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=== Adsorption Isotherms === | === Adsorption Isotherms === | ||
− | One very important tool for studying adsorption is an adsorption isotherm, a | + | One very important tool for studying adsorption is an adsorption isotherm, a curve of surface coverage <math>\theta</math> versus pressure (or some related variables) at a constant temperature. An application of these models is calculating the internal surface area of the adsorbent. |
==== Langmuir Isotherm ==== | ==== Langmuir Isotherm ==== | ||
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where b is the constant given in [Ho] as | where b is the constant given in [Ho] as | ||
<center><math>b = \frac{e^{-Q/kT}/\tau_0}{N\sigma_0/\sqrt{2\pi MPT}}</math></center> | <center><math>b = \frac{e^{-Q/kT}/\tau_0}{N\sigma_0/\sqrt{2\pi MPT}}</math></center> | ||
+ | here T is the temperature in K, Q is the activation energy in eV, k is the Boltzmann constant, N is the number of sites, <math>\sigma_0</math> is the adsorption site area, M is the adsorptive molecular weight, and <math>\tau_0</math> is the adsorption time. | ||
+ | |||
+ | The Langmuir Isotherm is too much of a simplification to find much practical use. Instead, it is more common to use the BET isotherm, which allows for multiple layers. | ||
==== BET Isotherm ==== | ==== BET Isotherm ==== | ||
+ | |||
+ | The BET isotherm is given by [Ho]: | ||
+ | <center><math>\frac{P}{P_0-P}\times\frac{1}{\theta} = \frac{1}{C\theta_m}+\frac{C-1}{C\theta_m}\times\frac{P}{P_0}</math></center> | ||
+ | where P is the adsorptive pressure, <math>\theta</math> is the surface coverage, <math>\theta_m</math> is the surface area covered by a monolayer only, Q is the heat of adsorption, and <math>C\propto\exp(Q/kT)</math>. The BET isotherm is a much more powerful tool because it can yield two desirable quantities: the head of adsorption, and the surface area of the adsorbent. Define | ||
+ | <center><math>Y = \frac{P}{P_0-P}\times\frac{1}{\theta}, \quad X = \frac{P}{P_0} \quad \Rightarrow \quad Y = \frac{1}{C\theta_m}+\frac{C-1}{C\theta_m}X </math></center> | ||
=== Vapor Pressure === | === Vapor Pressure === |
Revision as of 22:14, 2 January 2009
Nick Hutzler
I am a second year Physics graduate student in the Doyle Lab. I am working on an experiment to measure the electron electric dipole moment using a cryogenic molecule beam.
Contents
Final Project: Applications of Gas Adsorption
Introduction
Adsorption is the physical process where molecules (or atoms, though we shall use the word "molecule" to include those as well) in a fluid phase stick become bound to the surface of another solid or liquid. Adsorption is a very broad term that can include a gas or liquid adsorbing onto a liquid or solid through electrostatic attractions, chemical bonds, or some combination of both. In this discussion we will focus primarily on the adsorption of gases onto solid.
Before proceeding, let's introduce some terminology. In adsorption, the fluid-phase adsorptive molecules sticks onto the adsorbent and become adsorbate, or admolecules. When adsorbate unsticks from the adsorbent, whether spontaneously or induced, it is called desorption.
Brief Explanation of Adsorption
The fundamental reason why adsorption occurs is because surfaces have energy. The surface at the interface of a solid or liquid necessarily has some energy density associated with it that is higher than the energy density of bulk material. This is an empirical fact, because materials (both solid and liquid) tend to sacrifice surface area for bulk.
In liquids, surface energy arises from the fact that in the bulk, molecular (van der Waals) attractions between some molecule and all its neighbors balance out, resulting in a net force of 0. However, a molecule at a surface will feel a net attraction inwards, which gives rise to a surface energy. This force pulling the surface inward is balanced by the internal pressure of the liquid.
In solids, surface energy arises from disruption of the solid's lattice structure.
Types of Adsorption
Adsorption can be classified by whether the mechanism is "physical" (physisorption) or "chemical" (chemisorption) in nature. Because we will be focusing on physisorption, we then discuss several different adsorption mechanisms that would be classified as physical. The lists presented in this section are adapted from [Ke].
Physisorption vs. Chemisorption
- Physisorption is classified by weakly bound adsorbate, usually by van der Waals or London dispersion forces. An important feature of this type of adsorption is that it is reversible; specifically, desorption can be induced by raising the temperature of the adsorbent, or by decreasing the pressure of the adsorptive. The adsorptive does not suffer any type of chemical change when being adsorbed.
- Chemisorption is classified by strongly bound adsorbate, usually as a result of chemical bonds. Because chemical reactions occur, this process is typically irreversible, and the characteristics of the mechanism are strongly dependent on the species involved. Therefore, we will not focus on this type of adsorption.
