Nitrogen, phosphorous, antimony, and bismuth are all in the same group (column) of the periodic table.

The atomic radius increases from the top of a group to the bottom, due to increased principle shell number (n). As one travels down a group, another s shell is added, meaning that electrons are added in another orbit farther from the nucleus. This serves to increase the atomic radius of the atom.

Lithium, boron, oxygen, and neon are all in the same row (period) of the periodic table.

The atomic radius decreases from left to right along a period due to increased effective nuclear force. From left to right the atomic number increases, indicating that more protons are added. The addition of protons increases the positive charge in the nucleus, pulling in the outer electrons by increasing the effective nuclear force, decreasing the radius.

In math terms, we can equate effective nuclear force using the force equation between two charged particles.

We can see that the farther apart the electrons and protons are, the less the force is between them.

The second ionization energy of a neutral atom is always greater than the first.

Recall that ionization energy refers to the energy required to remove an electron from an atom. The first ionization energy will result in a cation. The reduced number of electrons in a cation allows them to be pulled closer to the nucleus by the positively charged protons, effectively decreasing the atomic radius and increasing the attractive force between the electrons and the nucleus. The energy required to remove a second electron must overcome this increased affinity, and will thus be greater than the first ionization energy.

The rest of the answers can be examined with a few simple trends kept in mind. Atomic radius increases to the left of a period and down a group. If two ions have the same electron configuration (like the chlorine anion and potassium cation), then the anion will have a larger radius because it has fewer protons to draw electrons inward. Electronegativity and electron affinity increase to the right across a period and up a group.

Atomic radius increases down each group of the periodic table and toward the left of each period. Since the elements listed are all in the same group, iodine would have the greatest atomic radius because it farther down the period compared to the others. 

Energy level increases moving down a group of the periodic table. As energy level increases, the outer valence shell becomes more distant from the nucleus, causing atomic radius to increase.

Energy level remains constant across a period, but electrons are added within the same orbitals. When new electrons are added within the same orbital, additional protons are also added to the nucleus. This increases the effective nuclear charge, pulling the electrons closer to the nucleus. The trend for atomic radius is to decrease as we move right along a row.

This means that the general trend for atomic radius is to increase as one moves to the left and downward on the periodic table.

Chlorine has a great thermodynamic desire to capture an electron, thus taking on the electronic structure of a stable noble gas. This causes chlorine to release energy when it captures an electron as it becomes more stable.

Sodium, on the other hand, would prefer to lose an electron and gain the configuration of a noble gas. Adding an electron would however award some stability to sodium, due to the complete s orbital that this would ensue.

Second electron affinity is usually encountered for such elements as oxygen and sulfur, which form anions with the addition of two electrons. The first electron affinity gives you O^- or S^-, and so it takes significant energy to add another electron to an already negative ion.

Alkali metals are much less dense than other metals due to their large radii, which results from having a single loosely bound valence electron. Some of the alkali metals have such low densities that they can float on water.

Alkaline earth metals are found in the second group of the periodic table and include beryllium, magnesium, calcium, strontium, barium, and radium. These compounds are not as reactive as the alkali metals (found in group 1), but still participate in many reactions due to their electron configuration. Alkaline earth metals carry two valence electrons, located in the s orbital. Loss of these two electron leaves the alkaline earth metals with a full octet, giving them a stable oxidation state of +2. In contrast, the alkali metals have a stable oxidation state of +1. Both compounds have very low first ionization energies, but the second ionization energies of the alkaline earth metals are much lower than those of corresponding alkali metals. Removing a second electron from an alkali metal removes it from a stable octet, while removing an additional electron from an alkaline earth metal results in a stable octet.

While all alkali metal salts are soluble, the alkaline earth metals result in several exceptions to the solubility rules. For example, is not soluble in aqueous solutions.

Halogens are in the group next to the noble gasses. They have seven valence electrons, and therefore have a high electronegativity. The addition of only a single electron (production of an anion) generates a full valence octet.

Their diameters vary within the group. The diameter can be very small, like fluorine, or large, like iodine. They do not conduct electricity well, as they are non-metals.

The halogens are the second to last column in the periodic table, meaning that they have an affinity for a single additional electron. Halogens would be most likely to react with alkali metals, which contain only one loosely bound electron in the valence shell. Alkali metals have very low ionization energy, readily losing an electron, while halogens have very high electronegativity, readily gaining an electron. This interaction allows the alkali metals to form ionic bonds with the halogens.