Satiate means to satisfy (a desire or an appetite) to the full; to fully satisfy a need or desire, often related to hunger or cravings.

Gratify means to give pleasure or satisfaction to (someone). In the context of fully satisfying a desire or appetite, it is the closest synonym.

Why the Other Options Are Incorrect:

Mollify: To appease the anger or anxiety of (someone). (Related to calming, not fully satisfying a desire.)

Perpetuate: To make something continue indefinitely. (Unrelated to satisfaction.)

Vanquish: To defeat thoroughly. (Unrelated to satisfaction.)