Types of Physisorption
- Monolayer/multilayer adsorbates: When the adsorbent is uniformly covered with sites on which the adsorptive can stick, the adsorbate will generally be a roughly uniformly distributed layer. When the adsorption sites are homogeneous and the adsorptive pressure is much smaller than the saturation pressure, this layer is typically one molecule thick. When the adsorptive pressure approaches the saturation pressure, there are typically multiple layers of adsorbate. The admolecules in monolayer often form a lattice gas, where there is typically one admolecule per site but they can hop from site-to-site (see for example [Mo])
- Pore fluids: An adsorbent can have holes or troughs ("pores") that can become filled with adsorbate, and under the right conditions can become a gas or liquid. In fact, some very interesting things can happen because of the complicated thermodynamics of these systems; for example, liquid water a few molecular layers thick has been observed at the surface of certain porous solids... at 77K! [Ro]
- Steric Adsorbates: Adsorbents can have site that can sterically attract certain atomic groups. This mechanism plays is important, for example, in the adsorption of large biomolecules onto activated carbon.
- Ionic Adsorbates: Adsorbent surfaces can be covered with ions that can exchange with ionic parts of a molecule, binding the molecule to the adsorbate.
- Quantum Adsorbates: Adsorption can occur due to purely quantum mechanical effects. One example is when an adsorbent has pores that are comparable to the de Broglie wavelength of the adsorptive, allowing them to enter the pores.
In real life, adsorbents typically have many of the above features.
Adsorption Thermodynamics
Equilibrium
Adsorption is often a rate-equation governed process, so "true" equilibrium would only occur at <math>t=\infty</math>. However, for real-life industrial applications, [Ke] defines a technical equilibrium as follows: an adsorbtion process is at equilibrium if <math>\Delta m/m < \varepsilon</math>, where <math>\Delta m</math> is the mass change in the total mass <math>m</math> over some time <math>\Delta t</math>, and <math>\varepsilon</math> is some parameter that depends on the application. Typical values are <math>\varepsilon=10^{-5}</math> and <math>\Delta t = </math> 30 minutes.
Industrial processes are often cyclical, and involve adsorption/desorption cycles on some time scale <math>t_c</math>. The distance from equilibrium of a process can then be characterized by a Deborah number:
For <math>De\approx 0</math> the system is at equilibrium.
Adsorption Isotherms
One very important tool for studying adsorption is an adsorption isotherm, a curve of surface coverage <math>\theta</math> versus pressure (or some related variables) at a constant temperature. An application of these models is calculating the internal surface area of the adsorbent.
Langmuir Isotherm
A simple yet powerful model was developed by Langmuir. This model assumes that the adsorptive forms a monolayer on the adsorbent, and that adsorbent covered with a monolayer is passivated. A quick derivation can be worked out by considering the equilibrium coefficient K of the adsorption reaction <math>M+S\leftrightharpoons MS</math> between molecules M and adsorption sites S:
If <math>\theta</math> is the fraction of adsorbent that is covered by a monolayer, then <math>[S]\propto 1-\theta</math> and <math>[MS]\propto\theta</math>. In the gas phase, the concentration is proportional to the pressure P, so
where b is the constant given in [Ho] as
here T is the temperature in K, Q is the activation energy in eV, k is the Boltzmann constant, N is the number of sites, <math>\sigma_0</math> is the adsorption site area, M is the adsorptive molecular weight, and <math>\tau_0</math> is the adsorption time.
The Langmuir Isotherm is too much of a simplification to find much practical use. Instead, it is more common to use the BET isotherm, which allows for multiple layers.
BET Isotherm
The BET isotherm is given by [Ho]:
where P is the adsorptive pressure, <math>\theta</math> is the surface coverage, <math>\theta_m</math> is the surface area covered by a monolayer only, Q is the heat of adsorption, and <math>C\propto\exp(Q/kT)</math>. The BET isotherm is a much more powerful tool because it can yield two desirable quantities: the head of adsorption, and the surface area of the adsorbent. Define
Vapor Pressure
Enthalpy
Pore Diameter
References
[Ke] Keller & Staudt, Gas Adsorption Equilibria
[We] Welch, Capture Pumping Technology
[Ro] Robens, E. "Some Remarks on the Interface Ice/Water." Proceedings VIII Ukrainian-Polish Symposium: Theoretical and Experimental Studies of Interfacial Phenomena and their Technological Applications (2004)
[Mo] Morgenstern, K. and Rieder, K. H. PRL 93 (2004) 056102
[Ho] Hoffman, Singh, and Thomas, Handbook of Vacuum Science and Technology.