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.


[![enter image description here](https://i.stack.imgur.com/Ez1nV.jpg)](https://i.stack.imgur.com/Ez1nV.jpg)


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.




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):


[![enter image description here](https://i.stack.imgur.com/hYWsa.png)](https://i.stack.imgur.com/hYWsa.png)


And here is a Desmos file [with sliders](https://www.desmos.com/calculator/6qhghxyn66).


 6

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)

f = interp1d(x, y, kind="quadratic")
y_smooth = f(x_new)

plt.plot(x_new, y_smooth)
plt.scatter(x, y)
plt.show()

```

[![enter image description here](https://i.stack.imgur.com/945HH.png)](https://i.stack.imgur.com/945HH.png)


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