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Because the element is in Period 3 and has an expanded octet, you can deduce it will most probably be Silicon, Phosphorus, Sulfur or Chlorine. The other elements in Period 3 are more likely to form ionic bonds (Na and Mg), do not have an expanded octet (Al), or are not common in bonding reactions (Ar). By calculating the formal charge of the central atom of each of the above elements, you can find that Phosphorus is the only element with a formal charge of 0.
Alternatively, an easier method is to calculate the total number of electrons in the Lewis Structure. And then subtract the valence electrons of every known element, to find the mystery element's valence electrons (which should be 5). And Phosphorus is the only element in Period 3 with 5 valence electrons!
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