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Overview of BET Theory
The BET theory was developed by Stephen Brunauer (Figure (PageIndex{1}) ), Paul Emmett (Figure (PageIndex{2}) ), and Edward Teller (Figure (PageIndex{3}) ) in 1938. The first letter of each publisher's surname was taken to name this theory. The BET theory was an extension of the Langmuir theory, developed by Irving Langmuir (Figure (PageIndex{4}) ) in 1916.
The Langmuir theory relates the monolayer adsorption of gas molecules (Figure (PageIndex{5}) ), also called adsorbates, onto a solid surface to the gas pressure of a medium above the solid surface at a fixed temperature to ref{1} , where θ is the fractional cover of the surface, P is the gas pressure and α is a constant.
[ Theta = frac{alpha cdot P}{1 + (alpha cdot P)} label{1} ]
The Langmuir theory is based on the following assumptions:
- All surface sites have the same adsorption energy for the adsorbate, which is usually argon, krypton or nitrogen gas. The surface site is defined as the area on the sample where one molecule can adsorb onto.
- Adsorption of the solvent at one site occurs independently of adsorption at neighboring sites.
- Activity of adsorbate is directly proportional to its concentration.
- Adsorbates form a monolayer.
- Each active site can be occupied only by one particle.
The Langmuir theory has a few flaws that are addressed by the BET theory. The BET theory extends the Langmuir theory to multilayer adsorption (Figure (PageIndex{1}) ) with three additional assumptions:
- Gas molecules will physically adsorb on a solid in layers infinitely.
- The different adsorption layers do not interact.
- The theory can be applied to each layer.
How does BET Work?
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Adsorption is defined as the adhesion of atoms or molecules of gas to a surface. It should be noted that adsorption is not confused with absorption, in which a fluid permeates a liquid or solid. The amount of gas adsorbed depends on the exposed surface are but also on the temperature, gas pressure and strength of interaction between the gas and solid. In BET surface area analysis, nitrogen is usually used because of its availability in high purity and its strong interaction with most solids. Because the interaction between gaseous and solid phases is usually weak, the surface is cooled using liquid N2 to obtain detectable amounts of adsorption. Known amounts of nitrogen gas are then released stepwise into the sample cell. Relative pressures less than atmospheric pressure is achieved by creating conditions of partial vacuum. After the saturation pressure, no more adsorption occurs regardless of any further increase in pressure. Highly precise and accurate pressure transducers monitor the pressure changes due to the adsorption process. After the adsorption layers are formed, the sample is removed from the nitrogen atmosphere and heated to cause the adsorbed nitrogen to be released from the material and quantified. The data collected is displayed in the form of a BET isotherm, which plots the amount of gas adsorbed as a function of the relative pressure. There are five types of adsorption isotherms possible.
Type I Isotherm
Type I is a pseudo-Langmuir isotherm because it depicts monolayer adsorption (Figure (PageIndex{6}) ). A type I isotherm is obtained when P/Po < 1 and c > 1 in the BET equation, where P/Po is the partial pressure value and c is the BET constant, which is related to the adsorption energy of the first monolayer and varies from solid to solid. The characterization of microporous materials, those with pore diameters less than 2 nm, gives this type of isotherm.
Type II Isotherm
A type II isotherm (Figure (PageIndex{7}) ) is very different than the Langmuir model. The flatter region in the middle represents the formation of a monolayer. A type II isotherm is obtained when c > 1 in the BET equation. This is the most common isotherm obtained when using the BET technique. At very low pressures, the micropores fill with nitrogen gas. At the knee, monolayer formation is beginning and multilayer formation occurs at medium pressure. At the higher pressures, capillary condensation occurs.
Type III Isotherm
A type III isotherm (Figure (PageIndex{8}) ) is obtained when the c < 1 and shows the formation of a multilayer. Because there is no asymptote in the curve, no monolayer is formed and BET is not applicable.
