by Tim James
This means there are lots of different forms of the Schrödinger equation and different ways of writing it. The most straightforward one is called the generalized time-dependent Schrödinger equation and it looks like this:
i represents the square root of minus one. Any normal number, either positive or negative, always generates an answer that is positive when multiplied by itself. −2 × −2 isn’t −4, it’s +4. But that means −1 would have no square root, so there must be another type of number which multiplies by itself to generate negatives. These numbers are called i numbers. The reason the letter i is chosen will muddle things at this point so don’t worry about it: it’s a historical convention and it’s meaningless. It may seem strange that we have to use freaky numbers in our equation because it seems like a cheat but that’s not what’s happening. Nature does things outside of our normal experience so we have to use numbers outside our normal experience in order to make sense of it. When we try using regular numbers, the equation gives answers that don’t match reality. It would seem that nature uses i numbers so we have to as well.
H is called the Hamiltonian and it refers to the total energy of the thing we’re examining. We’ve written it here with one letter but that’s a shorthand. Written in full, the Hamiltonian is a long-winded term that takes into account the particle’s mass, its kinetic energy, how far it is from the nucleus (called its potential), and so on, but it still just means how much energy the particle has.
Ψ: This symbol (psi) represents something called the wavefunction. It can mean a great many things, but in the context of chemistry it refers to the fact that the probable location of a particle ripple. Rather than having a specific coordinate in space, an electron’s location has a wavelike character when it’s not being interfered with. The wavefunction takes this into account.
|> is called a ket vector. What it refers to, generally speaking, is the state that something is in. In this case the state of the particle’s wave-function. The left side of the equation, read in full, is now telling us that if we calculate the total energy of the wavefunction’s state and multiply the answer by negative i, we’ll get something useful.
ħ is called Planck’s constant and represents 1.055 × 10−34 Joule-seconds. This number is a property of the Universe that relates the energy of a particle to its frequency. Frequency is how many times something will ripple per second and since all particles have ripply movement, we need a term that relates the two. Specifically, if we divide the energy of a particle by its frequency, we get a number called h, which is 6.626 × 10−34 Js in SI units. Planck’s constant is arrived at by dividing h by 2π. We do this because 2π is often used when measuring frequencies, so we include it as part of our constant to make the equations neater.
∂ is called a partial differential. It’s a symbol that tells us to measure how one property changes when you’ve got lots of other stuff going on and you only want to focus on one thing. In this case is telling us to compare the change of something compared to t (time).
So, the whole equation is telling us that if we can work out the total energy a particle has in a particular state (left-hand side), we can work out how its behavior will change with time (right-hand side).
If you know what energy an electron has, you can predict where it’s likely to be at any moment. Do this for all three spatial dimensions and you’ll end up with a description of where an electron is likely to be around its nucleus—the orbital.
APPENDIX IV
Neutrons into Protons
We’ve already met quarks in Appendix II and they can be tricky things. They come in many varieties but they all have an electric charge that adds up to the charge of a proton or a neutron.
An “up” quark has a charge of +2/3 while a “down” quark has a charge of −1/3. When two up quarks and a down quark occur in a trio, the charges combine to create an overall +1, a proton. If an up quark occurs with two down quarks, however, the charges cancel and the result is a neutron.
But quarks don’t stay as one type. An up quark can turn into a down quark and vice versa. A neutron is an udd (up down down) combination, but if one of the downs turns into an up we get an uud (up up down)—a proton. It’s the quark inside that flips and turns a neutron into a proton.
When this happens the amount of overall charge has changed and for some reason this is a big no-no for the Universe. Rather than creating a charge imbalance, the Universe prefers to keep things neutral so it does a particle shuffle.
When the −1/3 down quark changes character, it emits a particle called a W− boson, which carries away a −1 charge, leaving a +2/3 charge behind.
The W− quickly splits into an electron, which retains the charge, and another particle called an anti-neutrino, which has none. And there’s no simpler way of describing the whole process.
APPENDIX V
The pH and pKa Scales
You may have met the pH scale in school. The more acidic something is, the lower its number. Acids tend to have values below 7 while non-acidic things tend to be 8 and upward. The reason the scale was introduced was because the numbers involved in acid chemistry are often extremely small.
Suppose we had 1 × 105 hydrogens in a 1-liter bottle and 1 × 104 in another. The first contains 100,000 and the second contains 10,000. Clearly, the first is ten times more concentrated than the latter, but they are both extreme numbers. So we invoke the laws of logarithms.
A logarithm is the number of times you have to multiply something by itself in order to get a particular result. Say you had the number three and multiplied it by itself four times. That would be written as 3 × 3 × 3 × 3 or, more simply, 34. The answer is 81.
But suppose you want to do your calculation the other way around. You want to know how many times you had to multiply three by itself to reach 81. You would write it like this:
Log3 81 = 4
In other words, what number do I have to raise three to in order to reach 81. The answer would be 4.
In our earlier example, we had one solution containing 100,000 hydrogens. If we express this logarithmically we would write:
Log10 100,000 = 5
Five is a much easier number to work with so we might describe this acid as a “5 solution.” The number isn’t telling us the concentration directly but it’s telling us the order of magnitude with which we’re dealing.
