If there is a variation (ie. regular repetition) between atomic radii and atomic number of the elements, how would you describe it?

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We are asked about the relationship between atomic radius and atomic number.  First, let's define those terms.  The atomic radius of an atom is defined as the distance between the nucleus and the outermost electrons surrounding it.  It is usually expressed in picometers (10^-12 meters).  The atomic number is the number of protons in an atom's nucleus.

In general, atomic radius decreases as you move along a period (left to right horizontally) and increases as you move down a group (top to bottom vertically).  The atomic number increases in both of those directions (left to right and top to bottom).  The reason for this is that as you move left to right along a period, you are increasing the number of electrons in the valence (outermost) shell of the atom and also the number of protons in the nucleus.  The electrons therefore experience a larger effective nuclear charge from the nucleus.  Since this is a force of positive attraction (positive and negative), the electrons are held more closely to the nucleus, thus decreasing the atomic radius.  But as you move down a group to the next level, you add an entirely new shell of electrons to the atom, and the size of the atomic radius increases suddenly.

These trends are depicted graphically on the periodic table in the links below.

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