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# Math OERThe Angry Chemist

I get a certain number of people who are put off by my no-nonsense, tell-it-like-it-is tone that some people think is "egotistical." They think I should "tone it down." But, f\$&% that. Humility is for people who aren’t the best at things...I was born a cartoon character. That’s just my accent. And I can’t get rid of it. Honestly, I don’t want to. Stop oppressing my culture.

Okay, sit up straight. I owed your instructor a favor, so here I am, ready to teach some basic chemistry ideas to math students.

On the interwebs I am called the The Angry Chemist. I have some very famous YouTube videos, such as Chem Lab Lesson One: Five Tricks to Stop Subconsciously Touching Your Face and Chem Lab Lesson Two: How to Use the Eye Wash Station Because You Didn't Listen. Usually I shout valuable chemistry advice and complain about clueless chemistry students because, as the saying goes, "You're either part of the solution or part of the precipitate." But not today. Today I am going to math at you.

## Logs

To start, remember what you learned about decimal point scoots.

• Multiply by powers of ten.
• (a) 123.456 × 10
• (b) 123.456 × 100
• (c) 123.456 × 1,000

No, I am not giving you buttons to click on for the answer. Go read that topic's lecture notes again if you need to.

There is a fancy word for the decimal scoot pattern.

Logarithm

The logarithm of a power of ten is how many decimal scoots from 1 it has.

We write log(10) = 1, log(100) = 2, log(1,000) = 3, and so on.

To be confusing we could write this pattern as log(10n) = n but we won't do that.

The Greek lógos roughly translates into the English "reasoning". It can mean "speech" (as in dialogue) or "thinking" (as in logic) because in ancient Greece reasoning involved talking to people about your thoughts. Maybe you once heard a pastor claim it means "word" but that's as much as oversimplification as everything else in his sermon. The Greek arithmós translates into the English "number" (as in arithmetic).

In the early 1600s John Napier combined those two greek words to invent a cool new word that means "reasoning about numbers". What a delightfully vauge phrase! It could have meant anything in math! But Napier and everyone else have been using it for 400 years to talk about something as unexciting as decimal point scoots, that's what it means.

Some annoyingly hoity-toity readers might say, "That's not what logarithm means!" Oh yeah? Do you still carry a slide rule? If not, stop trying to distract us and go find someone else that cares that log(m × n) = log(m) + log(n). And if you do carry a slide rule we will avoid you for other reasons.

By the way, the English language has another way to say the same thing.

Orders of Magnitude

A number is one order of magnitude bigger than another if it is about 10 times bigger.

A number is two orders of magnitude bigger than another if it is about 100 times bigger.

A number is three orders of magnitude bigger than another if it is about 1,000 times bigger.

And so on...

So the logarithm of a power of ten is how many orders of magnitude it is bigger than 1. Different language, same meaning.

Ever since 1977 the best way to admire orders of magnitude is the video Powers of Ten. You probably watched it in school, unless your school was lame. Time to watch it again.

Most people who know only a little about logarithms think they know about earthquakes. Big earthquakes are not twice or quadruple the size of small earthquakes, but up to nine orders of magnitude bigger.

The Richter Scale measures an earthquake based on the logarithm of the wave heights recorded by a seismograph. But geologists don't use the Richter Scale much any more. There are newer and better ways to think about earthquake "bigness". You can read more if you wish to elevate yourself above the unwashed masses who think they know at least a tiny amount of useful information about earthquakes but actually don't.

I also want to check that you remember the pattern that goes in the opposite direction.

• Multiply by powers of one-tenth.
• (a) 123.456 × 0.1
• (b) 123.456 × 0.01
• (c) 123.456 × 0.001

Math people sometimes write 0.1 as the exponent 10−1. This allows us to make a pleasing table.

NumberNumber as ExponentLogarithm of that NumberName
10,0001044ten thousand
1,0001033thousand
1001022hundred
101011ten
1 0one
0.110−1−1tenth
0.0110−2−2hundredth
0.00110−3−3thousandth
0.000110−4−4ten thousandth

We will use logarithms again in a moment to build another pleasing table. But for the sake of completeness this section should end with the only logarithm joke that is even slightly funny. "When newlywed vipers build a house together, why do they put the rough wooden table in the bedroom and the pile of soft grass in the dining room?"

No, I'm not going to tell you the joke's answer. Go find another chemist.

Test Yourself!

What does it mean to multiply 10 by itself zero times? Nonsense, of course. But give it a numerical answer anyway.

Why is it true that log( 1n ) = − log(n)?

Explain XKCD comics 2091, 558, and 1162. (Answers are here, here, and here.)

Enough math and geology. It's finally time for chemistry.

