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I'm having some trouble understanding a part of Flory's model for linear polycondensation reactions. In his 1936 paper, he derives an expression for the probability of the existence of a particular $x$-mer as

$$\Pi_x = xp^{x-1}(1 - p)^2\tag{2}$$

Later, he states the following:

In order to locate the $x$ value which gives a maximum in $\Pi_x,$ let

$$\partial\Pi_x/\partial x = (1 - p)^2(p^{x - 1} + xp^{x - 1}\ln p) = 0$$

Not being the most mathematically savvy person, can someone explain to me how he got to this expression?

Reference

  1. Flory, P. J. Molecular Size Distribution in Linear Condensation Polymers. J. Am. Chem. Soc. 1936, 58 (10), 1877–1885. DOI: 10.1021/ja01301a016.
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  • $\begingroup$ "Not being the most mathematically savvy person" - How much do you know about differentiation? Does mytutor.co.uk/answers/5998/A-Level/Maths/… Make any sense? Do you know the Product Rule for differentiation? $\endgroup$
    – Ian Bush
    Mar 4 at 10:07
  • $\begingroup$ @Justus Have a look at my answer and if it answers your question, please click on the tick mark next to the answer. You also get +2 points on doing this :) $\endgroup$ Mar 5 at 11:00

1 Answer 1

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Given $\Pi_{x}$= $π‘₯*𝑝^{π‘₯βˆ’1}$ *$(1βˆ’π‘)^2$

$\partial$ refers to partial differentiation.

$\frac{\partial \Pi_{x}}{\partial x}$ means differentiating $\Pi_{x}$ with respect to x.

Therefore $(1βˆ’π‘)^2$ is a constant when differentiating with respect to x. So during differentiation the constant stays as a constant multiplier.

Now see that there are two terms involving x, therefore we need to use the product rule of differentiation.

$\left(\frac{\partial \Pi_{x}}{\partial x}\right)$ = $(1βˆ’π‘)^2$ * ($𝑝^{π‘₯βˆ’1}* \frac{\partial x}{\partial x} + x*\frac{\partial p^{x-1}}{\partial x}$)

$\frac{\partial x}{\partial x}$ = 1 and for calculating $\frac{\partial p^{x-1}}{\partial x}$ look at exponentiation rule of differentiation as suggested by @Ian Bush in the comments.

Therefore we finally have :

$\left(\frac{\partial \Pi_{x}}{\partial x}\right)$ = $(1βˆ’π‘)^2$ * ($𝑝^{π‘₯βˆ’1}* 1 + x*ln(p)*p^{x-1}$)

And since he is trying to locate the π‘₯ value which gives a maximum in $\Pi_{x}$, he equates the equation to 0.

$\left(\frac{\partial \Pi_{x}}{\partial x}\right)$ = $(1βˆ’π‘)^2$ * ($𝑝^{π‘₯βˆ’1}* 1 + x*ln(p)*p^{x-1}$) = 0

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