On purpose - purposely - purposeful

Someone's purpose is, according to OED (at moaning 1. a.: "That which a person sets out to do or attain; an object in view; a determined intention or aim." In other words, one's purpose is one's motive, or intention. When one does something with the intention of achieving the effect that it does in fact produce, one has acted on purpose. A child will be more severely admonished if it has hurt a sibling on purpose, rather than accidentally.

• The single adverb that carries the meaning of this adverbial phrase is purposely. OED defines this as "On purpose, by design; intentionally, deliberately" (AWE's emboldening).

This should not be confused with the adverb purposefully, nor the adjective purposeful.

• Purposefully is an adverb of manner - that is, it describes how an actor acts, "In a purposeful manner'.
• Purposeful is defined in two ways by OED:
  • • 1. Having a purpose or meaning; indicating or evincing purpose or plan; designed, intentional.

• • • 2. Having a definite purpose in mind; resolute.

So purposeful shows what someone looks like, the impression s/he gives of being determined, and set upon a particular goal. Purposely has less to do with impressions and appearances, but about doing something deliberately.