# Is there an algebraic form for the textbook reaction coordinate curves?

So here is a curve often used in textbooks to illustrate a reaction coordinate.

It's a nice short-hand device to learn the relationship between kinetic and thermodynamic parameters that control the progression and equilibrium of chemical reactions.

I was thinking whether there is an analytical expression for that curve, one that I could use to draw several curves like that (using, say, Desmos) for distinct values of free energy contents of reactants and products and activation energies.

• You know they are just some "pretty lines"? OK, there might be somewhere a case where curvature there is meaningful, but probably not in any textbook. Feb 10, 2023 at 19:28
• @Mithoron, yes I know. But I was looking for a way to systematically draw the pretty lines. Perhaps in a way that could be a tad more useful in an educational setting. Feb 10, 2023 at 19:32
• As nice as such a reaction coordinate looks like, it only is a projection (hence, a simplification) of a potential surface. Only with two degrees of freedom one can draw the later like a map. Feb 10, 2023 at 20:11
• See Wikipedia article on reaction coordinates, "en.wikipedia.org/wiki/Reaction_coordinate" Feb 10, 2023 at 20:13
• As has already been pointed out, the exact shape of these curves is usually meaningless. In practice you should calculate it (by calculating the energy $E$ as a function of some reaction coordinate $r$, which represents how a combination of bond lengths/angles changes over the course of a reaction), and then you can plot $E(r)$ versus $r$. But, If you just want the shape in this picture, I think you can define the energies $E(r)$ at the three points, and using that $\mathrm{d}E/\mathrm{d}r = 0$ at these points, fit it to a polynomial curve. Feb 10, 2023 at 20:45

Here is a function that works: https://www.desmos.com/calculator/cqwgfj2nzt

The general function would be

$$a \left( \frac{b}{1 - e^{5-x}} - \frac{c}{1 - e^{10-x}} \right)$$

I'm sure you could rearrange this such that the three parameters directly correspond to initial, activation and final energy, but as this is conceptual only anyway, I did not.

Here is an example graph (smoother than the one posted by the OP):

And here is a Desmos file with sliders.

• Well now that's just showing off. I thought a quartic would do the job! :) Feb 11, 2023 at 0:12
• The classic would be a Bézier spline.
– Karsten
Feb 11, 2023 at 1:17

You can simply use matplotlib to plot a smooth line connecting points

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d

x = np.array([0.1, 0.3, 0.5, 0.7, 0.9])
y = np.array([0.57, 0.85, 0.66, 0.84, 0.59])

x_new = np.linspace(x.min(), x.max(), 500)