While a theoretical approach is a good indication as to a process, it is of very little use if practically the theory produced fails to work. To determine practically the shape of the bridge and how the liquid condenses, two very different methods are employed; neutron scattering and physically filling a sample with a vapour and then drying while measuring the mass change over time.
Prior to this, it is important to understand the methods by which the build up of liquid occurs, namely the theory of BDDT and Langmuir isotherms.
Isotherms
In any system, there are two types of molecule; bulk and surface.
For the bulk any forces on a molecule are balanced when averaged over all directions.
A surface molecule differs in that in one face, there are no molecules to counter other forces leading to an increased potential energy compared to the bulk. This, incidentally is the reason why there is a meniscus on a liquid.
Molecules on a surface behave differently to those in the bulk. In addition the concentration of cations and anions may differ at a surface (compare to the bulk) leaving a charged imbalanced area. It is important to understand the nature and quantity of these surface sites.
Any adsorption properties depend on the the atoms or molecules on the surface and their dispersion. Dispersion is defined here as the ratio of molecules on a surface to the total number and is denoted by D.
n molecules | Dispersion on surface |
27 (33) | 0.963 |
64 (43) | 0.825 |
125 (53) | 0.784 |
216 (63) | 0.704 |
limit | 0.38 |
There are two forms of adsorption (the process of surface bonding); physisorption and chemisorption.
Physisorption is when molecules are held to the surface by relatively weak interactions (Van der Waals forces). These are typified by ∆Hads of the order of -20 kJ mol-1.
Adsorption is usually exothermic. If adsorption is observed, it must be a
spontaneous process, ∆Go = -ve.
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For a simple molecular adsorption, T∆o
must be negative. ∆Go will be negative,
therefore ∆Ho must be
more negative that the T∆So (as this
figure is preceeded by a minus operator, the right hand side of the equation
will become positive). ∆So can be
positive when a molecule breaks and adsorbs onto two sites (for example,
hydrogen onto glass)
Chemisorption occurs when the bond is formed between an incoming atom, ion, radical or molecule and the surface involves a re-arrangement or transfer of electron density. ∆H here is high (typically between -150 and -300 kJ mol-1). There is a limit to the amount of chemisorption possible; it is governed by the availability of sites because it is limited to just one atomic / molecular layer. In contrast, physisorption may involve multilayered adsorption.
Adsorption can be quantified in terms of an adsorption isotherm (diagram 4).
There are 5 forms of isotherm in the Brunnauer, Deming, Deming and Teller (BDDT) classification.
Diagram 5a - Langmuir Isotherm. This is found for the formation of a single monolayer system. When the monolayer is complete, there can be no further chemisorption. A similar shape is observed for physical adsorption on a microporous material but the plateau coverage is much greater (BDDT type 1)
Diagram 5b. BDDT type 2. Here a monolayer is formed with a subsequent adsorption of layers. Bulk condensation may then occur. An example is N2 on Fe (77K). ∆Hads (1st layer) is significantly greater than the next layers
Diagram 5c. BDDT type 3. No full monolayer is formed before adsorption starts in the subsequent layers (∆H1st layer is similar to subsequent layers). It is very rare. An example is Br2 on Si (350K).
Diagram 5d and 5e. BDDT types 4 and 5. These are equivalent to BDDT types 2 and 3, but for mesoporous solids (20 - 500Å). The distinguishing characteristics are a hysteresis upon adsorption - desorption (due to capillary condensation and its consequences)
Diagram 5e. BDDT type 5.
Not all isotherms fit the BDDT classification
Diagram 5f. A stepwise adsorption. This is thought to have different types of site available at the surface. An example is Kr on carbon black at 90K
The amount adsorbed is conveniently described in terms of surface coverage, θ.
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When θ = 0, there is no surface covered. When θ = 1, the surface is fully covered.
The Langmuir theory of monolayer adsorption
This was developed in 1916 and makes 5 assumptions
- Only a monolayer is formed
- adsorption is localised
- ∆Hads is independent of θ.
- There is only one type of site
- The gas phase may be treated as idea.
There are three factors which affect the adsorption rate.
At an equilibrium state,
Factor 1.
From the kinetic theory of gases :
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therefore, the collision of gas molecules with the surface area is proportional to the pressure of the gas.
Factor 2.
It is proportional to the probability that a site is vacant (1 - θ)
but θ = Vn / Vm (volume of the complete monolayer)
Factor 3.
There is an activation energy type term which describes the temperature dependence.
As a reaction scheme :
therefore
11a
11b
11c
11d
11e
Using (11e) allows a crude modelling of the experimental isotherm. For ease of analysis linearisation is achieved by inversion of the equation, multiplication by PA and subsequent division by Vm (volume of the monolayer). V is the volume of vapour.
12a
12b"
12c
Therefore, a plot of Pa/V vs. PA gives a gradient of 1/Vm and intercept of kdes/kadsVm.
Langmuir concerned himself only with monolayer formation. For multiple layers, BDDT has to be considered. There is a small degree of overlap in that a BDDT type 1 can be applied for a Langmuir isotherm.
BDDT theory
This assumes that the first step is the adsorption of the gas molecule onto the surface
As this is an equilibrium process, it is described in terms of an equilibrium constant, K1.
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If there was no further adsorption, θ v=1-θv i.e. a Langmuir isotherm system. For a multilayered adsorption, it should be considered as process of the type
which gives an equilibrium constant of
with the third layer
and the nth layer
One of the fundamental rules of BDDT is that K1 is different to all subsequent K values as only the first layer interacts with the surface and these other K values are effectively the same. All other interactions are of gases with their own type.
The influence of the surface decreases very rapidly as the number of layers increase. For very high levels of K, it is assumed that these resemble the equilibrium constant for A(g) + A(g) Ø2A(l).
The activity of the condensed phase = 1.
When Vp equals the atmospheric pressure (i.e. po) condensation occurs, that is
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The various fractional surface coverages can be calculated from the previous fraction surface coverages.
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Each of the qi layers are fractional areas, therefore the sum of all of these plus the vacant sites in the first layer must equal one.It can be shown that the series inside the brackets is (1-x) -1 (for an infinite number of layers).