Type IV Isotherm
Type IV isotherms (Figure (PageIndex{9}) ) occur when capillary condensation occurs. Gases condense in the tiny capillary pores of the solid at pressures below the saturation pressure of the gas. At the lower pressure regions, it shows the formation of a monolayer followed by a formation of multilayers. BET surface area characterization of mesoporous materials, which are materials with pore diameters between 2 - 50 nm, gives this type of isotherm.
Type V Isotherm
Type V isotherms (Figure (PageIndex{10}) ) are very similar to type IV isotherms and are not applicable to BET.
Terms And Conditions
1. This promotion is only valid for new customers.
2. A 'Qualifying Bet' is a losing first settled real money single bet of at least £10.00 or more at odds of Evens (2.0) or greater.
3. Multiples will not count towards this offer.
4. Free sportsbook bets will be credited within 24 hours of your qualifying bet settling and must be used in full within 14 days. After the 14 days the free bet will expire.
5. Any returns from a free bet will be paid back into your account minus the free bet stake.
6.The £5 casino bonus will be credited to all new customers on the settling of the first qualifying bet. This will be irrelevant of whether the sportsbook bet wins or loses.
7. Casino bonuses will be credited within 24 hours of the first qualifying bet settling.
8. Players must play through their bonus 20 times before you can withdraw/use on the sportsbook.
9.The rollover requirements must be met within 30 days of the bonus being awarded.
10. Bonus funds and winnings will be forfeited upon expiry of the bonus.
11. The £5 free sportsbook bet and the £5 casino bonus can not be used in conjunction with any other Black Type promotion.
12. Only one Black Type casino promotion can be entered into at any one time.
13. The bonus can be used on all games. However, progressive slots do not contribute towards the turnover requirements of the bonus.
14. The maximum stake that will contribute towards turnover requirements in any single bet will be £100.
Casino no gamstop. 15.The maximum amount that can be gained from the bonus is £1,000. If you have a balance in excess of this when meeting the wagering requirements only a balance of £1,000 will be credited by way of withdrawable funds.
16. The stake contribution of all games categories at Black Type are; Roulette (all varieties) 10%, Blackjack 10%, Poker 5%, scratch cards 100%, Progressive slots 0%, Baccarat 0%. For example, A £5 bet on the slots (non progressive) will be calculated as £5 for wagering requirements whilst a bet on roulette will be calculated as £0.50.
17. This offer is limited to one free bet award per person/household/IP address per day.
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Adsorption is defined as the adhesion of atoms or molecules of gas to a surface. It should be noted that adsorption is not confused with absorption, in which a fluid permeates a liquid or solid. The amount of gas adsorbed depends on the exposed surface are but also on the temperature, gas pressure and strength of interaction between the gas and solid. In BET surface area analysis, nitrogen is usually used because of its availability in high purity and its strong interaction with most solids. Because the interaction between gaseous and solid phases is usually weak, the surface is cooled using liquid N2 to obtain detectable amounts of adsorption. Known amounts of nitrogen gas are then released stepwise into the sample cell. Relative pressures less than atmospheric pressure is achieved by creating conditions of partial vacuum. After the saturation pressure, no more adsorption occurs regardless of any further increase in pressure. Highly precise and accurate pressure transducers monitor the pressure changes due to the adsorption process. After the adsorption layers are formed, the sample is removed from the nitrogen atmosphere and heated to cause the adsorbed nitrogen to be released from the material and quantified. The data collected is displayed in the form of a BET isotherm, which plots the amount of gas adsorbed as a function of the relative pressure. There are five types of adsorption isotherms possible.
Type I Isotherm
Type I is a pseudo-Langmuir isotherm because it depicts monolayer adsorption (Figure (PageIndex{6}) ). A type I isotherm is obtained when P/Po < 1 and c > 1 in the BET equation, where P/Po is the partial pressure value and c is the BET constant, which is related to the adsorption energy of the first monolayer and varies from solid to solid. The characterization of microporous materials, those with pore diameters less than 2 nm, gives this type of isotherm.
