At a given N2 adsorption quantity, i.e., the same total surface area N2 can cover, Langmuir (where N2 is adsorbed as a monolayer) will give a bigger value. Hope I have made myself clear.

It depends on your material. If you measure physisorption on a porous sample, there is not much physical meaning for the Langmuir isotherm (it assumes that surface saturates when adsorption sites are full), in contrast to BET which assumes multilayer coverage.

There is no meaning in comparing both areas. You need to employ the surface area measurement procedure which is adequate for your system.

I would fully agree with Fernando. Assume you are testing a microporous material, then Mangmuir is probably only an approximation of your true surface area, so maybe a characteristic surface area. The concept cannot really distinguish between monolayer coverage and pore filling and this is where you might get an over-estimation. BET developed this concept further to multilayer sorption, assuming that at low p/po the high energy sites get filled, which form more or less a straight line up to p/po~0.3 (depends a bit on the sample). This can be used to extrapolate to specific surface area which might be more accurate than Langmuir. I think it is also much more widely used than Langmuir. There are other concepts in place and there is endless literature on these concepts.

I might be partly repeating what Fernando and Andreas have already said, and overcomplicating matters, but I think I would express it slightly differently.

If you are considering a microporous material that exhibits Type 1 behaviour - which must be the case in order to fit the Langmuir equation - then the isotherm shape is generally due to pore-filling rather than monolayer adsorption. However, calculating a surface area using the Langmuir equation involves assuming that the saturation uptake (the amount of N2 in mol/g adsorbed when the isotherm flattens at high relative pressures) is due to the N2 forming a single monolayer over the surface of the material.

It is likely that the pore-filling mechanism, which results in the saturation uptake being dependent on the total pore volume, rather than surface area, will lead to more N2 being adsorbed in the material than would be the case for monolayer adsorption. I can imagine some scenarios where the two uptakes could theoretically be the same, if exactly two monolayers of N2 could fit in the pores of a material, for example, if their geometry was exclusively slit-like. However, in reality this will not be the case.

The BET model of adsorption, on the other hand, at least allows, in theory, for some of the adsorbed N2 to be present in the second, third, fourth, etc, layers, so that not all the adsorbed N2 is attributed to the first layer (and hence "surface area"). This picture of adsorption does not reflect reality, but that is a simple explanation of why the BET "surface area" is likely to always be lower than the Langmuir "surface area". I think this point is implicit in the question.

In addition, the BET method is used to calculate the position of the shoulder of the isotherm (historically known as "Point B"), which is assumed to represent the completion of the first monolayer. The position of Point B is then used to calculate a surface area. I think this will always be at a lower uptake (in mol/g) than the uptake used in the Langmuir case, if I understand the two approaches correctly. This again will result in a smaller calculated "surface area".

Regardless, it is important to recognise that neither model accurately represents the real adsorption process. It's just that the BET method, as Andreas says, might be a bit closer to reality because it at least allows for the possibility of more than just one monolayer adsorbing on the surface...

The two isotherm equations have different error functions and so their resolutions cannot entirely match. 

Thankyou for your replies.

to answer this question lets know

if you have 10 molecules and u wanted to adsorbed onto the surface of a solid material , and u want to measure the surface area.

surface area by Langmuir theory

it will be formation a monolayer on surface of material.so it will take a wide area like this

@@@@@@@@@@ the 10 molecules

|ــــــــــــــــــــــــــــــــــــــــ|

while BET theory

its based on formation multilayers on the surface of material ,it would be stacked on top of one another .

@@@@

@@@@@@

|ـــــــــــــــــــــــ|

then when u compare between the two surface area u can see the difference between them .