Wednesday, July 31, 2013

Science Clock Series: Part VII

Today's number comes from geology, and is given by:

\[\text{Quartz (Mohs scale, SiO}_2)\] The Mohs scale of mineral hardness is a scale developed by the German geologist Friedrich Mohs in 1812 to help classify minerals based their relative hardness. Its purpose is to show which minerals can scratch other minerals, and it doesn't represent the absolute hardness difference between them. (There are ten levels on the Mohs scale, but the absolute hardness difference between level one and level ten is a factor of 1600.) For any mineral, it can (in theory) be scratched by anything higher on the scale than it, and can scratch anything lower on the scale than itself.

There are ten levels on the Mohs scale, defined by ten specific minerals. The ten levels of the Mohs scale are defined by:
\begin{align}1&\dots\dots \text{Talc}\\
2&\dots\dots \text{Gypsum}\\
3&\dots\dots \text{Calcite}\\
4&\dots\dots \text{Fluorite}\\
5&\dots\dots \text{Apatite}\\
6&\dots\dots \text{Orthoclase Feldspar}\\
7&\dots\dots \text{Quartz}\\
8&\dots\dots \text{Topaz}\\
9&\dots\dots \text{Corundum}\\
10&\dots\dots \text{Diamond}
\end{align}As you can see, the hardness of quartz is the definition of hardness level 7 on the Mohs scale. For comparison, your fingernails have a hardness of about 2.2–2.5 on the Mohs scale, a copper penny is about 3.2–3.5, a pocket knife is about 5.1–5.5, and a steel file about 6.5.

As an interesting aside, the enamel your teeth are made of is basically a variant of apatite (called hydroxyapatite) which, as you can see, is the definition of hardness level 5. This suggests that you probably don't want to be scratching your teeth with anything higher on the scale than a 5. (Tooth enamel also happens to be the hardest substance in the human body, in case you were wondering.)

Quartz itself is an interesting mineral. It's the second most abundant mineral in the Earth's continental crust after feldspar, and has been used in jewelry and handicrafts throughout history. It is made up of silicon dioxide (also known as silica) with the chemical formula \(\text{SiO}_2\). Silica can solidify in many different arrangements, or even mixtures of them in an amorphous structure; two of these arrangements are known as \(\alpha\)- and \(\beta\)-quartz.

Quartz/silica is a pretty tough material and is relatively resistant to erosion (it's number 7 on the scale after all). Being the second most common mineral in the Earth's crust, silica shows up in many places. Most sand in inland deserts is made up of tiny particles of silica, and the type of glass used to make up windows and drinking glasses for the last few centuries (soda-lime glass) is composed of about 75% silica. Many marine organisms construct skeletons or homes for themselves out of it (sponges and diatoms [single-celled plankton] in particular). Silica even has a slight connection with our number 4: it is used to help extract DNA due to its ability to bind to it.

Anyway, check back next time for a look at another number from astronomy! Click here to jump directly to it.

Monday, July 29, 2013

Science Clock Series: Part VI

Today's number, like the previous one, comes from chemistry and is given by:

\[\approx\rho\text{ of Zn }(\times10^{-8}\,\Omega\cdot\text{m at }20^\circ\text{C})\] The letters Zn are the chemical symbol for the metal zinc, and at first glance you might think that the rho (\(\rho\)) in this equation is the same as the rho in the equation for the number one, standing for density. This is not the case. Confusingly, rho can also represent resistivity, as it does here.

Resistivity is the property of a substance to resist the flow of electricity. The maguscule omega (\(\Omega\)) stands for ohms, the standard unit of measure for resistance, which is a slightly different property than resistivity. Resistance depends on circumstances such as how a substance is shaped, while the resistivity of a substance is independent of the shape it takes. (For example, a short, fat copper wire has a lower resistance than a long, thin copper wire, but the resistivity of the copper making up the wire is the same in both cases.)

In one respect, resistivity and density are similar: they are both temperature dependent, which is why the resistivity is specified at \(20^\circ\)C. If we look up the resistivity of zinc at \(20^\circ\)C, we find it to be \(5.90\times10^8\,\Omega\cdot\text{m}\).

Tune in next time for a number from geology! Click here to jump directly to it.

Monday, July 22, 2013

Science Clock Series: Part V

Today's number comes from chemistry, and is given by:

\[\approx\text{sp. gr. Fe}^{2+}\,\text{Fe}_2^{3+}\,\text{O}_4\] The letters "sp. gr." stand for the term "specific gravity." Specific gravity  is the ratio of the density of a substance to the density of another substance, usually a reference substance of some kind.  The most common reference substance is liquid water which, as you may remember from the first post in this series, has a density of one gram per cubic centimeter.

