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I am having trouble understanding this AP Chemistry Concept

Diagonal relationships in the periodic table  exist as well as the vertical relationships. For example, Be and Al are similar in some of their properties as are B and Si. Rationalize why these diagonal relationships hold for properties such as size, ionization energy, and electron affinity.

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J.R. S. | Ph.D. in Biochemistry--University Professor--Chemistry TutorPh.D. in Biochemistry--University Profes...
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Let's take one example at a time.  Looking at B and Si.  They are not only in different groups, but also in different periods. This will be true for the comparison of Be and Al as well.
Looking at atomic radii (size):  we know that size increases going left within a period and down within a group. So, B is just left of Si, but is above it.  So, you would expect the size to be similar. B has 5 electrons in level 2, and Si has 14 electrons in level 3.  Level 3 would be larger than level 2, but the extra protons pull the extra electrons making the sizes similar. The same arguments can be applied to ionization energy and electron affinity.  The simple answer would be that two elements separated by 1 period and 1 group aren't that different in properties.