# Can titration curves be modeled with a sigmoid function?

## Main Question or Discussion Point

Currently in chemistry, we are doing titrations and buffers for acid base equilibrium. Ive noticed that titration curves have the distinct characteristics of sigmoid curves. So, can titration curves be modeled with a sigmoid function?

Borek
Mentor
More or less, but as far as i remember without reasonable accuracy. Note that titration curves are not symmetrical; sigmoid function is symmetrical.

symbolipoint
Homework Helper
Gold Member
Are you viewing a titration as cyclic? Periodicity is not a characteristic of titrations, unless you are looking for a way to correlate back-titrations to a fraction of a period or cycle. Titration curves are generally treated from zero quantity of titrant, through equivalence point or points, to some 'reasonably' quantity beyond the final equivalence point. When you say "sigmoid", I interpret this as "cyclic" or "periodic". Maybe I misunderstood.

Gokul43201
Staff Emeritus
Gold Member
Actually, I think the more important thing is the functional form.

Going by the shape, the sigmoid only looks like the titration curve for a strong monoprotic acid vs. a strong monobasic base. Using the H-H equation, it should be possible to write down explicitly, the relation between pH and volume of base added. Does this turn out to be the same function as the sigmoid? I haven't actually written it down, but I'd be surprised if it was.

Instead of the sigmoid function: y=(1+e-x)-1, I naturally expect something like y=C + ln(p(x)/q(x)), where p,q are polynomials (likely no higher than quadratics).

Borek
Mentor
That's not that easy. HH equation describes properly only part of the curve. H+ concentration for the whole titration is a root of polynomial - 3rd degree for strong acid/base titration (see equation 6.9), 4th degree for the weak/strong combination and so on. Somehow I can't see it as p/q combination you propose. But then I am known to be occasionally wrong As I understand the original question pakmingki wants to fit sigmoid or something similar to the titration curve (that's why he selected sigomid - it has correct shape, just like arctg). I have tried to follow the same way over 20 years ago, but to no avail - titration curve is not symmetrical.

symbolipoint
Homework Helper
Gold Member
Pakmingki,

Acid-base equilibria is not commonly (or never?) studied in relation to any region of cyclic, periodic, or "sigmoidal" functions. You could do an exercise for this and have some fun trying to use such functions (I assume they would be tangent or cotangent, or some such); maybe you might find an interesting relationship. If you do this kind of exercise, what would be the parameter which produces the pH and the volume of titrant?

The typical study of titrations of weak acids and bases yield quadratic equations; therefore, quadratic equations and logarithms are the mathematical skills which are most relevant for understanding titrations of weak acids and bases.

Gokul43201
Staff Emeritus
Gold Member
That's not that easy. HH equation describes properly only part of the curve. H+ concentration for the whole titration is a root of polynomial - 3rd degree for strong acid/base titration (see equation 6.9), 4th degree for the weak/strong combination and so on. Somehow I can't see it as p/q combination you propose. But then I am known to be occasionally wrong In this case, I gladly defer to your judgement. So, still, I don't believe the sigmoid is the correct functional form. Further, I don't see the point of modeling by an arbitrary function when the actual functional form can be derived (at the very least, under various approximations).

FT!
$$[H^+]=0.5 \left( (\frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )+\sqrt{( \frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )^2+4K_{water}} \right)$$

$$p(x)=-\log ( 0.5\left( ( \frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )+\sqrt{ ( \frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )^{2}+4K_{water} } \right) )$$

where Vbase is the variable

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epenguin
Homework Helper
Gold Member
I am known to be occasionally wrong
I am known to be more than occasionally wrong especially when I come here at night and post in a hurry as now - and I know Borek has this all at his fingertips.

But firstly I think all the OP meant by 'sigmoid' was just 'S-shaped' like some other curves he has met that have that description.

Where he has met them I don't know. But he could have met them in positively co-operative binding curves, e.g. of oxygen to haemoglobin or other ligands to certain proteins. These have often been called 'sigmoid' although actually sigmoidicity is not the same as co-operativity but strongly co-operative curves are sigmoid. But watch out! - what that means is that the curve of amount bound as a function of oxygen (or whatever ligand) concentration is sigmoid. But what the OP means is that the amount of protons, let's say, bound to the base is sigmoid as a function of the log of the concentration of protons in solution! A simple non-cooperative binding or titration curve is sigmoid in that sense - and symmetrical.

Then, at the risk of lateness of the hour errors, aren't the people who are denying it's symmetrical and sigmoid doing it too exactly? Shouldn't we say to students 'if you're getting quadratic or cubic equations you're doing it wrong'? At least for 99% of the problems they come here with? (always the same ones). (One of the ways) they get stuck is when they get equations that look too complicated for them, do not seem like anything they remember from class, because they don't realise what terms they can and should consider negligible. Just once or twice here did I see a necessity for those complete equations, and that was with a good student who wanted to understand in depth, completely abnormal, and even then the corrections were numerically of pretty academic significance.

Borek
Mentor
I think all the OP meant by 'sigmoid' was just 'S-shaped' like some other curves he has met that have that description.
Been there. About thirty years ago I was looking for some equation that would let me describe titration curves using some simple function similar to the one FT! posted, but not limited to strong acids/bases. Obvious approach was to check some math handbook looking for functions of the correct shape and then to try to fine tune them. That meant sigmoid functions. arctan looks promising, but when it comes to fine tuning it is way too symmetrical. It took me several years to accept fact that best approach is to not look for some fancy functions, but to accept the solution as given by the polynomial root, even if that means numerical approach.

FT!
Borek
Mentor
from this thread point of view this is the most relevant phrase from the paper:

As a final point, we will discuss the characteristics of the logistic function and show in what ways this function is inadequate for modeling titration curves.
Logistic function is one of the most basic examples of sigmoid functions.

Edit: what I don't like about Eaker's approach is that he doesn't use a single function, but three different functions, and he quietly ignores the problem of when to use each one - all we know is that first function describes titration curve up to some point before the equivalence point. But there is no discussion of when we need to switch from one function to another.

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Currently in chemistry, we are doing titrations and buffers for acid base equilibrium. Ive noticed that titration curves have the distinct characteristics of sigmoid curves. So, can titration curves be modeled with a sigmoid function?
This is the really same question i asked myself a couple of days ago when doing some acid-base titration in the lab.

$$[H^+]=0.5 \left( (\frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )+\sqrt{( \frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )^2+4K_{water}} \right)$$

$$p(x)=-\log ( 0.5\left( ( \frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )+\sqrt{ ( \frac{[Acid]V_{acid}}{V_{acid}+V_{base}}-\frac{[Base]V_{base}}{V_{acid}+V_{base}} )^{2}+4K_{water} } \right) )$$

where Vbase is the variable
Does these 2 equations apply to strong base-strong acid titrations only? Or, on the contrary, is it ok also for weak base-strong acid (or strong base - weak acid) titrations?

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Borek
Mentor
Does these 2 equations apply to strong base-strong acid titrations only? Or, on the contrary, is it ok also for weak base-strong acid (or strong base - weak acid) titrations?
Only for a strong acid/strong base. Note the lack of the Ka/Kb in the formula.

Only for a strong acid/strong base. Note the lack of the Ka/Kb in the formula.