10 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| BET theory | 3/4 | https://en.wikipedia.org/wiki/BET_theory | reference | science, encyclopedia | 2026-05-05T10:03:51.162056+00:00 | kb-cron |
∑
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∞
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<
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{\displaystyle \sum _{n=1}^{\infty }\theta _{n}=\theta _{0}+c\theta _{0}x\sum _{n=0}^{\infty }x^{n}=\theta _{0}+c\theta _{0}{\frac {x}{1-x}},\quad |x|<1}
Lastly, defining the excess coverage as
V
excess
=
V
ads
/
V
mono
{\displaystyle V_{\text{excess}}=V_{\text{ads}}/V_{\text{mono}}}
, the excess volume relative to the volume of an adsorbed mono-layer becomes
V
excess
=
∑
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=
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∞
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θ
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x
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)
(
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)
{\displaystyle V_{\text{excess}}={\frac {\sum _{n=0}^{\infty }n\theta _{n}}{1}}={\frac {\sum _{n=0}^{\infty }n\theta _{n}}{\sum _{n=0}^{\infty }\theta _{n}}}={\frac {cx}{(1-x)(1+(c-1)x)}}}
where the last equality was obtained by making use of the series expansions presented above. The constant
c
{\displaystyle c}
must be interpreted as the relative binding affinity the substance A has towards a surface, relative to its own liquid. If
c
>
1
{\displaystyle c>1}
then the initial part of the isotherm will be reminiscent of the Langmuir isotherm which reaches a plateau at full mono-layer coverage, whereas
c
<
1
{\displaystyle c<1}
means the mono-layer will have a slow build-up. Another thing to note is that in order for the geometric substitutions to hold,
x
<
1
{\displaystyle x<1}
. The isotherm above exhibits a singularity at
x
∗
=
1
{\displaystyle x^{\ast }=1}
. Since
x
=
K
ℓ
P
{\displaystyle x=K_{\ell }P}
one can write
x
∗
=
K
ℓ
P
∗
{\displaystyle x^{\ast }=K_{\ell }P^{\ast }}
, implying that
K
ℓ
=
1
/
P
∗
{\displaystyle K_{\ell }=1/P^{\ast }}
. This means that
x
=
P
/
P
∗
{\displaystyle x=P/P^{\ast }}
must be true, ultimately resulting in
x
∈
[
0
,
1
)
{\displaystyle x\in [0,1)}
.
== Finding the linear BET range == It is still not clear on how to find the linear range of the BET plot for microporous materials in a way that reduces any subjectivity in the assessment of the monolayer capacity. A crowd-sourced study involving 61 research groups has shown that reproducibility of BET area determination from identical isotherms is, in some cases, problematic. Rouquerol et al. suggested a procedure that is based on two criteria:
C must be positive implying that any negative intercept on the BET plot indicates that one is outside the valid range of the BET equation. Application of the BET equation must be limited to the range where the term V(1-P/P0) continuously increases with P/P0. These corrections are an attempt to salvage the BET theory, which is restricted to type II isotherms. Even while using this type, use of the data itself is restricted to 0.05 to 0.35 of
P
/
P
0
{\displaystyle P/P_{0}}
, routinely discarding 70% of the data. This restriction must be modified depending upon conditions.
== Limitations of BET == Terrell L. Hill described BET as a theory that is "... extremely useful as a qualitative guide; but it is not quantitatively correct". Although BET adsorption isotherm is still extensively used for different applications and is used for specific surface area determinations of powders whose calculation is not sensitive to the simplifications of the BET theory. Both Hackerman's and Sing's group have highlighted the limitations of the BET method. Hackerman et al. noted the potential for 10% uncertainty in the method's values, with Sing's group attributed the significant variation in reported values of molecular area to the BET method's possible inaccurate assessment of monolayer capacity. In subsequent studies using the BET interpretation of nitrogen and water vapor adsorption isotherms, the reported area occupied by an adsorbed water molecule on fully hydroxylated silica ranged from 0.25 to 0.44 nm². Other issues with the BET include the fact that in certain cases, BET leads to anomalies such as reaching an infinite amount adsorbed at
P
/
P
0
{\displaystyle P/P_{0}}
reaching unity, and in some cases, the constant C (surface binding energy) can be determined to be negative.
== Applications ==
=== Cement and concrete === The rate of curing of concrete depends on the fineness of the cement and of the components used in its manufacture, which may include fly ash, silica fume and other materials, in addition to the calcinated limestone which causes it to harden. Although the Blaine air permeability method is often preferred, due to its simplicity and low cost, the nitrogen BET method is also used. When hydrated cement hardens, the calcium silicate hydrate (or C-S-H), which is responsible for the hardening reaction, has a large specific surface area because of its high porosity. This porosity is related to a number of important properties of the material, including the strength and permeability, which in turn affect the properties of the resulting concrete. Measurement of the specific surface area using the BET method is useful for comparing different cements. This may be performed using adsorption isotherms measured in different ways, including the adsorption of water vapour at temperatures near ambient, and adsorption of nitrogen at 77 K (the boiling point of liquid nitrogen). Different methods of measuring cement paste surface areas often give very different values, but for a single method the results are still useful for comparing different cements.
=== Activated carbon === Activated carbon has strong affinity for many gases and has an adsorption cross section
s
{\displaystyle s}
of 0.162 nm2 for nitrogen adsorption at liquid-nitrogen temperature (77 K). BET theory can be applied to estimate the specific surface area of activated carbon from experimental data, demonstrating a large specific surface area, even around 3000 m2/g. However, this surface area is largely overestimated due to enhanced adsorption in micropores, and more realistic methods should be used for its estimation, such as the subtracting pore effect (SPE) method.