[실험] The BET method/Theory

What is BET (Brunauer-Emmertt-Teller) theory?

  1) used to measure the total surface area of solid or porous materials 
  2) Extended from the Langmuir theory, developed by Irving Langmuir in 1916 
     - based on the absorption of certain molecular in the gas state on the surface 

NOTE 1: terminologies used frequently 
1. Absorption: the enrichment of particles (such as molecules and atoms) in the vicinity of an interface  
  - absorbate: the material in the adsorbed state 
  - adsorptive: the same component in the liquid 
  - Adsorption space: the space occupied by the adsorbate 
  - 
  - Chemisorption: 
1. porosity: the volume ratio of free space to the porous material 
2. physisorption: physical absorption 
  - Van der Waals interaction between the surface and adsorbate molecules 
  --> Recommend using inert gas
  -- N2: Not spherical and has an electrical quadrupole moment. common probe for physisorption due to the availbility of liquid N2  

NOTE 2: Classification of porous materials 
1. Micropores: the size of the pores less than 2nm 
    - examples: Zeolite, etc 
    - HK theory is used to analyze the distribution of microporosity 
2. Mesopores: the size of pores in the range between 2nm and 50nm 
    - Examples: mesoporous silica 
    - BJH theory is used to analyze the distribution of mesoporosity 
3. Macropores: larger than 50nm 
    - Examples: Sintered metal and ceramics 

NOTE 3: Measurement methods used in porous materials 
1. Image-based: SEM and TEM 
2. Dispersion 
3. Mercury porosimetry: used to analyze materials with pore size around 3nm to 250um  
4. Absorption: Freundlich, Langmuir, Temkin 
5. BEM theory 

Measurement	Calculation methods	Notes
Surface aere	BET, Langmuir, Temkin, Freundlich	can be calculated from section of isotherm (generally P/P_0 = 0.05-0.35)
Total pore volume	t-plot, Kelvin equation	Generally carried out at P/P_0=0.99 although theoretically all pores should be full at P/P_0 =0.995
Mesopore volume, area, and distribution	BJH, Dollimore-Heal	Require full adsorption and desorption isotherm
Micropore distribution	- Dubinin-Radushkevich and Astakhov - Horvath-Kawazoe - Saito-Foley	Require full adsorption isotherm
Pore size modeling	DFT	Require full adsorption isotherm
Surface energy	DFT	Require full adsorption isotherm

NOTE 4: The types of hysteresis in the absorption 

 

Types	Descriptions
Type 1	▪ depicts monolayer adsorption     - the amount of adsorption $n_a$ approaches to limiting value  ▪ can be described by the Langmuir theory  ▪ The characterization of microporous materials with pore diameters less than 2mm gives this type of isotherm ▪  when c > 1 in BET theory
Type 2	▪ The flat area in the middle: the formation of monolayer  ▪ Unrestricted mono-multilayer adsorption  ▪ The most common isotherm obtained when using the BET method  ▪ At high pressure, capillary condensation occurs ▪ Example: Non-porous or macroporous
Type 3	▪ show the formation of a multilayer ▪ No monolayer is formed and BET is not applicable  ▪ Not common feature  ▪ Adsorbate-adsorbate interaction  ▪ When c < 1 in BET theory
Type 4	▪ Capillary condensation occurs:   - The gases condense in the tiny capillary pores of the solid below the saturation pressure of the gas     - The capillary condensation occurs in a mesopores  ▪ Show the hysteresis loop   - Initial loop: mono-multilayer adsorption    - 2nd loop: desorption  ▪ BET surface area characterization of mesoporous materials gives this type of isotherm plot
Type 5	▪ Not common feature  ▪ Weak adsorbate-adsorbate interactions  ▪ Porous adsorbets  ▪ not applicable to BET
Type 6	▪ Stepwise multilayer adsorption on a uniform non-porous surface

 

 

 

 

 

 

 

 

 

Langmuir theory

  1) derived from the physical absorption law of the gas molecules on the solid surface 
  2) The assumption used in the theory 
   ▪ The surface of the solid body: completely homogeneous surface 
   ▪ The n-th molecure layer start to be covered after the (n-1)th layer was completly filled 
   ▪ The gas molecules act as ideal behavior where there is no interaction between gas molecules and the absorbent surface 
   ▪ The rate of arrival of adsorption is equal to the rate of desorption 
   ▪ Heat of adsorption was taken to be constant and unchanging with the degree of coverage $\theta$ 

  3) Langmuir theory 

\begin{equation}
\theta = \frac{KP}{1+KP}
\end{equation}

 ▪ P: 

BET theroy

1. Assumptions with Lanmuir's theory 
  1) Gas molecules will physically adsorb on a solid infinitely 
  2) The different adsorption layers do not interact with each other 
  3) The theory can be applied to each layers 