Kids let go of helium balloons to learn that helium gas is lighter than air. Maybe you know that propane furnaces can be dangerous in basements because propane gas is heavier than air, so a leak would make a nice odorless, explosive pool of gas under your house. Imagine we fill one balloon with helium and another balloon to the same volume with propane. The propane balloon is of course heavier and does not float in air. (It does float on water. You can try this at home.)

But why is the propane balloon heavier than the helium balloon? Does it have more molecules of gas (does propane have "more stuff")? Or does it have the same number of molecules but each weighs more (is each bit of propane "heavier stuff")? Maybe both are true and propane has both "more stuff" and "heavier stuff"?

The fellow whose picture is to the right is Amedeo Avogadro. His big three posthumous regrets are that only one painting of him remains, that his name sounds too much like the word avocado, and that in that painting he looks a bit like an avocado. Mouse-over his image to check for yourself.

All chemists secretly think about avocados when they learn about Avogadro. It's okay that you do too. Just don't tell anyone. Or maybe you only speak one of the very few languages in which that fruit's name does not sound similar, in which case you can feel morally superior and also write poems about oranges.

Avogadro was a nobleman, revolutionary, and scientist from the Kingdom of Sardinia (now part of Italy). He answered the balloon question in 1811. He theorized that all gasses with the same volume, temperature, and pressure will have the same number of molecules, so that gasses like propane are more massive only because each molecule is "heavier stuff" compared to lighter gasses like helium.

Avogadro did not know how many molecules were in a certain amount of gas. He just theorized it would be the same number for any gas.

Chemists now know the answer. Buckle up for more history.

## Moles

The simplest atom is hydrogen, which has a single proton and electron (but no neutron). In 1805, a few years before Avogadro's limelight, an English scientist named John Dalton published first table of standard atomic masses, which declared a hydrogen atom had a mass of 1 unit, and the bigger atoms for the other elements went up from there. Sensible, right?

Unfortunately, atoms are complicated. Dalton did not know about the "binding energy" that converts mass into energy whenever protons and neutrons get bound together. An oxygen atom with eight each of protons and neutrons weighs about 0.03% less than those sixteen suckers do in isolation.

Chemists did a lot of work before figuring out how binding energy works. They knew that the common atoms of hydrogen have one proton, carbon has 6 each of protons and neutrons (for a total mass of 12), and oxygen has 8 each of protons and neutrons (for a total mass of 16). For about eighty years they knew carbon was about twelve times the mass of hydrogen, and oxygen was about 16 times the mass of hydrogen. Was it exactly ×12 and ×16? They thought so, but could not tell for sure.

In the late 1800s and 1900s the invention of the mass spectrometer and some early improvements finally allowed more careful measurements of atomic mass. Whoah! Carbon is not exactly ×12 and oxygen is not exactly ×16.

Panic! Almost a hundred years of data is in trouble because of binding energies.

With the increased accuracy from the new technology, which of these three options was actually closest to the old "one unit of atomic mass" that everyone had been using?

• one common hydrogen atom
• one-twelvth of a common carbon atom
• one-sixteenth of a common oxygen atom

It turns out the last was closest. Sorry, hydrogen. Let's change to calling oxygen the "typical" element. By redefining one unit of atomic mass as one-sixteenth of a common oxygen atom, the chemists could keep their past century of data.

Next the chemists asked how many oxygen atoms are needed for a total mass of 1 gram? The answer is an important number for the SI measurement system. It connects the counting of atoms with mass and weight.

Counting atoms is hard! It would take many years (and a Nobel Prize in 1926) to find an accurate answer. But during that time that number needed a name. Annoying mathematicians had already claimed the terms "group" and "set". So scientists needed another name.

Two names, actually. The German chemist Wilhelm Ostwald picked the name mole, which sounds a lot more respectable (and like the word molecule) in German. The French physicist Jean Perrin called it Avogadro's Number.

A pair means two of something. A trio means three of something. A dozen means twelve of something. A score means twenty of something. A gross means 144 of something. A mole means an Avogadro's Number of something.

Test Yourself!

What number do mathematicians think is the most disgusting?

(To get the joke's answer, highlight below and read it out loud.)

288 because it's two gross!

Eventually the counting was complete. There are about 6×1023 atoms in one gram of common oxygen. (We're rounding.)

Wha ha! Thunder and Lightning!

Or we can say a mole of things is about 6×1023 of those things.

People use moles for counting itsy bitsy atomic stuff like atoms, molecules, and electrons. For anything else a mole is too ridiculously big.

A mole of grains of sand would cover the beaches of about 100,000 earths. In other words, all the sand on earth is about five orders of magnitude smaller than a mole.

