A negative value for the intercept (in essence the uptake of the monolayer is physical nonsense. By linearisation of the Langmuir isotherm as you do it, you overemphasize data points at a low concentration (typically those with a high associated error). There are different ways of linearizing the Langmuir expression, e.g. 

Ce/Qe = 1/Qm *Ce+ (1/bQm)

Qe = Qm - (1/b)*(Qe/Ce)

The best way would be to perform a non-linear regression on your data (with as the initial guess one of the solutions to the linear regressed models - which is physically sound)

Thank you Professor Eric Van Steen

I agree with the first part of the answer given by the PhD Van Steen, because obtaining a negative intercept, in this case, it's a physical impossible. However, to try to look for the solution of the "error" using another way of linearizing the Langmuir expression or using nonlinear regression techniques is the wrong proposal. Firstly, because whatever form of linearization of the function are equivalent and the same result would be obtained. On the other hand, neither you should use technical of non lineal regression, because the techniques of lineal regression are always more potent fitting technical that those non lineal, for this reason if the function is linearizable, first you linearized the function and later fitting experimental date using lineal regression technical. This is justified statistically and could be found in the literature.

In my opinion you should, first of all, whether you get isotherm corresponds to an isotherm of type I, in which case you may use the Langmuir isotherm to explain the adsorption process. If indeed the isotherm is of type I, you should carefully review everything done and find your error; otherwise unfortunately you should repeat the experiments. Now, if the isotherm you get is not the type I, then you cannot use the Langmuir isotherm to explain the adsorption process. In this case, you must determine what type of isotherm is and in correspondence with this to use a model to fitting yours isotherm. Even you can propose your own model.

As mentioned above, a negative intercept means nothing. If you have performed experiments in duplicates/triplicates, you may try to plot each set individually to check the consistency. Sometimes, if you do not have many data points, outliers or large error bars can change the model fitting significantly. 

Thank you  very much 

I have got similar results when plotting Ce/qe  vrs Ce I got the intercept  negative 

y = 4.8994x - 11.412   and  R2 = 0.9947. 

Does it mean results of the experiments are not valid??

The results may be explained as follows:  The negative value of Langmuir constant (b) for Pb(II) suggest that it gives RL value more than 1 and adsorption of Pb(II) on synthesized resins were very difficult. It may be due to, As ionic radius increases, charge density decreases, the adsorption capacity of synthesized resins decrease as for Pb(II).

my regression value is 1 '? what it means 

negative intercept why ? in langmuir plot 

Hi Sami Abdullatif, 

I also have same result like you...

y = 4.573x - 7.546, y = 0.479x - 0.031 and y = 0.513x - 0.020 and regression value is 0.999.

so, I think your explanation is helpful.

Nice

Same here...I also got the similar result..

Thanks for all of explanations...

I get negative value for Langmuir constant (KL) , jow de can explain this ??????

Hi Sami Abdullatif,

I also have same result like you

This means that one should be careful when using various graphical methods, here an analogue of L-B plot. Here, presumably, the equilibrium curve is convex, but it should be concave. This could be the result of unidentified side effects e.g. adsorbent particle aggregation, pore blocking, poisoning etc.

hello, could you help me, please. I need to do a regression with the breakthrough curve data, but I don't know how to do it. I don't know how to solve the differential equations involved in the development of the curve.

I obtain the equation y=-0.2234x + 4.0833

R²=0.3511, what does the negative value is slope means

The negative values of intercept can be possible depending upon your starting point of applied concentration.

For example, if your first Ce is near 0 there will be chances of positive Kl while if you started your work from a higher concentration (let's suppose 10 mg L-1) then you will end up Ce a little bit away from 0 which may cause intercept (Value of Y when X=0) to be negative. It's just the way you have made your graph and presented data.

You can explain the negative results while keeping in mind this point.

There are tonnes of paper in which they have reported negative intercept and even slope in some cases (even Qmax that can be of discussion).

Find the following once for your references

https://www.scielo.br/j/eq/a/rtQnfy9kPyg6fnVVM3M75cM/?format=pdf&lang=en