The use of method names does not explain much. Both methods are numerical methods. In the linear regression method, the model must be linear with respect to parameters, but the nonlinear regression method may or may not. The linear regression method is based on the objective function in the form of the sum of the squared differences, in the case of the nonlinear regression method the objective function can be arbitrary. In fact, nonlinear regression is a contractual term, usually connected with the Levenberg-Marquardt minimization approach. Regardless of the method, parameter values can be correlated (usually forgotten). Their values may also vary.

A few questions:

Are you talking about LDF models (dq / dt = k (qe-q) and QDF (dq / dt = k (ge-q)2) and their linearized form?

How many experimental points and what is sampling time?

Adsorption in hydrogel - do you consider the influence of internal diffusion?

Regards,

Yes, I am talking about LDF and QDF models. The experiments take 10 experimental points during 240 min and I have not considered the influence of internal diffusion.

Thanks for your attention.

Let's clarify some terminology first. Linearization is the process whereby an equation is massaged to a form Y = m X + b. A common example is to linearize the Arrhenius equation r = A exp(-E/RT) to a form ln(r) = -(E/R)*(1/T) + ln(A) where Y = ln(r), X = 1/T, m = -E/R and b = ln(A). Linear-regression and non-linear regression are by comparisons methods to fit a theoretical equation to experimental data.

You have two different theoretical models to fit to your experimental data. One is called linear and the other is called quadratic. The ONLY PROPER way to compare the confidence of one model over the other is to avoid linearizing either model before fitting them to the data. In your case, the two equations to obtain q versus t are

k*t = ln(qe - q)    // linear

k*t = - 1 / (qe - q)   // quadratic

Plot q versus t. Fit the data to the equation q = qe - exp(k*t) where qe and k are the two fitting constants. Then, fit the data to the equation q = qe + (1/k*t) where qe and k are the two fitting constants. From this, you can determine from the chi-squared results which model is best.

Or you can simply plot the results to see which one fits the data best.  If your eye can't tell the difference both fits are either very good or very bad..

Thanks all for the help.

One problem is a mathematical model describes reversible adsorption process as for example:

dq/dt = ka(1-q) – kdq,   for  t=0,     q=0

or in equivalent forms:

dq/dt=k(qe-q), k = ka+kd, qe=ka/(ka+kd),     (LDF model)

q=qe(1-exp(-k t)).  (*)   

But you can also use the approximation equation in the form:

q=qet/(1+l t).   (**)

I wonder, if this equation will also properly describe your data? – In my opinion, it should.

The other problem is the experimental data: Another result we get, if most of the data refers to the state of equilibrium, and the other if the minority. Hence my question about sampling time - was it constant or variable rising. It is best to include a figure with the kinetic curve.

Should you convert the equations (*) and (**) to y = a + bx (mathematically ease), this is another problem that involves changing the probability distribution of random variables from usually normal to ??? ...... What is also forgotten ...... as for correlation between values of model parameters.  -  but almost everyone is doing it.

Regards,

In my opinion the results will be different

Equations:

q=qe(1-exp(-kt)) (solution of LDF model, q=0 for t=0)

q=qe(kqet/(1+kqet))  (solution of QDF model, q=0 for t=0)

q=qe(t/(1+kt))   (transformed form of M-M model)

represent similar behaviors, and therefore, faster or slower increase of q to asymptote: q = qe. With a few experimental points (fast growth of q) each of these equations may give a similar value of r2 close to 1. More differences should occur in the case of average relative error.

I believe that the whole problem arises from the fact that various models are used to describe one kinetic curve (obtained for only one initial concentration of the active ingredient). In fact we have here  to do with approximation of a kinetic curve using some function. Such studies have little in common with real kinetic studies.

Regards,