A mole of kilograms (or gallons) of water would cover the oceans of about 100 earths. In other words, all the jug-fulls of water on earth are about two orders of magnitude smaller than a mole.

A mole of stars approximates all the stars in the entire universe. Astronomers are not sure which number is bigger.

One last example for the parents out there: if a million babies cried once per second for 19 billion years, that would be about a mole of baby cries.

We can tie together Avogadro's idea with his number. A mole of atoms of an element weighs about as many grams as that element's atomic mass.

You young kids like videos, right? Have another, mostly because I respect this guy's tie.

## Gunpowder, Treason, and Plot

By the way, I was sloppy on purpose.

Oxygen atoms can have more or less than 16 total protons and neutrons. Some of these "flavors" of the atom are radioactive and others are stable. When I talked about a "common" oxygen atom earlier, did I mean the most common flavor in nature? Or did I mean the blend of flavors found in normal air?

A physicist uses the first definition. Although pure oxygen-16 is more theoretical than practical the physicist does not care. As the saying goes, physics is the study of frictionless elephants as massless strings.

But as a chemist I meant the second. Call me biased, but I actually like the air I breathe, which fills my lab.

This eventually led to a big fight. In 1961 the scientific community unsuccessfully tried to settle the dispute by redefining the "typical" element as carbon-12 instead of oxygen-16. For any college chemistry class it won't matter, because all of those three options we listed above are within 0.04% of each other. You newbie chemists have enough trouble not lighting my lab on fire, let alone measuring powders and doing experiments to even 1% accuracy.

But I advise learning some martial arts before attending a IUPAC meeting. I'm convinced that's actually a cannon hiding in the logo.

Yes, I really am biased. Ask a physicist for her side of the story.

Since the world has (for now) finally settled on carbon-12 as the "typical" element, everone who is allergic to rounding can finally say that Avagadro's Number is exactly 6.02214076×1023.

## Acid

Now I can tell you how chemists measure mixing stuff in water. They use "moles of stuff per liter of water".

Concentration

The terms molar concentration and molarity both refer to how many moles of stuff are added per liter of water.

What we care about today is how much a mixture is acidic. I don't care whether what stuff we dissolve in our liter of water. I just want to know how much it bubbles or explodes when mixed with baking soda.

Remember that hydrogen atom? No neutron, only a proton and electron.

If that electron goes away, the result is a lone proton. This is also called a hydrogen cation, or a hydron for short. It is abbreviated as H+

At your level of chemistry knowledge, hydrons are what makes a mixutre acidic. The more hydrons, the stronger the acid. I am lying to be simple. I don't want to define or talk about ions.

Acids are like earthquakes. A strong acid does not have double or quadruple the hydrons of a weak acid. It has orders of magnitude more. So lets put lots of hydrons in a liter of water.

Now, you can't just scoop up handfuls of hydrons and dump them into a flask of water. (I can as a fictional hero, but you don't have super-chemist powers.) The simpler technique is to bubble certain gasses into the water. Here are three that work well. Remember that it takes as many grams of a gas as its molecular mass to make one mole of hydrons in our liter of water.

• hydrochloric acid (HCl) with a mass per mole of 1 + 35 = 36 grams
• hydrobromic acid (HBr) with a mass per mole of 1 + 80 = 81 grams
• nitric acid (HNO3) with a mass per mole of 1 + 14 + (16 × 3) = 63 grams

The bubbles of these gasses will completely dissolve. Each molecule will separate into H+ and its other part, either Cl, Br, or NO3.

If we bubble 36 grams of hydrochloric acid into our liter of water, that one mole of HCl molecules separates into one mole of hydrons and one mole of Cl atoms.

If we bubble 81 grams of hydrobromic acid into our liter of water, that one mole of HBr molecules separates into one mole of hydrons and one mole of Br atoms.

If we bubble 63 grams of nitric acid into our liter of water, that one mole of HNO3 molecules separates into one mole of hydrons and one mole of NO3 atoms.

This is like shopping by weight in the bulk food section of the grocery store, or when buying nails by the pound at the hardware store. Nobody wants to count out individual peanuts or nails. So we use weight as an approximate for counting.

Here is a table that summarizes acidity when bubbling HCl into our one liter of water. Distilled water has some H+ in it already, which the first row and column show. To go down from the top row to the second row we add only a miniscule amount of gas. To move to the next row we add ten times the current amount: another order of magnitude. The equivalents of basic or acidic strength are only approximations, depending upon what brand of coffee, wine, or battery acid you drink.