The chemical formula \(\text{Fe}^{2+}\,\text{Fe}_2^{3+}\,\text{O}_4\) stands for the chemical compound iron(II,III) oxide with the chemical name ferrous-ferric oxide, found in nature as the mineral magnetite. (“Fe” and “O” being the chemical symbols for iron and oxygen, respectively.) The Roman numerals II and III refer to the oxidation state of the iron atoms in the compound, which are represented in the formula by the superscript +2 and +3 respectively. The oxidation state is basically how many electrons an atoms gains or loses while in a compound. Positive numbers indicate that an atoms has lost electrons (which have a negative charge), and negative means an atom gains electrons.

The subscript 2 and 4 refer to the number of atoms of that kind, so there are two \(\text{Fe}^{3+}\) atoms and four oxygen atoms. In a stable compound the oxidation numbers should come out to zero (no net electrical charge). Since oxygen atoms almost always have a \(-2\) oxidation state, they add up to give \(-8\) to the oxidation state of the compound. There is one iron atom giving +2, and two iron atoms giving +3, for a total of \(2+(2\times3)=+8\) to the oxidation state, which nicely balances the oxygens and helps ensure the compound is balanced and stable.

Ferrous-ferric oxide as it appears in nature in the form of magnetite has a blackish-brown color with a metallic sheen and has a density of approximately 5.17 grams per cubic centimeter, which gives it a specific gravity of 5.17 (relative to water). Magnetite is the most magnetic naturally-occurring material, and is also where the name magnetism comes from.

Check back next time for another number from chemistry! Click here to jump directly to it.

Monday, July 15, 2013

Science Clock Series: Part IV

Today's number comes from biology, but before we get to it, I just want to say “sorry” for the long delay between posts. I usually try to discipline myself to write more frequently, and although I've been a bit busy and had some trouble settling on the scope for this post, those are petty excuses. I did have some difficulty deciding how much to write for this post (given its subject), and began writing a lengthy dissertation before eventually deciding to cut back somewhat for conciseness. Anyway, without further ado:

Today's number comes from biology, and is given by:

\[\text{# of bases in DNA}\] First of all, what does the word “base” even mean in this context? It does not (as I at first naively assumed) have anything to do the use of the word base in mathematics (specifically in exponentiation where it refers to the number b in the expression \(b^{\,n}\)). It is actually a contraction of the word “nucleobase” and its use is mainly historical, having to do with the properties of nucleobases in acid-base reactions. In this case it relates to the use of the word “base” in chemistry, in reference to substances that neutralize acids.

Although the use of the word “base” in this instance doesn't come from math, it does have a curious appropriateness. Going back to the mathematical side of things for a moment, numeral systems (such as the decimal system in place in most of the world today) can be specified as “base-X”, where X refers to the number of distinct symbols that can, in principle, express all the natural numbers. Thus the decimal system in use throughout most of the world today (which uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) is a base-10 system. Binary, the system used by computers, is base-2, because it uses only 0 and 1. Any number can be used as the base of a numeral system, and many different numbers have been used by various people groups throughout history. Anyway, the point of this diversion is that DNA can be considered to be a form of a base-4 system, since it uses a collection of four different (nucleo-)bases to encode genetic information.

Just what are these mysterious nucleobases, however? They're four small molecules (containing between 9 and 15 atoms each) known as adenine, guanine, cytosine, and thymine, and abbreviated A, G, C, and T. (There's also a fifth molecule, uracil, that substitutes for thymine in RNA, but we're only concerned with DNA here.)

Just as information can be converted to base-2 and transmitted and stored digitally as a long string of 0’s and 1’s, the information in a creature's genetic code is stored in base-4, which we could represent using long strings of the numerals 1-4 (or as they actually do in genetics, as longs strings of A’s, G’s, C’, and T’s).

The details of how exactly ordered strings of tiny molecules are used by certain proteins to create all the other proteins in a living creature are truly fascinating, absolutely mind-boggling, and far, far too vast for me to get into in this post. Suffice to say, you should go read up about it on your own.

Anyway, tune in next time for a number from chemistry! Click here to jump directly to it.

Monday, July 1, 2013

Science Clock Series: Part III

Today we turn to the field of cosmology for our number, which happens to be three, and is given by:

\[\approx\ \text{background radiation of space (K)}\] The “background radiation of space” refers to the Cosmic Microwave Background Radiation, usually abbreviated “CMBR,” or just “CMB.” The CMBR is a diffuse ocean of electromagnetic radiation which pervades space evenly from every direction and which has its peak energy in the microwave portion of the spectrum. It has an almost uniformly frigid temperature of \(2.72548\pm0.00057\) K (about \(-270.42^\circ\)C, or \(-454.76^\circ\)F).

