# How to determine Langmuir adsorption isotherm constants and use the models

I am doing some adsorption tests on bio materials and would like to test the data with Langmuir and Freundlich adsorption isotherm models. Assume these values:

adsorbent(g).   initial con (µg/L)  final con.(µg/L)
0.1                   1                0.65
0.1                   5                2.1


(Sample volume is 20ml. In both cases the dose of adsorbent was the same (0.1g) and in the first initial concentration is 1ppb and, after adsorption, is 0.65ppb. In the second initial concentration is 5ppb and, after adsorption, is 2.1ppb.)

Using these data how can I use Langmuir and Freundlich isotherms models to get the constants and check whether my adsorption data fit the model or not. (If any other parameter is needed you can assume some reasonable value.)

For langmuir isotherm:

Let say we are using linear langmuir isotherm.

The formula is : $\frac{1}{q_e} = \frac{1}{q_m} + \frac{1}{q_mKL}$

find the value of $q_e$. Using the formula $q_e = \frac{(C_i-C_f)V}{\text{mass}}$ of adsorbent, and then plot the graph $\frac{1}{q_e}$ versus $\frac{1}{C_f}$. It would be easier by using excel. From the graph, we can calculate the slope. The linear graph will produce $y=mx+c$ equation. m=intercept=1/qe, hence qe=1/m. C=1/qm.KL. From the C value, we can actually find KL.

From the graph generated using excel, it will shown the R2 value. This value is the important indicator to determine which isotherm model fit the system. The highest R2 value will fit the system. Let say R2 value langmuir is 0.8 and R2 for freundlich is 0.9. So, we can conclude that, Langmuir model fit your system.

For Freundlich isotherm

Plot the graph log qe versus log cf. From the graph you will get linear equation, y=mx + c and the R2 value. Compare the R2 value with langmuir's R2. Highest R2 value indicate that those model fit with the system.

Cf=final concentration Ci= initial concentration