Measurements

1. How to measure the surface area? 

  1) Degas 
   - The solid sample is heated (around 110 c) and vacuumed to remove any adsorbed contaminants from the atmosphere 
   - The system has to be flushed with N2 gas to remove any chemical contaminations (e.g., water vapor on the sample)
   - The sample mass has to be reweighted to take into account any mass loss during degassing 
   - The reference volume also has to be treated in the same way 
   - IUPAC recommendation is that the sample should be degassed for at least 16 hours 
   - A minimum of 0.5 g of sample is required for the BET to successfully determine the surface area 
  2) Evacuation
   - The sample and the reference volume have to be vacuumed 
  3) Measuring Volume   
   - Note: the sample and the reference tube have similar dead volumes 
  4) Adsorption 
   - 

  2) the solid sample is cooled to liquid nitrogen temperature under vacuum (typically at cryogenic temperature) 
  3) Nitrogen gas is dosed to a solid sample in controlled increments 
     ▪ Nitrogen gas: Adsorbate 
     ▪ Porous material: Adsorbent  
  4) After each dose of N2 gas, the relative pressure is allowed to equilibrate, and the weight of the nitrogen adsorbed is determined 
     ▪ The relative pressure and quantity of absorbed gas are measured to give an adsorption isotherm 

NOTE: Why $N_2$ gas is used in BET test? 
 a. high purity 
 b. strong interaction with most solids 
 c. the interaction between liquid and solid phase of the nitrogen is weak 

NOTE: The BET equation 

\begin{equation}
\frac{1}{W(P_0/P-1)}=\frac{1}{W_m C}+\frac{C-1}{W_m C}\left(\frac{P}{P_0}\right)
\end{equation}

  ▪ C: the BET constant, a fixed amount which depends on the enthalpy of absorbed gas ( positive-definitive) 
  ▪ $W_m$: the weight of nitrogen constituting a monolayer of surface coverage 
  ▪  W: The weight of nitrogen introduced in the experiment 
  ▪ P: the experimental pressure 
  ▪ $P_0$: 

NOTE: The above equation is simply $y=a x +b$. To be specific,

\begin{neqarray}
y &=& \frac{1}{W(P_0/P-1)}\\
a &=&  \frac{C-1}{W_m C}\\
x &=& \frac{P}{P_0}\\
b &=&  \frac{1}{W_m C}
\end{ neqarray }

 
NOTE: C and $W_m$ are calculated from the slope and intercept of BET-plot 

  - The weight of nitrogen constituting a monolayer of surface coverage is determined. 
  - The total surface area of the sample can be calculated from the slope and intercept of the BET plot 
  - This theory well fits to type 2 and 4 isotherms at the relative pressure between 0.05 and 0.35 

Note: Troubleshooting 

  (1) If there is a negative intercept
    - If C is above 200: may be indicative of the presence of micropores
    - If C is below 100: may indicate strong adsorbent-adsorbate interaction

 

Note: Why does nitrogen gas use commonly in this process 

  (1) At the LN2 temperature (77K), the equilibrium vapor pressure of N2 is 1 bar 
  - the whole range of thermodynamic potential can be written bewteen vacuum and atmospheric pressure 

$\mu = \mu_0 _ RT ln\frac{p}{p_vac}$

 

Advantages of BET theory 

   - have the ability to porosity of the material whose pore size is 0.4 - 50 nm 
   - easy measurement 
   - low cost in the experiment 
   - Non-destructive method 

Limitation of BET theory 

   - Assume that the surface is homogeneous and smooth 
   - 

 

2. Specific surface and Porosity percentage

2.1. The total surface area $S_{t}$ and the specific surface area $S_{BET}$

\begin{eqnarray}
S_{t} = \frac{v_m N_A \sigma}{V}\\
S_{BET} = \frac{v_m N_A \sigma}{V m}
\end{eqnarray}

where $v_m$ is the monolayer volume obtained from BET equation, $N_A$ is the Avogadro's number, $m$ is the mass of the sample, $V$ is the molar volume of the gas, and $\sigma$ is the cross-section of adsorbate molecules. 

\begin{eqnarray}
 
\end{eqnarray}

The total pore volume: 
 - calculation is based on Kelvin equation 

 

The limitation of BET methods

  - Essentially, it is an estimated methods, e.g., 
  (1) critically criticized for assuming that absorption in the n-layer occurs when the n-1 th layer is completely filled
  (2) 
In the pressure range $P/P_0\propto$ ~ 0.3-0.05, the absorption data is well suited to BET equation. 

 

BJH (Barrett-Joyner-Halenda) method 

- A popular method for estimating pore volume and pore diameter of a porous material 
1) Assumption
  - Pores have a cylinderical shape 
  - Pore radius is equal to the sum of the Kelvin radius and thickness of the film adsorbed on the pore wall 
2) 
  - Produces an average pore volume and a pore diameter 
  - 

The advantages of the BET method 

enable to measure the porosity in the range between 0.4nm to 50nm with low cost 

 

Appendix: Pros and Cons of the BET and BJH methods

	Limitations	Advantages
BET analysis	• Doesn't account for the effects of surface heterogeneity  • Results are sensitve to the choice of adsorbate	• Provide accurate measurements of the specific surface area of porous materials  • relatively simple, reliable, and standardized in surface science
BJH analysis	• relies on the assumption that pores are cylindrical which may not be accurate for all materials  • Only applies to pores where capillary condensation occurs, limiting its use to materials with specific pore sizes and structures	• Accurately determines the pore size distribution of mesoporous materials  • particularly effective for materials with mesoporous structures such as catalysts and adsorbents

