This is the fourth post in the math/science part of my "Gen Eds in a Nutshell" series. It's a series of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.
Thus far in the math/science subjects:
1. From the previous post, we know that we distinguish the elements from each other by the number of protons an atom has in its nucleus. In general, this number does not ever change. Since the sun fused these elements together and spit them out, they have stayed the same for most elements.
We call the number of protons in an atom the "atomic number" and this tells us what the element is. Hydrogen has one proton in its nucleus and has the atomic number 1. Helium has two protons in its nucleus and is atomic number two. Lithium is atomic number 3 and has 3 protons in its nucleus. Atomic number 4 has four protons in its nucleus and is the element Beryllium.
2. The "atomic mass," in effect, is the number of particles in the nucleus. Remember from the previous post that the nucleus of an atom has protons and neutrons in it. Although the mass of these two is slightly different (a neutron is just a sliver more massive than a proton), they are close enough for us to assign them both the same mass--1 "atomic mass unit" (amu).
So the atomic mass is the number of amus of the nucleus, which is the sum of the number of protons and neutrons in the nucleus of an atom. There is no obvious way to predict how many neutrons will be in a particular atom's nucleus. In fact, the same element can have different forms with different numbers of neutrons (we call these different "isotopes" of an element).
3. In 1869, Dmitri Mendeleev put together the discoveries of the century before him and created the first "periodic table." He identified recurring patterns in the masses of elements.  We now know that these recurring patterns relate to the way electrons fill up the space around the nucleus of an atom. In particular, the most fundamental elements follow a kind of "octet rule" or rule of eight.
The octet rule is based on the idea that electrons fill up layers or "shells" around the nucleus (it's a little more complicated--see the previous post). When an element has a layer that is full, that is a very stable element, what we call a "noble gas." In the periodic table, the eighth column is the column of noble gases. These elements are at peace with the universe because their outer shell is completely full.
These elements are helium (it is unique in that it only has two spaces in its outer shell, so it is full at atomic number 2), neon (10), argon (18), krypton (36), xenon (54), and radon (86). You'll notice that from helium to neon and from neon to argon, 8 additional protons (and thus 8 additional electrons) are added to the atom to get from the one atom to the next.  When the shell reaches 8 electrons, it becomes very stable, an element that doesn't react with other elements.
Notice that hydrogen only has one proton/electron (the neutral atom always has the same number of electrons as protons) and is somewhat unstable. Thus the Hindenberg explosion. But helium is now used in blimps and such, because as a noble gas, it doesn't explode like hydrogen.
4. The eight basic columns of the periodic table relate to elements that have the same number of electrons in their outer shell when they are neutral (that is, when the number of electrons balance out the number of protons in the nucleus, yielding a 0 charge). So column one are elements that have one electron in their outer shell when they have a neutral charge. Column seven are elements that have seven electrons in their outer shell when they have a neutral charge.
5. The elements in column 1 are called the alkali metals. They are lithium (3), sodium (11), potassium (19), rubidium (37), cesium (55), francium (87). Notice that the numbers follow the same pattern as the noble gases in column 8. After the first shell of 2 electrons is filled (helium), lithium has one electron in its outer shell (2 + 1). Then add 8 electrons to get to sodium. Add eight more to get to potassium. See note 2 to explain how to get to the other numbers.
So a neutral sodium or potassium atom has one electron. But if eight electrons in an outer shell is the ideal, then sodium can get there most easily by giving its one electron away. Then the full shell of eight underneath is left. If sodium, which has 11 electrons in its neutral state, gives its one outer electron away, it then has the same number of electrons left as neon, a noble gas.
For this reason, column one elements tend to "give up" one of their electrons in reactions so that they can prune down to a full shell of eight electrons. With one electron gone, they lose that negative 1 charge and end up with a "plus 1" charge (because they now have one more proton than they have electrons).
The alkali metals of column one thus are quite eager to get rid of an electron. That makes them very reactive. Sodium in water--boom. Potassium in water--BOOM.
6. If column 1 elements are eager to get rid of an electron, column 7 elements are eager to steal an electron from somewhere else (thus being prone to take on a -1 charge from the extra electron). These are the "halogens." They are fluorine (9), chlorine (17), bromine (25), iodine (53), and astatine (85).
Elements from column 1 and column 7 like each other. Sodium wants to give up an electron and chlorine wants to get one. So salt (sodium chloride) is formed.
7. In the middle of the periodic table is a section of ten columns that are called the transition metals. They don't fit exactly into the scheme above because they have an extra "orbital" called the "d" orbital. A lot of the metals that humans have been using for a long time fall into this part of the periodic table: gold, silver, iron, copper, nickel, zinc, titanium, platinum, mercury.
8. Just to the right of the transition middles, between these metals and non-metals on the far right of the table, are "semiconductors" like silicon and germanium. Metals conduct electricity and heat readily. Non-metals tend to resist the flow of electricity. Semiconductors are used extensively in electronics to make the components of computers and such.
10. Tucked into the bottom of the transition metals section of the periodic table are two series that relate to yet another orbital that comes into play with very heavy metals. The Lanthanide series are fourteen elements from 57-71. Then the actinide series are elements tucked in from 89-103. Most of these, and those with even higher numbers, are man-made elements, only created in the laboratory.
The two most important actinides, though, which occur in nature, are uranium and plutonium. These radioactive elements decay into other elements. The "half-life" of a radioactive element is the amount of time it takes for half of a sample to decay into other elements, like lead.
Next Week MS5: Molecules and Ions
 Mendeleev spoke of "weights," but we no longer talk of atomic weights because, in physics, weight is a force that relates to gravity. Mass applies no matter whether you are on earth or in outer space.
 The reason it does not add neatly by eight going on to krypton, xenon, and radon is because once the atom gets this big, two other "orbitals" become involved. The "d" orbital interjects an extra 10 protons/electrons to get from argon to krypton and from krypton to xenon. Then to get to radon, there is an "f" orbital which adds 14 more.
The number 8 comes from the addition of 2 electrons in an "s" orbital (the basic spherical one) to 6 electrons in a "p orbital (perpendicular). See the previous post.