I have been trying to solve a problem i posed to myself in the applied sciences, and technically, i did (though it is not of any practical use). But the problem is that the solution is, well, not pretty to look at, and more importantly, difficult to solve and work with. To cut staight to the point, I am looking either for a way to simplify the following equation:
$$y=a\frac{\sum_{i=0}^nx^{n-i}f(i)(-x^2+ibx+c)}{\sum_{i=0}^{n}x^{n-i}f(i)(x^2+dx-c)}$$
Where
$$f(i)=f(i-1)\cdot k_i$$
Given a list of $n$ values $k_1,k_2,...k_n\in\mathbb{R}$, and that $f(0)=1$.
I am truly at a loss for how to do this.
(Btw, this formula is "translated" into normal mathematical notation. If anyone is curious about the orginal acid/base-chemistry problem, is it basically the following: Derive a formula that describes a titration curve.)
Litterally any help will be greatly appreciated.
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$\begingroup$ Welcome to MSE! As a chemist, I'd love to see how you got to this from a titration curve, in particular wondering where the quadratic functions are coming from (are you accounting for fugacity etc? That might be the source). I can also say that I cannot imagine this expression has a "happy" solution; partial summations usually don't, and the necessity of specific values of $k_i$ mean you can't use a CAS easily either. The "ideal" titration curve ought to be a sigmoid of some sort, though; given it's usually pH on the $y$ axis, I'd expect one of the exponential-based sigmoids. $\endgroup$– Eric SnyderCommented Mar 15 at 0:03
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$\begingroup$ @EricSnyder, I thought it couldn't be simplified any further, but i keep waking in the middle of the night, thinking i might finally have figured it out, but sadly my ideas never amount to any good solution. I thought it would be a sigmoid too! And it turns out that around the equivalence point, it can be approximated by an inverse sigmoid, as given by the Henderson-Hasselbach equation. But this idea sadly falls apart when moving away from the equivalence point, and especially when working with polyhydronic/polyprotonic acids or bases. $\endgroup$– redibCommented Mar 15 at 8:55
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1$\begingroup$ @EricSnyder, Based on a couple of equations from the book ”Principles of Quantitative Chemical Analysis” by Robert de Levie, 1997, you can derive the following equations, that describe a polyprotonic acid titrated with NaOH: $$V_b=V_a\frac{\sum_{i=0}^nK_t(i)[H^+]^{n-i}(-[H^+]^2+iC_a[H^+]+K_w)}{\sum_{i=0}^{n}K_t(i)[H^+]^{n-i}([H^+]^2+C_b[H^+]-K_w)}$$ $$\sum_{i=0}^{n}K_t(i)[H^+]^{n-i}\left(V_t[H^+]^2+\left(V_bC_b - iV_aC_a\right)[H^+]-K_wV_t\right)=0$$ The derivation I have made for these formulas is currently written in Danish, but I will happily translate it into English, and send it to you $\endgroup$– redibCommented Mar 15 at 8:57
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1$\begingroup$ @EricSnyder, I actually made a program to calculate the titration curves, and it can be found here: Chemistry Calculators. (Be aware that it is "pay what you want".) $\endgroup$– redibCommented Mar 15 at 9:00
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