Langmuir Surface Area

The Langmuir model was developed by Irving Langmuir in 1918 while he worked at the research laboratory of The General Electric Company. The model assumes that gas molecules can bind to a material at fixed individual sites on the surface of the material with one molecule binding to one site, and with each site being identical to all other sites in nature regardless of the occupancy of adjacent sites. The model assumes only a single layer of molecules will form on the surface. With these assumptions, the Langmuir model produces the following relationship for the isotherm,

$\displaystyle \frac{Q_{ads}}{Q_m} = \frac{b P}{1 + b P}$

where $b$ is a constant with units of inverse pressure and $Q_m$ is the monolayer quantity of gas adsorbed. The left-hand side of this expression is more generally the fraction of occupied surface sites in the Langmuir model. The Langmuir isotherm expression can be rewritten in the form

$\displaystyle \frac{P}{Q_{ads}} = \frac{1}{b Q_m} + \frac{P}{Q_m}$

in which we refer to the the left-hand expression $P/Q_{ads}$ as the Langmuir transform. The Langmuir transform is convenient because it is linear in the gas pressure $P$ for this model. One can easily extract the constants $b$ and $Q_m$ from an Isotherm data set with a simple line fit using the Langmuir transform. The surface area can then be determined as the number of molecules in the monolayer multiplied by the area per Langmuir site, which in the case of physical adsorption is just the cross-sectional area of the gas molecule $\sigma$. Putting this together, the Langmuir surface area is then

$\displaystyle A_{Langmuir} = N_m \sigma = N_A \frac{Q_m}{Q_{mole}}\sigma$

where $Q_{mole}$ is the quanity of gas in exactly one mole, and $N_A$ is the Avagadro constant. Using a linear fit to the Langmuir transform data one may determine $Q_m = 1/m$ and $b = 1/(m Y_0)$ where $m$ and $Y_0$ are the slope and y-intercept of the best fit line. The Langmuir Surface area is then,

$\displaystyle A_{Langmuir} = \frac{N_A \sigma}{Q_{mole} m}$

Langmuir: Example Calculation

As an example of the Langmuir calculation, we use the reference calculation from the report-models-python on github from Micromeritics. We use an ZSM-5 sample analyzed with Argon with the absolute pressure as the dependent variable, and selecting a limited range of the isotherm data set.