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Explain why three ${sp^2}$ hybrid orbitals result when one s atomic orbital is blended with two p atomic orbitals.

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When some number of atomic orbitals are mixed together to produce hybrid atomic orbitals, the same number of hybrid atomic orbitals must result. So, when N atomic orbitals are mixed together, N hybrid atomic orbitals are produced. In the case you describe, mixing one s atomic orbital with two p atomic orbitals means that three hybrid atomic orbitals must result.

These three hybrid orbitals will each contain 33.333% s character and 66.666% p character because that is the ratio of the one s orbital to two p orbitals that we are mixing together. We can describe such an orbital as $\ce{s^{33.333}p^{66.666}}$, but if we reduce that notation to the lowest common denominator we get $\ce{s^1p^2}$, or the commonly used description $\ce{sp^2}$. This simply means that this hybrid orbital is composed of one part s character to two parts p character.

To summarize if we mix 3 atomic orbitals, one s and two p orbitals, then 3 $\ce{sp^2}$ atomic hybrid orbitals will result.

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