Likewise, a solution ten times more dilute would be called a “4 solution.” This is useful because, when we’re dealing with something huge like a concentration of 1,000,000,000,000,000,000, it’s easier to call it an “18 solution” rather than write the whole thing out.
So why does the scale run backward? The answer is that most acids, even very concentrated ones, only contain a small number of hydrogens per liter.
Most of the acids you’re likely to encounter fall somewhere in the region of 0.00001 to 1. Writing this in standard form, we use negative powers, i.e. 10−6 to 10−1. In this case, the 10−1 is the more concentrated solution.
It was a Danish chemist named Søren Sørensen who suggested we use negative logarithms when writing our acid concentrations, purely because it looks neater. The less concentrated acid gets a value of:
−Log10 0.00001 = 6
While the more concentrated acid ends up as:
−Log10 1 = 1
He referred to the negative logarithm of a number as the “potenz” of it, which really means “the power you have to raise the number ten to.” And so we define the pH scale as:
pH = −Log10 (concentration of hydrogen ions in a liter)
On this scale, most acids fall between 1 and 6 but the scale can go in either direction. An acid with a concentration of 10 would have a pH of 0, while an acid with a concentration of 100 would be pH −1.
The pKa scale works in exactly the same way and uses the same system. Only this time, rather than measuring the concentration of hydrogens, we’re measuring the strength of an acid, i.e. how willing it is to release a proton into solution. The bes
t way to express this is to say what fraction of original acid ends up dissociating.
Suppose you have 100 acid molecules and only one of them splits. We would say the strength for this acid is 1 percent. For reasons that we won’t get into (an appendix of an appendix is a bit silly), we express the strength of an acid as a fraction called the Ka. And, once again, these numbers are typically extremely small.
Only a tiny amount of hydrogens have the gumption to dissociate, so we get Ka values with high negative numbers. Using the p method, we take the negative logarithm of the Ka (how strong it is) and voilà, the result is our pKa scale.
APPENDIX VI
Groups of the Periodic Table
Going from left to right across the periodic table, several of the columns (groups) have names that are largely historical. Groups 3 to 10 are named after the top element in the group e.g. group 10 is called “the nickel group,” but all other columns have informal monikers.
Group 1 Alkali metals
Group 2 Alkaline earth metals
Group 3 Scandium group
Group 4 Titanium group
Group 5 Vanadium group
Group 6 Chromium group
Group 7 Manganese group
Group 8 Iron group
Group 9 Cobalt group
Group 10 Nickel group
Group 11 Coinage metals
Group 12 Volatile metals
Group 13 Icosagens
Group 14 Crystallogens
Group 15 Pnictogens
Group 16 Chalcogens
Group 17 Halogens
Group 18 Noble gases
Acknowledgments
One thing I’ve learned during the course of writing my first book is that while my name appears on the cover, what you read is a collaboration of many minds. I’d like to thank several of them.
First and foremost, the students of Northgate High School, for giving me a reason to get up in the morning and helping me get the whole thing off the ground. This book wouldn’t exist without them. Rock and roll, guys. Rock and roll.
Huge and heartfelt thanks to my agent Jen Christie for taking a chance on an awkward writer, being patient when I was difficult, and for guiding me through the insane world of publishing and marketing—something far more complex than science.
Duncan Proudfoot from Little, Brown Book Group is wonderful and I want to thank him for immediately “getting” what I was trying to do. I really hope the book doesn’t let him down.
In terms of the book’s content, the person I need to thank most is Ella Catherall who edited every page, fact-checked the science, and corrected all 385 errors. If the book is any good, it’s down to her.
Thank you to the Science department at Northgate (there’s about twenty staff so I can’t namecheck everyone) for all the tolerance they have shown me while I’ve learned to be a teacher. It’s a privilege working with them. And in particular, a very special thanks to Hazel and David—not just for advice, but for their inspiration and friendship.
I need to thank the great Seishi Shimizu for teaching me how to think and write like a scientist. He was a great mentor and it was an honor being his student.
Thank you to Karl Dixon for reading and critiquing the manuscript, making me laugh when I stopped believing it was any good, and for being the Sherlock Holmes to my Watson these many, many years.
A huge thank you to Mandalyn King for always giving it to me straight and being pretty much amazing. I respect her opinion more than she knows.
From an earlier time in my life, I want to thank Mr. Evans for being that teacher, and John Miller for persuading me to become one myself.
I obviously need to thank my wife who is probably the most patient person in the world. Thanks for letting me spend so much time doing this.
And finally, I want to thank my father, who taught me the importance of asking questions and turned me into a scientist.
Notes
INTRODUCTION: A RECIPE FOR REALITY
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CHAPTER 1: FLAME CHASERS
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3. J. D. Clark, Ignition! An Informal History of Rocket Propellants (New Brunswick, NJ: Rutgers University Press, 1972).
4. “Eastern Germany 2004,” Bunker Tours. Available from: http://www.bunkertours.co.uk/germany_2004.htm (accessed August 18, 2017).
