# The BET method for measurement of surface area.

The BET method uses a measurement of the physisorption of a gas to derive a value of “surface area” for a sample.

The gas molecules can pass between particles and into all pores, cracks, and surface roughness, so that the measurement probes the full microscopic surface area of the sample.

Most often, the sample in the form of powder or granules, and the result is stated as a Specific Surface Area, in units of *area per unit mass*. It may also be given as *area per unit of volume*, or as the absolute area for an object.

## A short description of the BET method

The BET method uses:

- a physisorption measurement
- the BET model, which describes the quantity of adsorbed gas as a function of the relative pressure, \(P/P_0\)
- fitting the model to the measured data, to obtain the amount of gas corresponding to a single layer over the whole sample
- using the
*area per molecule*to find the total area of the sample - usually, converting the area to a Specific area, using the mass of the sample

## The BET measurement

BET analysis requires measurement of an “Adsorption isotherm”.

Because the BET model uses the *relative pressure* of the adsorptive, it is necessary that the gas be condensible at the adsorption temperature - in other words, the «gas» is really a *vapour*.

A typical measurement involves :

- putting a known amount of sample into a sample cell or container.
- outgassing or other treatment of the sample, to remove impurities, moisture.
- increasing the pressure of the gas, while measuring the amount adsorbed on the surface of the sample. For the best precision, this is done at a number of discrete pressures, and with a wait for equilibrium and measurement of the amount adsorbed at each point.
- often, the saturation vapour pressure, \(P_0\) is measured at the same time, or it may be calculated from knowledge of the temperature.

BET is most widely performed using adsorption of Nitrogen gas at 77 K, the boiling point of liquid nitrogen, but other species and temperatures are used.

- Argon at 87 K (liquid Argon temperature)
- Krypton at 77 K
- Carbon dioxide, CO
_{2}at 0 °C or at 25 °C - Water at 20 °C using DVS instruments, such as the IGAsorp and the SPS.

## The adsorption isotherm

For BET, the adsorption isotherm is measured as a graph of “amount adsorbed”, \(n_{adsorbed}\) versus “Relative Pressure”, \(P/P_0\), where \(P\) is the pressure of the adsorbtive, and \(P_0\) is its saturated vapour pressure at the fixed experimental temperature.

Instruments like the **BELSORP mini X sorption analyser**, and the **BELSORP MAX II** sorption analyser, can easily measure the adsorption isotherm of gases such as Nitrogen, Argon, Krypton, often at cryogenic temperatures (as well as other gases and vapours at a range of temperatures).

Adsorption of Water and other volatile substances is often performed by Dynamic Vapour Sorption instruments, such as the **IGAsorp** from Hiden Isochema, and the multisample **SPS from ProUmid**.

## The BET model

The model gives the amount of adsorption, *at a fixed temperature*, relative to the amount in a single monolayer, \(n_{m}\). It is a sorption isotherm model :

\[ \newcommand{\ppo}{P/P_0} \frac{ n_{adsorbed} }{ n_{m} } = \frac{C\,\ppo}{ \left( 1 - \ppo \right) \left( 1 + \left(C-1\right)\,\ppo \right) } \]

- \(P_0\) is the saturated vapour pressure of the adsorbate.
- \(C\) is a constant considered to relate the adsorption strength of the first layer to the enthalpy of vaporisation of the liquid adsorbate.

The BET model, named for its creators (Brunauer, Emmett, and Teller ), describes how an increasing pressure of gas causes the progressive formation of :

- a first layer of molecules, which interact directly with the surface, followed by
- subsequent layers which behave as if they were condensing pure liquid on top of the first layer.

The model has two parameters :

**\(n_m\)**, the**monolayer coverage**: the number of atoms in a single filled layer**\(C\)**, the C constant, which is interpreted as describing relative strength of the interaction between the surface and the first layer, compared with the following layers.

For a large value of \(C\), the first layer is almost complete before the next layers begin to form so that there is a sharp knee at low pressure. For small \(C\), the monolayer does not form until high values of \(P/P_0\), giving a Type III type of isotherm.

In BET measurement of surface area, we begin with a series of values of \(n_{adsorbed}\) and \(P/P_0\). We do not yet know the values of \(C\) and \(n_m\).

## The BET fit

Rearranging the BET isotherm model gives the linear form :

Multiply by \(\newcommand{\ppo}{P/P_0} \frac{ \left( 1 + \left(C-1\right)\,\ppo \right) }{ C\;n_{adsorbed} }\)

\[ \newcommand{\ppo}{P/P_0} \frac{\ppo}{ n_{adsorbed} \left( 1 - \ppo \right) } = \frac{1}{ n_{m}\,C } + \frac{C-1}{ n_{m}\,C } \,\ppo \]

A simple treatment of the isotherm data, and plotting the left-hand term, gives the **BET plot**.

In the region where the BET model is valid, the BET plot is linear, and the slope, \(\mathbf{s}\) and intercept, \(\mathbf{i}\) are measured from a straight-line fit.

\(\mathbf{s}\) and \(\mathbf{i}\) are used to calculate the values of \(n_m\), as the number of molecules in a monolayer, and the constant \(C\). Using a known value for the amount of surface area occupied by each molecule in the monoloayer, the surface area of the sample is obtained.

### Single point BET analysis

For applications where the surface areas of the samples vary, but the substance remains unchanged, there is a the possibility of using only a single point on the BET isotherm. This simplified measurement is very rapidly performed by the mono-point BELSORP‑MR1

## Practical BET analysis

The BET model is based on a very simple model for an idealised adsorption process on a planar surface, where the effect of the surface affects only the first layer of adsorbed molecules, and where there are no lateral interactions within the adsorbed layer.

Although it would be unusual for a material-adsorptive system to have these properties, it is found for many non-microporous materials that a fit to the BET plot is possible in the range of \(P/P_0\) from 0.05 to 0.3 .

However, a simple straight-line fit is insufficient to ensure validity of the model, or even consistency of the analysis. A number of recommendations have been formalised in international standards to ensure that BET analyses are reproducibly and meaningfully obtained and reported.

In addition, careful sample preparation and correct use of the analysis instrument are necessary.

When microporosity is present, the fitting is normally linear in a lower pressure range, and caution is advised in the use of the term “surface area” with respect to BET results for these materials.