Water's
H+ Molarity
Moles of H+
Total
H+ Molarity
Logaritm of
H+ Molarity
p[H+]EquivalentGrams of
0.000000100.0000001−77neutral—distilled water0
0.00000010.00000090.000001−66normal tap water0.0000324
0.00000010.00000990.00001−55black coffee0.0003564
0.00000010.00009990.0001−44wine0.0035964
0.00000010.00099990.001−33vinegar0.0359964
0.00000010.00999990.01−22lemon juice0.3599964
0.00000010.09999990.1−11battery acid3.5999964

Look at the table columns.

The first two columns add to get the third.

For the fourth column we take the logarithm of the third column, because we want to change an itty bitty decimal that varies over orders of magnitude into a nicer number (just like with earthquakes). But the result is negative, which is awkward.

Thanks to a historical accident from the silly physicists, everyone already has to use a negative amount for something as fundamental as the charge of an electron. Enough of that garbage. To avoid having yet more needless negativity in our labs, we chemists smartly decide to remove the negative from that logarithm value and name it the p[H+].

Chemists argue about what the letter p in p[H+] stands for. I will tell you the real answer. In your mind pronounce p[H+] as "pitifulness of hydrons" because it gets bigger as the solution gets less acidic. If let anyone tells you otherwise, nod your head agreeably while ignoring them, like Matthew Cuthbert whenever Marilla talks to him.

The right-most column is simply the second column multiplied by 36, because adding a mole of H+ requires 36 grams of HCl.

Test Yourself!

Imagine our HCl bubbler only works at 0.000324 grams per second. The first step, from p[H+] 7 to 6, takes 1 second. The second step, from p[H+] 6 to 5, takes 11 seconds. The third step, from p[H+] 5 to 4, takes almost 2 minutes. How long would each of the remaining three steps take?

Please realize that p[H+] is substitute for counting or weighing . Nobody wants to count how many H+ are in a liter of water. (Only I can, but I get bored rather quickly.) Even I have trouble weighing the H+ while it is floating in the water. So we measure the p[H+] instead, and it tells us about how many moles or grams of stuff is dissolved.

Test Yourself!

How would the table change if we were bubbling HBr or HNO3 instead of HCl?

We just told a story about how bubbling an acids into water adds more H+ to the baseline amount naturally found in water. We could tell a different story in which we do similar bubbling with a different chemical (called a base) that reduces the amount of H+ naturally found in water. The combined result would be a table like this:

Total H+ MolarityLogaritm of H+ Molarityp[H+]Equivalent
0.00000000000001−1414lye
0.0000000000001−1313oven cleaner
0.000000000001−1212household bleach
0.00000000001−1111household ammonia
0.0000000001−1010liquid soap
0.000000001−99baking soda
0.00000001−88sea water
0.0000001−77neutral—distilled water
0.000001−66normal tap water
0.00001−55black coffee
0.0001−44wine
0.001−33vinegar
0.01−22lemon juice
0.1−11battery acid

Please remember that this is a table of two different stories. We go down from the middle by bubbling an acid: slowly at first, then orders of magnitude more quickly. We go up from the middle by bubbling a base the same way.

Can we keep bubbling more acid? We can and the table will keep going down. You can have a negative p[H+]. But it's like tractors and pet bears: some people own them, but not in a college chemistry lab. Similarly having a p[H+] above 14 is possible but not something seen in a college chemistry lab.

You are now smart enough to read about how an electronic p[H+] meter works.

Since I'm a jerk, I will conclude with something really nit-picky. You will often see pH instead of p[H+]. Strictly speaking, pH is about something called "activity" taught in a second- or third-year college chemistry class for reasons you don't care about. But everyone—and I mean everyone—writes pH instead of p[H+] to be lazy. It's okay. Don't judge them. You can do that too. Or you can be highfalutin and write p[H+] properly while holding out your pinky finger in a dainty manner. Bonus points if you do this at a tea party while wearing a t-shirt with my picture on it.

Test Yourself!

Distilled water has about 55 moles of water molecules per liter. If we use the bottom row of the table and bubble 3.6 grams of HCl into one liter of water, we are putting in 3.6 moles of H+ and an equal amount of the other Clpart. What percentage of particles in this solution are H+?

For my second conclusion, know that I am pranking you when teaching you the word hydron. No one but me uses that word any more. When talking about the sthingy exchanged between acids and bases, instead call it hydrogen ion. When talking about the same thingy floating around by itself in solution and getting a little too friendly with some of the water molecules, call it a hydronium ion. When talking about atomic structure, call it a proton. All three mean the same thing in different ways, similarly to how my students call me "teacher", my kids call me "father", and my buddies call me "oh no it's that guy again".

Angry Chemist signing off. Hopefully I got a reaction out of you.