 

 

 

Appendix: Derivation of BET equation

Before deriving the BET equation, the theory assumes the following things; 

  (1) The gas molecules undergo multilayer adsroption on solid surface
  (2) The principle of the Langmuir theory can be applied to each layer 
  (3) A dynamic equilibrium exists between successive layers. The rate of evaporation from a particular layer is equal to the rate of condensation of the preceding layer 
       - The uppoermost layer is in equilibrium with vapor phase 
       - Adsorption on the adsorbent occurs in infinite layers 
  (4) The enthalpy of adsorption of the first layer (E1) is a constant 
  (5) Condensation forces are the principal forces of attraction 

The approximations made by BET; 
  a. $E_2=E_3=\cdot=E_i=E_L$
  b. $\frac{b_i}{a_i}=g$

• $S_i$: The surface area of the adsorbent covered by i-th layers of adsorbates 
• The rate of condensation on the bare surface = $a_1 P S_0$
• The rate of evaporation from the first layer = $b_1 S_1 e^{\frac{E_1}{RT}}$   
 → From the assumption (3),
     - $a_1 P S_0$ = $b_1 S_1 e^{\frac{E_1}{RT}}$
        ...   
     - $a_{i} P S_{i-1}$ = $b_{i} S_{i} e^{\frac{E_i}{RT}}$  
• Total surface area of the adsorbent: $A = \Sigma S_i$ 
• Total volume of the adsorbent gas: $V = V_0 \Sigma (i S_i)$ 

\begin{equation}
\frac{V}{A} = \frac{V_0 \Sigma (i S_i)}{ \Sigma S_i } \rightarrow \frac{V}{A V_0} = \frac{\Sigma (i S_i)}{ \Sigma S_i } = \frac{V}{V_m}
\end{equation}

• $V_m$: The volume of the gas for monolayer coverage 

  From the assumption (3),

\begin{eqnarray}
S_1 &=& \frac{a_1}{b_1}P e^{\frac{E_1}{RT}} S_0\\
S_2 &=& \frac{a_2}{b_2}P e^{\frac{E_2}{RT}} S_1 = \frac{P}{g}e^{\frac{E_L}{RT}} S_1 = x S_1 = xy S_0\\
S_i &=& x^{i-1}yS_0 = x^{i} c S_0,   \left(c = \frac{y}{x} = \frac{a_1}{b_1}g e^{\frac{E_1-E_L}{RT}}\right)
\end{eqnarray}

Therefore, using the above relation, the volume ratio can be rewritten as below;

\begin{eqnarray}
\frac{V}{V_m} &=&  \frac{\Sigma (i S_i)}{ \Sigma S_i } = \frac{c S_0 \Sigma (i x^i) }{ S_0 + c S_0 \Sigma (x^i) }\\
&=&\frac{c \frac{x}{(1-x)^2}}{1+ c \frac{x}{(1-x)} } = \frac{cx}{(1-x)(1+(c-1)x)}
\end{eqnarray}

Therefore, the final expression is 

\begin{equation}
\frac{x}{(1-x)V} = \frac{1}{cV_m} + \frac{c-1}{c V_m} x, (x = \frac{p}{p_0})
\end{equation}

• $p$: the equilibrium pressure of the gas over the surface 
• $p_0$: the saturated vapor pressure of the gas at experimental conditions 

1. The relation between the BET equation and the Langmuir equation

Let's suppose that in the BET equation, the finite layers covered by the absorbents exist. Then if we assume n=1, then the BET equation returns to the Langmuir equation. 

2. The Explanations of isotherms by BET theory

2.1. Type 1 isotherm 
When $E_1>>E_L$, the BET equation returns to Lanmuir isotherm for monolayer adsorption 
2.2. Type 2 isotherm 
When $E_1>E_L$, the BET equation gives type 2 adsorption isotherm, indicating multilayer adsorption 
2.3. Type 3 isotherm 
When $E_1<E_L$, it gives the type 3 adsorption that the multilayer begins even before the completion of monolayer adsorption  

Type	c and P/P_0	Notes
Type 1	▪ c > 1  ▪ $P/P_0 < 1$
Type 2	▪ c > 1

3. Surface area and the BET equation

The surface area of the adsorbent can be obtained by calculating the volume of monolayer coverage ($V_m$) using BET and Langmuir isotherms;

• $V_m$: the volume of the gas at STP required to cover the whole surface of adsorbent by monolayer adsorption 
• The number of gas molecules in $V_m$ at STP;

\begin{equation}
n = \frac{N_0}{22.414}V_m
\end{equation}

• The total monolayer surface area = $\frac{N_0 V_m S}{22.414}$ 
S: the area of single adsorbate gas molecule

Note that the number of gas molecules at STP in 22.414 is Avogadro number.