BET theory expands on Langmuir adsorption, allowing for multilayer gas adsorption on solid surfaces. It's crucial for determining surface area and pore size distribution in porous materials, especially those with mesopores or macropores.

The BET equation, derived from adsorption-desorption equilibrium, helps calculate specific surface area. This is vital for characterizing materials used in catalysis, adsorption, and energy storage, where high surface area often means better performance.

BET Theory for Multilayer Adsorption

Principles and Assumptions

• BET theory extends the Langmuir adsorption model to multilayer adsorption
  • Allows for the determination of surface area and pore size distribution of porous materials
• Assumes gas molecules can adsorb onto a solid surface in an infinite number of layers
  • No interaction between each adsorption layer
• First adsorbed layer has a heat of adsorption equal to the heat of condensation of the adsorbate
  • Subsequent layers have a heat of adsorption equal to the heat of liquefaction
• Introduces the concept of a statistical thickness of the adsorbed film
  • Ratio of the amount of adsorbate in each layer to the amount required to form a monolayer

Applicability and Adsorption Isotherms

• BET theory is most applicable to Type II and Type IV adsorption isotherms
  • Exhibit multilayer adsorption and capillary condensation in mesopores (pores with diameters between 2 and 50 nm)
• Type II isotherms are characteristic of non-porous or macroporous materials (pores with diameters greater than 50 nm)
  • Show unrestricted monolayer-multilayer adsorption
• Type IV isotherms are characteristic of mesoporous materials
  • Exhibit a hysteresis loop associated with capillary condensation in mesopores

BET Equation Derivation and Limitations

Derivation of the BET Equation

• Derived by considering the equilibrium between the rates of condensation and evaporation for each adsorption layer
• Resulting equation: $1/[v((P_0/P)-1)] = (c-1)/(v_mc) (P/P_0) + 1/(v_mc)$
  • $v$: volume of gas adsorbed at pressure $P$
  • $P_0$: saturation pressure
  • $v_m$: volume of gas required to form a monolayer
  • $c$: BET constant related to the heat of adsorption
• BET equation can be linearized to obtain $v_m$ and $c$ from the slope and intercept of a BET plot
  • Plot of $1/[v((P_0/P)-1)]$ versus $P/P_0$

Assumptions and Limitations

• Assumes the surface is energetically homogeneous
  • No lateral interactions between adsorbed molecules
• BET equation is valid only for a limited range of relative pressures
  • Typically between 0.05 and 0.35, where multilayer adsorption is dominant
• May not accurately describe adsorption in microporous materials (pores with diameters less than 2 nm)
  • Pore filling rather than layer-by-layer adsorption occurs in micropores

Surface Area Determination Using BET

Specific Surface Area Calculation

• Specific surface area of a porous material can be calculated from the monolayer volume $v_m$ obtained from the BET equation
• Surface area is given by: $S_{BET} = (v_m * N_A * σ) / (V m)$
  • $N_A$: Avogadro's number
  • $σ$: cross-sectional area of the adsorbate molecule
  • $V$: molar volume of the gas
  • $m$: mass of the sample
• Nitrogen is the most commonly used adsorbate for BET surface area measurements
  • Typical cross-sectional area of 0.162 nm2 per molecule at 77 K

Experimental Procedure

• BET method requires the measurement of an adsorption isotherm
  • Typically using a volumetric or gravimetric technique
  • Obtains the volume of gas adsorbed at different relative pressures
• Sample is degassed to remove adsorbed contaminants
  • Cooled to liquid nitrogen temperature (77 K) for nitrogen adsorption
• Adsorption isotherm is measured by incrementally dosing the sample with nitrogen gas
  • Measuring the equilibrium pressure after each dose

Importance of Specific Surface Area

• Specific surface area is an essential characteristic of porous materials
  • Influences their performance in various applications (catalysis, adsorption, and energy storage)
• High specific surface area materials (activated carbons, zeolites, and metal-organic frameworks) are desirable for adsorption and catalysis
  • Provide more sites for adsorption and reaction
• Specific surface area can be used to compare the effectiveness of different porous materials
  • Optimize their synthesis and processing conditions

BET Isotherm Analysis and Surface Area Calculation

Adsorption Isotherm Interpretation

• BET adsorption isotherm plots the volume of gas adsorbed ($v$) against the relative pressure ($P/P_0$) at a constant temperature
• Shape of the isotherm provides information about the pore structure and adsorption behavior of the material
  • Type II isotherms indicate non-porous or macroporous materials
  • Type IV isotherms indicate mesoporous materials with capillary condensation
• Linear region of the BET plot (typically between relative pressures of 0.05 and 0.35) is used to determine the monolayer volume $v_m$ and the BET constant $c$
  • Slope and intercept of the linear fit are used to calculate $v_m$ and $c$

Surface Area Calculation from Experimental Data

• Specific surface area is calculated from $v_m$ using the appropriate values for the adsorbate cross-sectional area and molar volume
• Example calculation for nitrogen adsorption at 77 K:
  • $v_m$ = 10 cm3/g (from BET plot)
  • $N_A$ = 6.022 × 1023 mol-1
  • $σ$ = 0.162 nm2
  • $V$ = 22414 cm3/mol
  • $m$ = 0.1 g
  • $S_{BET} = (10 * 6.022 × 10^{23} * 0.162 × 10^{-18}) / (22414 0.1) = 43.8 m^2/g$

Additional Information from BET Analysis

• BET constant $c$ is related to the heat of adsorption
  • Provides information about the strength of the adsorbate-adsorbent interactions
  • Higher values of $c$ indicate stronger interactions and a more energetically homogeneous surface
• Deviations from linearity in the BET plot at low or high relative pressures may indicate the presence of micropores or mesopores, respectively
  • Alternative analysis methods (Langmuir, Dubinin-Radushkevich, or Barrett-Joyner-Halenda) may be required for accurate characterization of these materials
• Pore size distribution can be obtained from the adsorption isotherm using methods such as the Barrett-Joyner-Halenda (BJH) analysis
  • Based on the Kelvin equation and the assumption of cylindrical pores