BET Theory and How Its Used to Calculate Surface Area

HORIBA Scientific Particle Analysis

Carl Lundstedt

BET Theory and how its used to calculate surface area

June 25, 2019

Adsorbate

Adsorbent

monolayer absorbed gas volume (vm)

1 1 1 = + [( / ) – 1] 𝑐𝑐 − 𝑝𝑝 𝑣𝑣 𝑝𝑝0 𝑝𝑝 𝑣𝑣𝑚𝑚𝑐𝑐 𝑝𝑝0 𝑣𝑣𝑚𝑚𝑐𝑐 = adsorbed gas quantity = saturation pressure of adsorbate 𝑣𝑣 = equilibrium pressure of adsorbate 𝑝𝑝0 = BET constant = exp 𝑝𝑝 1 𝐿𝐿 = heat of adsorption for𝐸𝐸 −the𝐸𝐸 first layer 𝑐𝑐 𝑅𝑅𝑅𝑅 = heat of vaporization 𝐸𝐸1 𝐸𝐸𝐿𝐿

1 1 1 = + [( / ) – 1] 𝑐𝑐 − 𝑝𝑝 𝑣𝑣 𝑝𝑝0 𝑝𝑝 = 𝑣𝑣𝑚𝑚𝑐𝑐+ 𝑝𝑝0 𝑣𝑣𝑚𝑚𝑐𝑐

1 𝑦𝑦 𝑚𝑚𝑥𝑥 𝑏𝑏 [( / ) – 1]

𝑣𝑣 𝑝𝑝0 𝑝𝑝 1

𝑐𝑐 − 1 𝑣𝑣𝑚𝑚𝑐𝑐

1 = 𝑐𝑐 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 intercept =𝑣𝑣𝑚𝑚𝑐𝑐 1 1 = 𝑣𝑣𝑚𝑚𝑐𝑐 + 𝑣𝑣𝑚𝑚 = 𝑠𝑠𝑠𝑠𝑠𝑠1 +𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 © 2018 HORIBA, Ltd. All rights reserved. 8 From the monolayer absorbed gas volume (vm), we can determine total and specific surface area

= 𝑣𝑣𝑚𝑚𝑁𝑁𝑁𝑁 = total surface area𝑆𝑆𝑡𝑡 of sample material 𝑉𝑉 = monolayer absorbed gas volume 𝑡𝑡 𝑆𝑆 = Avogadro’s number = 6.02 x 1023 molecules/mol 𝑚𝑚 𝑣𝑣 = cross-sectional area of adsorbed gas molecule 𝑁𝑁 = molar volume of adsorbed gas 𝑠𝑠 𝑉𝑉 = [=] m2/g 𝑆𝑆𝑡𝑡 𝑆𝑆𝐵𝐵𝐵𝐵𝐵𝐵 𝑎𝑎 = specific surface area = mass of sample 𝑆𝑆𝐵𝐵𝐵𝐵𝐵𝐵 𝑎𝑎 © 2018 HORIBA, Ltd. All rights reserved. 9 Can only assume linear relationship for adsorption isotherms in the range of 0.05 < / < 0.30

𝑝𝑝 𝑝𝑝0

Micropores Mesopores Macropores

Zeolite, Mesoporous silica, Sintered metals Activated Activated carbon and ceramics carbon, Metal organic framework 2 nm 50 nm

Macroporous/ Microporous non-porous (C > 150) (C > 80) (C < 2)

Mesoporous Step-wise (C > 80) (C < 2) multilayer

1st point 2nd point 3rd point

for BET constant (c) and Vm (monolayer volume)

BET constant = 145.84 Surface area = 11.36 m2/g