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 AlPO₄ 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 K_sp.

For AlPO₄ the expression is given by:

K_sp= [ Al ³⁺][ PO₄^3-],

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 PO₄^3- 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 K_sp expression becomes:

K_sp= [ x ][ x +0.050] . Since K_sp 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 :

b² >> 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 –b ±√( 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
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3162709/

The Quadratic Equation http://cse.unl.edu/~ylu/raik283/notes/Resoureces/Quadratic_equation.htm

Snow, Chemistry And The Spirit Of Christmas

Sometimes a view from an outsider is what’s needed to make us pause and reflect. Two of our students, one on an exchange program from Germany and one here from the southern U.S, have been enthralled this week at school not by the equations on the board or by the books we read but by the snow that they rarely see in their native environment.

One of them went out and ate the fluffy form of H₂O, like we did when we were children,and then he fell back and made snow-angels. The other student just couldn’t wait for the week’s second snowfall.

Too accustomed to snow, as Canadians we associate it with shoveling and difficult driving conditions. We dream of escaping the cold and forget what a privilege it is to live in a land of ice and snow at a point in history when thanks to technology and science we can enjoy and understand snow. Unlike our Nordic ancestors, we are not condemned to shivering in a storm with inadequate clothing and poorly insulated shelter. In a country endowed with hydroelectricity, it is so easy to stand in warmth and to watch a cascade of transient crystals land on a window.

We know why it is truly rare for two snowflakes to be alike because their patterns depend on their immediate environment. The slightest changes in pressure or temperature can affect the shape; as one drop of water crystallizes it releases energy, changing the destiny of an immediate neighbor. And yet all these differences represent variations upon a theme: all flakes have six sides and six needles. In the three dimensions of its solid form, snow molecules maximize the number of intermolecular attractions between themselves. Specifically, the oxygen of one H₂O molecule attracts two hydrogen atoms from two different molecules.

Conversely, each of water’s hydrogen atoms bonds to an oxygen atom from two different H₂O’s. A staggered hexagon results with a water molecule at each vertex. Even when, in the smallest of snowflakes, this arrangement is repeated about three billion trillion times, the basic pattern remains the same.

But enough said about chemistry. Snow is also the framework for the spirit of the holiday season. A few months ago my daughter was told there was no Santa and that parents were the ones filling the stockings on Christmas Eve. She approached me on the subject but seemed more curious than sad, so I told her a “transition-truth”. I told her that Santa is a spirit who gets into parents and makes them buy the gifts that children desire. In previous years we had written to Santa Claus together, but last week she wrote the letter, sealed the envelope, walked past me, and headed to the mailbox. I said, “Wait. Aren’t you going to tell me what you asked for?”

“No,” she replied. “You will be filled with the spirit of Santa and you’ll know what to get me.”

Let’s hope that the snow does not melt before Christmas.

Cancer, deep-sea partnerships and the resented carbon-tax

“We shouldn’t be looking for heroes, we should be looking for good ideas.”

-Noam Chomsky

Forgive me this indulgence, but rather than devote the usual 500 to 1200 words to a given topic, I would like to scan a trio of topics and ideas that have caught my attention in the past week.

The mantra “everything causes cancer” is far-removed from the truth. Out of the literally millions of known mixtures, compounds and elements, less than 500 are proven or probable carcinogens. As children and young adults, many of us never suspected city air, processed meat and alcohol to be among the culprits. This unfortunately leads some of us to wave the white flag and adopt the erroneous generalisation. But that defeatist attitude is music to the ears of the carcinogens’ users and producers who refuse to acknowledge responsibility. Speaking of music, sound is a form of energy, and along with radio waves and all the colours of the rainbow—none of those, unlike ultraviolet, x-rays and gamma, cause cancer.
Something a little newer and more original. If as a teacher or student you hear of an oxidation-reduction partnership, the words might conjure up an image of people partnering up to do a redox lab. But until recently, not even the most specialized biochemists imagined that two different organisms were symbiotically working together to oxidize methane and reduce sulfate, respectively, in layers of sediment and rock under the sea floor. One of the microbes, a methanotroph which uses methane (CH₄) as its energy and carbon source, reduces CH₄ to hydrogen carbonate ion (HCO₃^–) in the absence of oxygen. But the metal ions that the organism uses to pick up electrons lost by methane are not efficient enough. A sulfate-reducing bacteria comes to the rescue. It uses the electrons released by the oxidation of methane to reduce sulfate (SO₄^2-) to sulfide (S^2-). Its reward? It too obtains energy in the process.
There have been riots in Paris in early December 2018 over the proposed fuel tax. Since France does not generate very much CO₂ in generating electricity (70% of the country’s power comes from nuclear energy), it would make more sense for them to subsidise electric car purchases and taxing new gasoline- powered automobiles. I’m never one to defend internal combustion automobiles, but let’s bear in mind that the entire transportation sector including ships, trains, planes and automobiles account for only 14% of carbon emissions. And then what impact will higher fuel prices actually have on consumption and consequent emissions?
More carbon emissions from the violent protesters in Paris. From Newsweek, Dec 2 , 2018

Governments need to quickly use a comprehensive approach so that all emissions are addressed and abated. The current narrow-minded focus on taxation of fuel won’t cut it. This is exacerbated by the fact that environmental values are are not deeply held and shared by enough people. As a result every mantra: “eat less meat”, “drive less”, “preserve forests”, etcetera, gets misinterpreted as an authoritative order. And too many of us are so talented at cutting our noses to spite the face of government.As an alternative strategy, the federal government can nourish the roots of the environmental grass movement. I know schools are asked to do an awful lot these days. But most schools do next to nothing about climate change. Yet they seem to find the time to peddle an awful lot of electronics and student travel. Here’s a list of things schools can do to be proactive about climate change.(1) Encourage teachers to use public transit or electric cars and not fill the parking lot with 19th century inventions.

(2) Compost cafeteria waste.

(3) Reduce and reuse before dumping stuff into recycling bins.

(4) Reward students for coming up with green ideas.

(5) Serve beef sparingly in the cafeteria. Replace it with more chicken meals, which have a much lower carbon footprint.

(6) Encourage local travel instead of flying or driving to remote destinations.

Sources:
Nature Communications. Carbonate-hosted methanotrophy represents an unrecognized methane sink in the deep sea
www.nature.com/articles/ncomms6094
Scientific American The Mystery of the Missing Molecules scientificamerican/journal/v319/n5/box/scientificamerican1118-32_BX1.html
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December 8, 2018 0