Q7(a)(i). BET equation and the surface area of a solid

The equation

1. Brunauer, Emmett and Teller extended the Langmuir model to multilayer adsorption: the first layer is held by the surface, further layers form by condensation of the vapour on the layer below. The BET isotherm: VVₘₒₙ=cz(1-z)1-(1-c)z, in linear form, (z)/((1-z)V)=1cVₘₒₙ+(c-1)cVₘₒₙ z

The terms

1. V: volume of gas adsorbed (reduced to STP) at pressure p. p*: vapour pressure of the liquid adsorbate at the temperature of the experiment. z=p/p*: relative pressure.
2. Vₘₒₙ: volume of gas needed to cover the surface with a complete monolayer.
3. c: constant, c=e^(Δ_(des)H-ΔᵥₐₚH)/RT. It is large when the enthalpy of desorption from the first layer is much greater than the enthalpy of vaporisation of the liquid (strong surface binding); then the isotherm has a sharp knee at monolayer coverage (type II isotherm).
4. The isotherm rises without limit as p→ p*: the adsorbate then condenses to bulk liquid.

Surface area

1. Measure V at several pressures, usually N₂ at its boiling point (77 K). Plot z/(1-z)V against z in the range z≈0.05 to 0.35, where the plot is linear.
2. Slope s=c-1cVₘₒₙ, intercept i=1cVₘₒₙ. Hence Vₘₒₙ=(1)/(s+i), c=1+s/i
3. Number of molecules in the monolayer =Vₘₒₙ22 414 cm³ mol⁻¹N_A. Each N₂ molecule covers about 0.16 nm²: 3pt] surface area=Vₘₒₙ22 414 cm³× N_A×σ
4. Dividing by the mass of sample gives the specific surface area (m² g⁻¹), which includes the walls of all open pores.

Application. BET surface areas are the standard specification of catalysts, activated charcoal (several hundred m² g⁻¹), zeolites and silica gel, since catalytic activity and adsorption capacity scale with the area.