Page 52 - Adsorption Technology & Design, Elsevier (1998)

Fundamentals of adsorption equilibria 49

of the partial differential (c3G/tgA)p,T,n is termed the spreading pressure 7r
while the positive value of (tgG/tgn)p,T,A is the chemical potential p of the
adsorbate in the potential field of the adsorbent. The integrated form of
equation (3.22) is therefore
G = - 7rA +/an (3.23)

On general differentation we thus obtain
dG = - ~rdA + A dlr + ladn + ndp (3.24)

which shows how all the variables A, 7r, n and p contribute to a change in G.
Subtracting equation (3.24) from (3.20) and writing (OG/cOA)p,r,, = -
and (t3G/t~n)p,T,A =/,/, results in the equation
-- A dTr + nd/z = 0 (3.25)

Substituting the classic thermodynamic relation p(p) for the vapour phase at
pressure p,

dla = RgTd In p (3.26)
where p is the partial pressure of the adsorbate, one obtains

A dTr = nRgTd In p (3.27)
which is the differential form of the Gibbs adsorption isotherm. In its
integral form it becomes
P~ P.q
~r = R~T ~ (n/A) d In p = (RgT/MSg) ~ q(p) d In p (3.28)
0 0

in which q(p) is the mass of gas adsorbed per unit mass of adsorbent (a
function of partial pressure), M the molecular mass of adsorbate and Sg the
surface area per unit mass of adsorbent. Both Sg and q(p) are directly
measurable at a given temperature and the integral can be determined by
numerical integration. A value for the spreading pressure 7r may be thus
calculated. Alternatively, if the adsorbent surface is a liquid (such as a fatty
acid film on water) then the surface area of the liquid film can be found by
independent measurements of the isotherm q(p) and spreading pressure tr.
The latter quantity can be measured using an ingenious device known as the
Langmuir trough which enables the liquid film to be confined between
barriers on the liquid surface and hence defining its area; the force necessary
to keep the liquid film coherent is measured by means of a balance upon
which weights are placed.