From a Pesticide to an Alternate Quadratic Formula

In the English language, changing the suffix of a word like baked to baking might make it conform to the rules of grammar, but it won’t dramatically change the meaning of the word. That’s not the case when it comes to the language of chemistry.

Consider aluminum phosphide versus aluminum phosphate. Aluminum phosphide (AlP) is a common and inexpensive pesticide. But it releases deadly phosphine gas when it comes in contact either with moisture or with hydrochloric acid in the stomach , as shown by the following reactions:

AlP + 3 H2O → Al(OH)3 + PH3

AlP + 3 HCl → AlCl3 + PH3

Aluminum phosphide-poisoning is a serious problem in the Middle East and Asia. It has also led to rare but profound tragedies in the United States.  Four children were killed in Texas  last year (2017) after Fumitoxin pellets near their mobile home were hosed down by their father after neighbors complained about the smell.

Change phosphide’s suffix from ide to ate, and you get a totally different compound known as aluminum phosphate (AlPO4). This substance is formed from the reaction between aluminum sulfate and phosphate in waste water. Since AlPO is poorly soluble, the phosphate is unavailable to algae and eutrophication becomes less likely.

The maximum solubility of aluminum phosphate at a given temperature can be calculated from its equilibrium solubility product constant, known as Ksp.

For AlPO4  the expression is given by:

Ksp= [ Al 3+][ PO43-],

where the square brackets represent the equilibrium concentration of the respective ions in moles/L. Since the stoichiometric ratio of AlPO4  to Al 3+ is one to one,  the latter’s concentration is equal to how many moles of AlPO4 dissolved per liter of solution.

But what if there was already some PO43- present in a solution before AlPO4 appeared? In the same manner that any phosphate naturally present in an individual’s mouth makes his teeth enamel less vulnerable to bacterial acidic attack, phosphate’s present will decrease the solubility of AlPO4 .

Let’s say there are 0.050 M of phosphate already present, then the Ksp expression becomes:

Ksp= [ x ][ x +0.050] . Since Ksp for AlPO4 is 9.84 X 10-21. Expanding we get:

x2 + 0.050x – 9.84 X 10-21 = 0.

But if this is solved with a typical calculator, it will yield an answer of zero. this happens because in the quadratic formula based on ax2+ bx + c = 0 :

quadraticb2 >> 4ac, which means that the square rooted expression yields b to the typical calculator. ± b, when combined with –b from the formula’s numerator yields either the useless negative value and zero.

What’s the solution then?

In such cases we can use an alternate version of the quadratic formula, which can be obtained by multiplying the numerator and denominator by –±√( b2– 4ac). This will yield:


With this version of the formula, when b2 >> 4ac, we get either an undefined expression (division by zero) which we ignore, or approximately c/-b, which in our example gives us the solubility(x) to be not quite zero (although awfully close!) but 1.97 X 10-19 mol/L. More generally this version increases accuracy anytime the absolute value of one of the roots is much smaller than the other. A calculator can be easily programmed with conditional statements so that it can handle any situation. I wrote this one for a TI-83.



Other Sources:
Managing aluminum phosphide poisonings

The Quadratic Equation


Hot Chili Pepper Chemistry

The English word pepper is ambiguous. Botanically it could be refer to plants that produce black powdered pepper. The angiosperms (flowering plants) who serve as a source of that spice belong to the order Piperales and to the 3600-member family Piperaceae. (Kings play chess only for goodness’ sake is a mnemonic for remembering the order of biological classification: kingdom, a group of related phyla, which in turn, at least in singular form, phylum, is a group of related classes, and so on with orders, families, genera and species. ) Piperine is the compound mainly responsible for black pepper’s pungency, but as expected it does not appear in unrelated plants also dubbed as peppers.


In jalapeños,  dihydrocapsaicin and capsaicin, are the two  amides mainly responsible for their hotness.

Pepper can also refer to Capsicum chinense, the Red Savino haanero chili, a member of the tomato family, Solanacea.  That fruit scores in the hundreds of thousands of units on the Scoville scale of “hotness”.  Other hot  peppers and milder ones are found among the many varieties of another Capsicum species, known as annum, which includes sweet peppers, jalapeños, and New Mexico chili.


There are four different amides (specifically, capsaicioids) found in hot peppers of the Capsicum genus. A Brazilian study revealed that in these fruits capsaicin makes up anywhere from 24% (in jalapeños) to 95% (pimenta preta) of capsaicioids present. These in turn combine to account anywhere from 0.2 mg to 7 mg per gram of fresh pepper, in other words, less than 1%. The percentage of course soars when the pepper is dried. Except in jalapeños and fatalis, the equally potent compound, dihydrocapsaicin, is the second most abundant amide. The greatest concentration (9.2%) of nordihydrocapsaicin, a substance about half as hot, was found in pimenta de mesa. Finally another related compound, homocapsaicin, the least potent of the quartet, is entirely absent from most spicy peppers, but it makes up 12% of dedo-de-moça pequena’s capsaicioids.


nnordihydrocapsaicin 9 100 000


dihydrocapsaicin 16 000 000


homocapsaicin 8 600 000


capsaicin 16 000 000

pimenta de mesa

The hot amides in pimenta de mesa consist of about 53% capsaicin, 37% dihydrocapsaicin, 1% homocapsaicin and about 9% nordihydrocapsaicin . The latter’s concentration is not surpassed in 20 other species of Capsicum.

From examining the above structural diagrams notice that in comparison to capsaicin,  homocapsaicin’s double bond is slightly further away from the tail end of the molecule. On the other hand, nordihydrocapsaicin’s strength is compromised by being one CH2 shorter than its hotter counterpart, dihydrocapsaicin.

How do these compounds exert such a powerful reaction in our mouths? In all mammals they cause tingling and burning sensations by activating  a non-selective cation channel, called VR1, on nerve endings. It’s not a coincidence that the same channel also interacts with compounds released by inflammation from actual intense heat sources or acidic protons. Birds, however, have a variant of VR1, which is still sensitive to heat and acid but which does not interact with capsaicin or its analogues.  It’s likely an example of co-evolution between Capsicum plants and animals who can eat their fruits without suffering deterring consequences. They then fly to other destinations to spread ingested seed.
We get more evidence of coevolution thanks to Jordi Altimiras of  Linköping University who made me aware of a study revealing that chilli seed germination is decreased in the gastrointestinal tract of mammals but not by the passage through the tract of birds.

Finally why are peppers producing capsaicioids in the first place? Any biosynthesis is catabolic and thus consumes energy. But making capsaicinoids in fruits is a worthwhile investment; it reduces fungal infection and seed mortality. From that narrow perspective we mammals have more in common with fungi than birds.


  • Molecular basis for species-specific sensitivity to “hot” chili peppers.  2002 Feb 8;108(3):421-30.
  • Comparative Study of Capsaicinoid Composition in Capsicum Peppers Grown in Brazil
  • Directed deterrence by capsaicin in chillies. Nature 412: 403-404 and Tewksbury JJ, Reagan KM, Machnicki NJ, Carlo TA, Haak DC, et al. (2008)
  • Evolutionary ecology of pungency in wild chilies. ProcNatAcadSciUSA 105: 11808-11811 .