5. Diogenes Laertius, The Lives and Opinions of Eminent Philosophers, Vol. II, Books 6–10, trans. R. D. Hicks (Cambridge, MA: Harvard University Press, 1925).
6. “Protactinium,” Encyclopedia. Available from: http://www.encyclo-pedia.com/science-and-technology/chemistry/compounds-and-elements/protactinium (accessed August 18, 2017).
7. J. Emsley, The Shocking History of Phosphorus: A Biography of the Devil’s Element (London: Macmillan, 2000).
8. H. M. Leicester, H. S. Klickstein, A Source Book in Chemistry 1400–1900 (Cambridge, MA: Harvard University Press, 1952).
9. H. Muir, Eureka: Science’s Greatest Thinkers and Their Key Breakthroughs (London: Quercus, 2012).
10. Muir, Eureka.
11. M. Sędziwój, “Letters of Michael Sendivogius to the RoseyCrusian Society,” Epistle 54 (January 12, 1647), The Masonic High Council the Mother High Council. Available from: http://rgle.org.uk/Letters_Sendivogius.htm (accessed October 8, 2017).
12. I. Asimov, Breakthroughs in Science (Boston, MA: Houghton Mifflin, 1960).
13. R. Harré, Great Scientific Experiments: Twenty Experiments that Changed Our View of the World (Oxford: Phaidon, 1981).
14. I. Asimov, Words of Science (London: Harrap, 1974).
15. Isaiah 54:11.
16. “Periodic table—lithium,” Royal Society of Chemistry. Available from: http://www.rsc.org/periodic-table/element/3/lithium (accessed August 18, 2017).
17. B. C. Gibb, “Hard-luck Scheele,” Nature Chemistry, vol. 7 (2015), pp. 855–6.
18. Leicester and Klickstein, A Source Book in Chemistry.
CHAPTER 2: UNCUTTABLE
1. The Core (2003), dir. Jon Amiel, Paramount Pictures.
2. T. Irifune et al., “Ultrahard polycrystalline diamond from graphite,” Nature, vol. 421 (2003), pp. 599–600.
3. David Robson, “How to make a diamond from scratch with peanut butter,” BBC (November 7, 2014). Available from: http://www.bbc.com/future/story/20141106-the-man-who-makes-diamonds (accessed August 18, 2017).
4. B. Russell, History of Western Philosophy (Oxford: Routledge Classics, 2004).
5. D. Hurd, J. Kipling, The Origins and Growth of Physical Science (London: Penguin, 1958).
6. J. Dalton, A New System of Chemical Philosophy (London: R. Bickerstaff, 1808).
7. W. L. Masterson, C. N. Hurley, Chemistry: Principles and Reactions (Boston, MA: Cengage Learning, 2012).
8. R. Harré, Great Scientific Experiments: Twenty Experiments that Changed Our View of the World (Oxford: Phaidon, 1981).
9. A. Einstein, “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,” Annalen der Physik, vol. 322 (1905), pp. 549–60.
CHAPTER 3: THE MACHINE GUN AND THE PUDDING
1. “A Boy and His Atom: The World’s Smallest Movie,” IBM Research. Available from: http://www.research.ibm.com/articles/madewithatoms.shtml (accessed August 18, 2017).
2. E. T. Whittaker, A History of Theories of the Aether and Electricity (Harlow: Longman, Green & Co, 1951).
3. E. Rutherford, Nobel Lectures: Chemistry 1901–1921 (Amsterdam: Elsevier Publishing, 1966).
4. H. C. von Bayer, Taming
the Atom: The Emergence of the Visible Microworld (New York: Random House, 1992).
5. R. W. Chabay, B. A. Sherwood, Matter & Interactions, third edition (Hoboken, NJ: Wiley, 2002).
6. Man of Steel (2013), dir. Zak Snyder, Warner Bros.
7. H. P. Lovecraft, The Dunwich Horror and Other Stories (London: Pocket Penguin Classics, 2010).
8. Superman Returns (2006), dir. Bryan Singer, Warner Bros; P. S. Whitfield et al., “LiNaSiB3O7(OH)—novel structure of the new borosilicate mineral jadarite determined from laboratory powder diffraction data,” Acta Crystallographica Section B, vol. 63, no. 3 (2007), pp. 396–401.
CHAPTER 4: WHERE DO ATOMS COME FROM?
1. “The coldest place in the world,” NASA (December 10, 2013). Available from: https://science.nasa.gov/science-news/science-at-nasa/2013/09dec_coldspot (accessed August 18, 2017).
2. R. Sahai et al., “The coldest place in the Universe: Probing the ultra-cold outflow and dusty disk in the Boomerang Nebula,” The Astrophysical Journal, vol. 841, no. 2 (2017).
3. J. W. Park et al., “Ultracold dipolar gas of fermionic Na23K40 molecules in their absolute ground state,” Physical Review Letters, vol. 114 (2015).
4. Plato, Theaetetus, trans. J. McDowell (Oxford: Oxford University Press, 1999).
5. B. Russell, History of Western Philosophy (Oxford: Routledge Classics, 2004).
6. G. Dixon, P. Parsons, The Periodic Table: A Field Guide to the Elements (London: Quercus, 2013).