Type II Isotherm
A type II isotherm (Figure (PageIndex{7}) ) is very different than the Langmuir model. The flatter region in the middle represents the formation of a monolayer. A type II isotherm is obtained when c > 1 in the BET equation. This is the most common isotherm obtained when using the BET technique. At very low pressures, the micropores fill with nitrogen gas. At the knee, monolayer formation is beginning and multilayer formation occurs at medium pressure. At the higher pressures, capillary condensation occurs.
Type III Isotherm
A type III isotherm (Figure (PageIndex{8}) ) is obtained when the c < 1 and shows the formation of a multilayer. Because there is no asymptote in the curve, no monolayer is formed and BET is not applicable.
Type IV Isotherm
Type IV isotherms (Figure (PageIndex{9}) ) occur when capillary condensation occurs. Gases condense in the tiny capillary pores of the solid at pressures below the saturation pressure of the gas. At the lower pressure regions, it shows the formation of a monolayer followed by a formation of multilayers. BET surface area characterization of mesoporous materials, which are materials with pore diameters between 2 - 50 nm, gives this type of isotherm.
Type V Isotherm
Type V isotherms (Figure (PageIndex{10}) ) are very similar to type IV isotherms and are not applicable to BET.
Terms And Conditions
1. This promotion is only valid for new customers.
2. A 'Qualifying Bet' is a losing first settled real money single bet of at least £10.00 or more at odds of Evens (2.0) or greater.
3. Multiples will not count towards this offer.
4. Free sportsbook bets will be credited within 24 hours of your qualifying bet settling and must be used in full within 14 days. After the 14 days the free bet will expire.
5. Any returns from a free bet will be paid back into your account minus the free bet stake.
6.The £5 casino bonus will be credited to all new customers on the settling of the first qualifying bet. This will be irrelevant of whether the sportsbook bet wins or loses.
7. Casino bonuses will be credited within 24 hours of the first qualifying bet settling.
8. Players must play through their bonus 20 times before you can withdraw/use on the sportsbook.
9.The rollover requirements must be met within 30 days of the bonus being awarded.
10. Bonus funds and winnings will be forfeited upon expiry of the bonus.
11. The £5 free sportsbook bet and the £5 casino bonus can not be used in conjunction with any other Black Type promotion.
12. Only one Black Type casino promotion can be entered into at any one time.
13. The bonus can be used on all games. However, progressive slots do not contribute towards the turnover requirements of the bonus.
14. The maximum stake that will contribute towards turnover requirements in any single bet will be £100.
Casino no gamstop. 15.The maximum amount that can be gained from the bonus is £1,000. If you have a balance in excess of this when meeting the wagering requirements only a balance of £1,000 will be credited by way of withdrawable funds.
16. The stake contribution of all games categories at Black Type are; Roulette (all varieties) 10%, Blackjack 10%, Poker 5%, scratch cards 100%, Progressive slots 0%, Baccarat 0%. For example, A £5 bet on the slots (non progressive) will be calculated as £5 for wagering requirements whilst a bet on roulette will be calculated as £0.50.
17. This offer is limited to one free bet award per person/household/IP address per day.
18. If your free bet is subsequently voided it will be replaced within 48 hours.
19.This offer applies to bets placed on the desktop, mobile and Black Type app.
20. Black Type Bookmakers reserves the right to cancel an entry to the promotion, terminate any account and withhold any promotional benefits at any time and/ or take such other action as it deems appropriate in its absolute discretion, in particular but without limitation in respect if we suspect a player is abusing this offer.
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21. Fast withdrawal casino uk. Black Type Bookmakers reserves the right to void any bets pre-event and return money to players account that follow a predetermined pattern of suspected bonus abuse.
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