Fitting data with Arrhenius Equation

Question

I have the following data:

DataAZ400K = {{293, 1.98}, {313, 1.34}, {326, 1.58}, {344, 
    2.54}, {358, 3.8}, {373, 6.12}};

I want to fit this data with an exponential fit of the Arrhenius form $y \propto e^{-\frac{C}{k_{B}x}}$ where $k_B=1.38 \times 10^{-23}$ and $C$ is a fitting parameter. Ideally, I would also like to do the fitting in a $Log(y)$ v/s $1/x$ plot where the fitting should just be linear. How can I do the fitting in both linear plot and the second type $Log(y)$ v/s $1/x$.

Answer 1

NonlinearModelFit and LinearModelFit both work well. The one caveat is setting the scale of the parameters since the Boltzmann constant has such a small numerical value.

Alternatively, you can include kB in the fit by fitting to, e.g., A Exp[-Ck/x] with fit parameters A and Ck. A good initial guess for Ck is the typical size of your x values, i.e., around 300.

fit = NonlinearModelFit[DataAZ400K, A Exp[-Ck/x], {{A, 1}, {Ck, 300}},x]

This gives a fit with values for A and Ck. The C parameter in your fit equals Ck times kB. Compare fit and data:

Show[Plot[fit[x], {x, 290, 380}], ListPlot[fit["Data"]]]

For linear fit, convert your data to the form {1/x, Log[y]} e.g.,

linearData = {1/#[[1]], Log[#[[2]]]} & /@ DataAZ400K

and then a linear fit as a function of a variable x1 standing for 1/x:

fitLinear = LinearModelFit[linearData, x1, x1]

and plot:

Show[Plot[fitLinear[x1], {x1, 1/400, 1/290}], ListPlot[fitLinear["Data"]]]

To display the original y coordinates with the linear fit, exponentiate and plot with log scale:

 Show[LogPlot[Exp@fitLinear[x1], {x1, 1/400, 1/290}], ListLogPlot[{#[[1]], Exp[#[[2]]]} & /@ fitLinear["Data"]]]