which molecule or ion below has a lewis structure that does not obey the octet rule?

The option c should be CS₂ instead.

The structures of the options are shown below.

For IF₅ (a), the central I atom has 5 I-F single bonds and 1 lone pair of electrons. There are 6 × 2 = 12 electrons in the outermost shell of the central I atom, and thus it does not obey octet rule.

For PH₃ (b), the central P atom has 3 P-H single bonds and 1 lone pair of electrons. There are 4 × 2 = 8 electrons in the outermost shell of the central P atom, and thus it obeys octet rule.

For CS₂ (c), the central C atom has 2 C=S double bonds. There are 4 × 2 = 8 electrons in the outermost shell of the central C atom, and thus it obeys octet rule.

For CO₃²⁻ (d), the central C atom has 1 C=O double bond and 2 C-O single bonds. There are 4 × 2 = 8 electrons in the outermost shell of the central C atom, and thus it obeys octet rule.

For CH₄ (e), the central C atom has 4 C-H single bonds. There are 4 × 2 = 8 electrons in the outermost shell of the central C atom, and thus it obeys octet rule.

The answer: a. IF₅

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A simple short-cut to the answer:

In IF₅, the central atom has at least 10 electrons in the outermost shell (at least 5 I-F bonds).

The answer: a. IF₅

a. IF5 <--- for sure

c. CS2-2 <--- I have not heard of this ion. If it exists, it would not obey the octet rule.