People who are interested in the question of "how or why" chemical reactions happen (I'm one) don't necessarily end up in any particular branch of chemistry. I'm officially a "physical organic chemist" but I did my graduate work in silicon and phosphorus chemistry, which tends to fall within the area of "inorganic" chemistry. And when I did my Ph.D., in the early 1990s, Washington University didn't make much practical distinction between "organic" and "inorganic" chemistry. You had organic chemistry graduate students in inorganic groups, and vice versa.

I suppose whether you choose Physical, Organic or Inorganic chemistry in graduate school will depend on what sort of "how and why" questions you're interested in. But don't think that this choice will limit you at all; a Ph.D. is supposed to teach you to think properly about any problem. Within limits, of course; but (as an example) I know of people who started off as pretty straightforwardly "organic chemists" who are now doing most of their research in theoretical chemistry.

I'd say that you want to find a graduate advisor who is interested in the same types of questions you are. But (and here's a word of wisdom from my own doctoral advisor, Peter Gaspar): "physical organic" chemists can do anything; what they learn is a widely useful way of thinking about problems.

Tangentially, "biochemistry" per se almost doesn't exist any more as a separate sub-discipline, except as an undergraduate course with a big thick textbook. People working on "biochemical" problems are most often trained as physical, organic or inorganic chemists. This was certainly the case where I went to graduate school. (The rest of the folks in that area are calling themselves "molecular biologists" now.)

I hope this is helpful!

Dan Berger