Polanyi potential theory

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Polanyi potential theory is a simple model of how gases adsorb onto solid surfaces. It treats adsorption as an equilibrium between the chemical potential of the gas near the surface and the chemical potential of the gas far away from the surface. The attraction between the gas and the surface is mostly due to van der Waals forces and depends on how close the gas particle is to the surface. The gas is assumed to behave as an ideal gas until it condenses, which happens when the pressure is higher than the gas’s equilibrium vapor pressure.

History and development
- In 1914 Arnold Eucken introduced the concept of adsorption potential.
- Michael Polanyi followed with his own adsorption model, publishing in 1916 and then a more complete version with experimental support.
- The theory faced criticism for oversimplifying interactions, especially electrical effects and screening by other molecules.
- In 1930 Fritz London helped show how cohesive forces from quantum-mechanical polarization could align with Polanyi’s ideas, reinforcing the theory’s validity.
- Polanyi’s work influenced later models and remains a foundation for several adsorption theories, including the Dubinin–Radushkevich and Dubinin–Astakhov equations.

How the theory works (the core idea)
- At a fixed temperature, the gas molecules near a surface move in a potential that depends on distance to the surface, much like a gravitational or electric field.
- Adsorption occurs when the chemical potential of the gas at the surface (or near-surface position) equals the chemical potential far away from the surface.
- From this balance, one defines an adsorption potential that describes how strongly the surface can attract the gas.
- Using the ideal-gas relationship (pV = RT), the theory connects pressure, temperature, and distance from the surface to predict how much gas will adsorb.
- When the gas pressure reaches the liquid’s vapor pressure (p0), a thin liquid film can form on the surface. This is the basis for computing the adsorption potential in many cases.

Key equations and variants (simplified)
- The adsorption potential εs is related to concentrations in solution:
εsw = −RT ln( ce / cs )
where ce is the equilibrium concentration of the adsorbate in solution, cs is its saturated concentration, R is the gas constant, and T is temperature.
- The amount adsorbed (qe) is linked to this potential through forms that Dubinin and coworkers developed. The two well-known versions are:
- Dubinin–Radushkivech (DR): uses a fixed exponent b = 2.
- Dubinin–Astakhov (DA): uses an exponent b that can be adjusted to fit data.
- A common DA-like expression is:
log qe = log Q0 + (εsw / E)^b
where Q0 is the maximum adsorption capacity, E is the characteristic adsorption energy, and b is a fitting parameter.
- The ratio of a sample’s adsorption energy to that of a standard vapor can be described by an affinity coefficient β, helping compare different systems.

Applications and importance
- Polanyi potential theory is widely used to study activated carbons, carbon black, and other porous carbons.
- It has been applied to adsorb organic molecules such as polycyclic aromatic hydrocarbons (including nonionic and ionic forms like phenols and anilines), and to carbon nanotubes and carbon nanoparticles.
- In many cases, Polanyi-based models fit adsorption data better than some other classic models (like Langmuir or Freundlich), especially for nonuniform surfaces or multi-component systems.
- The theory also supports ideas about competitive adsorption, where different solutes compete for the same adsorption sites.

Limitations and later work
- The original model can oversimplify by neglecting electrical interactions and the screening effects of other molecules.
- Real surfaces can be curved or porous, which influences adsorption in ways not captured by a flat-surface version.
- Nonetheless, the Polanyi framework remains influential and is often the starting point for more refined adsorption models and for interpreting data from activated carbons and related materials.

In summary, Polanyi potential theory explains adsorption as a balance of chemical potentials driven by a surface-dependent potential, providing a practical way to relate pressure, temperature, and adsorbate amount through the concept of adsorption potential and its associated Dubinin equations. It remains a useful tool for understanding how gases and vapors stick to porous solids like activated carbon and carbon nanotubes.