Tuesday, July 30, 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.

Sunday, July 28, 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.

Sunday, July 21, 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.

Sunday, July 14, 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.