1. BET equation
Assumption for BET(Brunauer, Emmett and Teller) method
- Gas adsorption at the flat and uniform surface
- no lateral interaction between the absorbed molecules
- multi-layer adsorption is possible
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BET equation 1
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2. BET equation 2
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BET equation 3
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3. BET equation 4
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BET equation 5
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4. BET equation 6
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BET equation 7
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5. BET equation 8
V = Volume of gas adsorbed at pressure P
Vm = Volume of gas which could cover the entire adsorbing surface with a
monomolecular layer
Po = Saturation pressure of the gas, i.e. the pressure of the gas in equilibrium
with bulk liquid at the temperature of the measurement.
P/Po = relative pressure
Advanced Electronic Ceramics I (2004)
BET analysis
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6. BET analysis
V = Volume of gas adsorbed at pressure P
Vm = Volume of gas which could cover the
entire adsorbing surface with a
Typical range monomolecular layer
Po = Saturation pressure of the gas, i.e. the
1 pressure of the gas in equilibrium with bulk
liquid at the temperature of the
2 measurement.
Single point BET
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Validity and limit of BET analysis
Assumptions (And limit)
1. Gas adsorbs on the flat, uniform surface of the solid with a uniform
heat of adsorption
Non-uniform surface due to surface steps, cracks, edges, vacancies
and other defects (thereby, non-uniform adsorption energy)
2. No interaction between the adsorbed molecules.
Adsorbed molecules can interact with each other
3. From the 2nd layer, gas adsorb either on the remain the remaining
free surface or on top of the already adsorbed layer.
4. The adsorption of the 2nd and subsequent layers occurs with a same
heat of adsorption (= the energy of evaporation)
The heat of adsorption is likely to change gradually as additional
layers build up rather than in a single step.
5. There is no limit to the number of layers which can adsorb.
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7. BET analysis
Why multi-point adsorption at constant temperature?
In order to eliminate the changes in pressure from small changes in
temperature. (using the regression of multi points of data)
A small temperature fluctuation changes the saturation vapor
pressure considerably.
Ex) 0.1K increase changes the saturation pressure of nitrogen
from approx. 760 mm Hg to 800 mm Hg.
Argon or Krypton as an absorbent can be used for more sensitive
measurement when the surface area of the sample is very low.
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BET equipment
Surface Area:
Specific: 0.01 m2/g and higher
Total: 0.1 m2 and higher, +- 0.03 m2
Pore Volume: 4 x 10-6 cm3/g and higher
- Fully automatic operation
- High speed operation
- Multi-point analysis automatically
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8. BET equipment
Each pressure point completely equilibrates
before the next point is taken. The Gemini
has, as illustrated in Figure 1 (below), two
gas reservoirs (A) which are filled with equal
volumes of the desired adsorptive, usually
nitrogen. From the reservoirs, gas is dosed
into the sample and balance tubes. A
transducer (B) on the sample side monitors
for the target pressure. As the sample
adsorbs gas, the pressure would tend to
decrease in the sample tube were it not that
transducer (B) causes a fast response servo
valve (C) to hold the pressure constant.
Transducer (D) located between the sample
and balance tubes detects any pressure
difference between the two tubes and causes
another servo valve (E) to balance the
pressures in both tubes. A third pressure
transducer (F) monitors the pressure
between the two reservoirs to determine the
amount of gas that is adsorbed on the
sample.
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Scherrer equation
λ : wavelength
0.9λ t : crystallite size
B : broadening at half
t= of the peak height
B cosθB angular width of 2θ
θB : Bragg angle
λ = 1.5Å
d=1Å
θ = 49o
t = 1mm
B=2x10-7 rad (10-5deg)
t=500Å
B=4x10-3 rad (0.2 deg)
B. D. Cullity, Elements of X-ray Diffraction
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9. Comparison
Information Sedimentation Laser Microscopy BET Scherrer
(Photo-sedi.) Diffraction (SEM) equation
(X-ray sedi.) (TEM)
0.02-50 µm 0.01-50 µm > 0.0005 µm ~0.001 µm
size range no limit
particle primary + primary + primary primary + primary
secondary secondary secondary
dispersion Yes Yes usually no no no
medium
Main convenient sub-micrometer direct actual crystallite
advantage detection information contribution size
(size & morphology) of surface
Main cannot know cannot know
disadvantage the real particle the real particle small number
sizes when the sizes when the of observation
dispersion is dispersion is tends to lead
not good not good error in estimation
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