liquid commences to form in a capillary by the condensation of dn
moles of vapour, the work done against the liquid surface is (/~0-/~)dn and
the force stabilizing the liquid condensate is -- trdA, where cr is the surface
tension of the pure liquid and dA is the consequential decrease in surface
area as the annular ring of liquid increases. Equating the work done to the
stabilizing force
RgT In (ps/pa)dn = -- trdA (3.1)

If the pore in which condensation occurs is an open-ended cylindrical
capillary then dA is 2rcldr where I is the capillary length and dr is the increase
in the radius of the annular liquid film. The number of moles dn transferred
from vapour to liquid will be 27rlrdr/Vm where Vm is the molar volume of the
liquid. Substituting these quantities into equation (3.1) yields the relation
(3.2)
pa = ps exp (-- trVmlrRgT)
first formulated by Cohan (1938) to describe the gradual filling of a cylindrical
capillary. If the pore geometry is other than that of a cylinder, then dn/dA is
different and consequently the fight-hand side of equation (3.2) differs
accordingly. On desorption the free energy decreases and the pore, now full of
liquid condensate, will have a hemispherical meniscus at each end. The number
of moles transferred will be 41rr2drlVm and the corresponding decrease in area
is 81rrdr. Equating the stabilizing force to the gain in free energy (/~o- lt)dn
when desorption occurs at a pressure Pd, the equation
pd = P~ exp (--2xrVm/rRgT) (3.3)
results, similar to that first proposed by Thompson* (1871) and known as the
Kelvin equation. The relationship between the pressure on adsorption p~
and that on desorption pd from an open-ended cylindrical capillary is thus

(PJps) 2 = (Pd/Ps) (3.4)
Whenever pa and Pd are not coincident the relationship between them
depends on the pore geometry. For the ink-bottle shaped closed pores
described by McBain (1935) with a neck radius r~ < rb the radius of the wider
body, then (pJps) 2 > pd/Ps provided also that rb < 2rn. On the other hand for
open-ended pores with a wider body than neck at each end, (pa/ps) 2 < pd/ps.
The above arguments are more fully discussed elsewhere (Thomas and
Thomas 1967 and 1997, Gregg and Sing 1967, and Everett 1958) although it
should be noted that alternative theories such as that proposed by Foster
tw. T. Thompson (1871) was the distinguished physicist who succeeded to a peerage and took
the title Lord Kelvin.