Because the CMBR pervades all of space, as far as we can tell, it defines the coldest temperature you can find naturally in the universe (at the current epoch). This is what people usually have in mind when they think about the temperature of space. If you were to travel to the intergalactic void in the incomprehensibly large bubbles of empty space between the faintly glimmering gossamer filaments of galaxy clusters, far away from any stars or sources of heat, this is the temperature you would measure. That's what I mean when I say that the SCUBA-2 instrument on the James Clerk Maxwell Telescope is currently the coldest place in the known universe, because its sensor arrays operate at a temperature of a mere \(0.07\) K (about \(-273.08^\circ\)C or \(-459.54^\circ\)F), 70 millikelvin above absolute zero.

Now, although the CMBR is incredibly uniform in all direction, it's not perfectly uniform. It is, however, pretty close, having no variations larger than 1 part in 10,000. For comparison, according to the regulations of the World Pool-Billiard Assocation, pool balls need only be smooth to 1 part in 500. To put that in perspective, the CMBR is twenty times smoother than a pool ball.

Analysis of the variations in the CMBR is a very active area in cosmology. In the last 24 years there have been no fewer than three satellites (COBE, WMAP, and Planck) dedicated to measuring and mapping the minute variations found across the sky in the CMBR. On a more personal note, it also holds a bit of a special place in my heart because it's partly responsible for me going into astrophysics.

You see, I've always been interested in astronomy as long as I can remember, but when I was young I was interested only in the planets and moons of our Solar System. Stars, galaxies, and the wider universe held no interest for me. It wasn't until I came across a book on cosmology sometime around the age of eleven that I became interested in the universe in general.

You see, any modern theory of cosmology needs to explain the existence of the CMBR. It is generally taken as evidence of the Big Bang theory of the universe's formation, and is explained as light from diffuse, homogeneous clouds of (primarily) hydrogen from early in the universe's history that has been highly redshifted over time by the expansion of the universe down to the energies found today. It is extremely difficult (if not impossible) to explain in the Steady State model of the universe (a competing theory during the early 20th century which was generally considered to be dis-proven upon the discovery of the CMBR), and is generally taken as proof of a finite age (and thus a beginning) of the universe.

There are, however, some problems with this particular interpretation. For one, the CMBR is too smooth; for it to be as smooth as it is today in the Big Bang theory, it would have required various parts of it to exchange heat energy between themselves and equalize their temperatures. However, if we pick two areas of the sky on opposites sides from us, there hasn't been time for them to have exchanged energy. This led to the postulation of inflation, a period of greatly-accelerated expansion of the universe, providing time beforehand for everything to equalize; however, there has yet to be any concrete evidence for such a period of rapid expansion. There's also the problem of galaxy clusters not casting as much of a “shadow” on the CMBR through the Sunyaev-Zel’dovich effect as they should, on average, and the anomalously weak quadrupole moment in the distribution of variations in the CMBR.

The point of the above is not necessarily to show that the Big Bang theory is wrong, per se, merely to point out that there remain unsolved problems with it (as there do in pretty much every area of science. It wouldn't be the “search for knowledge” if we already knew everything, would it?). As scientists we must always keep in mind the possibility that there may exist alternatives that fit the data as well or better, and it would be prudent to keep an open mind.

I encountered one such alternative as a young lad in the previously-mentioned book Starlight and Time, by Russell Humphreys, which contained an alternate cosmological theory consistent with the book of Genesis, General Relativity, and everything known about the CMBR up to 1994 when it was published. I remember that as a child the equations of General Relativity scattered liberally throughout the text were incomprehensible to me – I'm not sure I had even begun algebra at that point – but the accompanying text explained fascinating concepts like time dilation, gravitational red-shifts, and the expansion of the universe in language that I could grasp. It was, in all honesty, a pivotal point in my life. I was hooked on physics, and knew that, someday, I too wanted to spend my life studying the fundamental mysteries of the universe.

(If you're curious, in Humphrey's theory the CMBR is the primeval light created on the first day of creation, just stretched and red-shifted into the microwave region. If you're curious and not afraid of math [though as mentioned the writing stands on its own], you can find the book on, as I discovered while writing this post. I never owned a copy myself and it's been years since I read it, so I now have the Kindle edition to look forward to re-reading on my phone.)

Anyway, you now know what the cosmic microwave background is, and that where it comes from depends on your starting assumptions. Regardless, it's a fascinating topic and I could say a lot more about it, but this post is long enough already. Tune in next time for a very important number from biology! Click here to jump directly to it.