I know only a few papers on the equilibrium study of the adsorption process, where the two parametric exponential equation, in this case called the Jovanovic equation or the Jovanovic isotherm, was used. The vast majority of researchers use the Langmuir equation. This equation is based on theoretical foundations and has a long tradition, but in fact it is one of the standard mathematical functions - hyperbolic function, which has the following general form: y = a (kx/(1 + kx)). This function increases gradually to a (asymptotic increase) as x increases. There are no inflection points and y (0) = 0. The exponential function in the form of y = a (1 - exp(- kx)) behaves similarly, but its increase to a, for the same k value, is faster. In the case of the description of the equilibrium in the adsorption process, this equation has no theoretical basis. But it is a function that refers to behaviors that occur more often in nature than those described by the hyperbolic function. It seems that this is a better alternative than eg. the power function (Freundlich isotherm) and the logarithmic function (Temkin isotherm) of which the first does not fulfill the boundary condition in infinity, and the second the boundary condition in zero and in infinity.
IMHO, one "technical" reason is that till now very many scientists like to linearize the model equations, and it is easy to do for the Langmuir and Freundlich models. Yes, at present, it is easy to use non-linear fitting procedures, and linearization procedures affect statistical analysis (due to the lack of normal distribution), but yet the papers are continuously published in established Journals in which data is linearizied. If a model does not allow linearization, a part of scientists will not consider it. A second reason is also "technical", in some sense: it is due to the multiple widely disseminated misjudging which model is better. When high r2 is obtained, then erroneously it is concluded that the model is Ok. How many studies are selecting the best models on comparing sum of (squares) of deviations and (!) examining the adequacy of the fitting? So, the pessimistic conclusion is that there should bea good reason to check any less standard model.
Interpretation of adsorption data with common models are relatively easy and can provide all necessary information while incorporation of new or relatively unknown function make the overall process more complex and unrealistic.
In two replies, a number of problems were raised that it is difficult to reffer to them all at once. One of them was the so-called "linearization".
Personally, I am against transforming equations into a form, which after introducing new variables can be presented in the form of a linear equation/function: Y = AX + B. There are several reasons for this. The statistical distribution of new variables changes from normal to ? (this issue was mentioned in first reply). The location of experimental points in the new coordinate system changes (eq. in case of Y = 1/y small values become large). There may be difficulties with clearly drawing a straight line. The error for the linear function (this is the most often given in papers) and for the source equation (often larger) can be different. Moreover, why do this if there is no problem with access to numerical methods, and the computer is on every desk. Of course, it should be remembered that in the case of nonlinear equations, and especially in the case of exponential and power equations, their parameters are numerically correlated. Then the graphical method with the use of equivalent linear form could be helpful in determining the starting point for the numerical method.
Unfortunately, the procedure, incorrectly called linearization, is commonly used in works on adsorption, and not only. Regards,
Often a nonlinear correlation to be fitted to data is 'somehow' linearized, even if just as a previous step before data fitting by nonlinear least squares.
The model correlation y(x) = a(1 - exp(- k·x)) can be transformed as: ln{1 - y(x)/a} = - k·x.
We can simply take just some 2 data points {(x1,y1), (x2,y2)} to solve the corresponding system of 2 equations and find first guess estimates for both parameters (a0, k0): {ln{1 - y1/a0} = - k0·x1, ln{1 - y2/a0} = - k0·x2}; {ln{1 - y1/a0}/ln{1 - y2/a0} = - x1/x2, k0 = - {ln{1 - y2/a0}}/x2}. First equation can be iteratively solved by Newton–Raphson method.
We can then depict in a 'linearized' plot the whole set of data points scattered around a straight-line trendline of zero intercept obtained by least squares. A better estimate for the parameter k (k1) can be taken from the slope of the trendline. A better estimate for the parameter a (a1) can be taken from the arithmetic mean a1 = ∑{y(xi)/(1 - exp(- k1·xi))}/n, i = 1, 2, ... , n.
We can now correct the 'linearized' plot for the whole set of data points, as ln{1 - y(x)/a1} vs. x. A better estimate for the parameter k (k2) can be taken from the slope of the trendline. A better estimate for the parameter a (a1) can be taken as a2 = ∑{y(xi)/(1 - exp(- k2·xi))}/n, i = 1, 2, ... , n.
Both the parameters could still be further iteratively refined, until some suitable convergence criterion is met. These would be unbiased estimates for the 'linearized' correlation ln{1 - y(x)/a} = - k·x; but not for the original correlation.
The last linearized plot is quite convenient to linearly display the set of data points as well as scattering around the trendline.
Last estimates of the parameters can still be refined, by iterative nonlinear least-squares regression, to find unbiased estimates for the original correlation.
This video shows how to fit Jovanovic isotherm model nonlinear equation easily using originpro. #aminulcheminnovation #chemcolony #aminulsir #aminulchem https://youtu.be/1w7kNbDabyg
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