The potential theory of Polanyi

Polanyi3 approached the phenomenon of adsorption in a completely different way. He assumes that the molecules close to the surface feel a potential — similar to the gravitation held of the earth. One cause for this potential is the van der Waals attraction. The potential compresses the gas close to the surface, isothermally. Once the pressure becomes higher than the equilibrium vapor pressure, it condenses. In equilibrium, the chemical potential of the gas at a distance x from the surface /t(.x, / ) must be equal to the chemical potential /t(oo, P) at a large distance away form the surface. Px is the local pressure at a distance x, P is the partial pressure far from the surface. The differential of the chemical potential is 

Sm is the molar entropy, Vm the molar volume of the gas at a distance x, and Um is the molar internal energy. Integrating the left side from infinite distance to a distance x leads to 

The integration is simple since the chemical potential is a state function. The integral is zero because in equilibrium the chemical potential should be the same everywhere. Integration of the right side of Eq. (9.50) at constant temperature leads to 

3 Michael Polanyi, 1891-1976. Hungarian physicist who worked in Berlin and Manchester. His son, John Charles Polanyi, received the Noble Price in chemistry in 1986. 

The internal energy at x = oo was set to zero Um(oo) = 0. Now we assume that the gas behaves like an ideal gas until it condenses. For an ideal gas we can use VmP = RT and solve the integral  

---

Surface Area and Permeability or Porosity. Gas or solute adsorption is typicaUy used to evaluate surface area (74,75), and mercury porosimetry is used, ia coajuactioa with at least oae other particle-size analysis, eg, electron microscopy, to assess permeabUity (76). Experimental techniques and theoretical models have been developed to elucidate the nature and quantity of pores (74,77). These iaclude the kinetic approach to gas adsorptioa of Bmaauer, Emmett, and TeUer (78), known as the BET method and which is based on Langmuir s adsorption model (79), the potential theory of Polanyi (25,80) for gas adsorption, the experimental aspects of solute adsorption (25,81), and the principles of mercury porosimetry, based on the Young-Duprn expression (24,25). 

---