Advanced Electronic Ceramics I (2004)

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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Advanced Electronic Ceramics I (2004)

BET equation 1

Advanced Electronic Ceramics I (2004)

BET equation 2

Advanced Electronic Ceramics I (2004)

BET equation 3

Advanced Electronic Ceramics I (2004)

BET equation 4

Advanced Electronic Ceramics I (2004)

BET equation 5

Advanced Electronic Ceramics I (2004)

BET equation 6

Advanced Electronic Ceramics I (2004)

BET equation 7

Advanced Electronic Ceramics I (2004)

BET equation 8

V = Volume of gas adsorbed at pressure PVm = Volume of gas which could cover the entire adsorbing surface with a monomolecular layerPo = 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

Advanced Electronic Ceramics I (2004)

BET analysis

1

2

Typical range

V = Volume of gas adsorbed at pressure PVm = Volume of gas which could cover the entire adsorbing surface with a monomolecular layerPo = Saturation pressure of the gas, i.e. the pressure of the gas in equilibrium with bulk liquid at the temperature of the measurement.

Single point BET

Advanced Electronic Ceramics I (2004)

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.

Validity and limit of BET analysis

Advanced Electronic Ceramics I (2004)

BET analysis

http://www.micromeritics.com/sa_gemini_at.html

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.

Advanced Electronic Ceramics I (2004)

BET equipment

http://www.micromeritics.com/sa_gemini_at.html

Surface Area:Specific: 0.01 m2/g and higherTotal: 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

Advanced Electronic Ceramics I (2004)

BET equipmentEach 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.

http://www.micromeritics.com/sa_gemini_at.html

Advanced Electronic Ceramics I (2004)

Scherrer equation

0.9λt =

B cosθB

λ : wavelengtht : crystallite sizeB : broadening at half

of the peak heightangular width of 2θ

θB : Bragg angle

λ = 1.5Åd = 1 Åθ = 49o

t = 1mmB=2x10-7 rad (10-5deg)

t=500ÅB=4x10-3 rad (0.2 deg)

B. D. Cullity, Elements of X-ray Diffraction

Advanced Electronic Ceramics I (2004)

Comparison

Sedimentation(Photo-sedi.)(X-ray sedi.)

0.02-50 µm

primary +secondary

Yes

convenient

cannot knowthe real particlesizes when the dispersion isnot good

Laser Diffraction

0.01-50 µm

primary +secondary

Yes

sub-micrometerdetection

cannot knowthe real particlesizes when the dispersion isnot good

Microscopy(SEM)(TEM)

> 0.0005 µm

primary

usually no

direct information(size & morphology)

small numberof observationtends to leaderror in estimation

BET

no limit

primary +secondary

no

actualcontributionof surface

Scherrer equation

~0.001 µm

primary

no

crystallite size

Information

size range

particle

dispersionmedium

Main advantage

Maindisadvantage