I just finished law school, am a science major, and will be doing IP law, so maybe I can offer you some insight.
The patent bar requires you to have a bachelors degree in science or engineering. There is a list of majors that qualify you to take the patent bar. If you don't have one of those majors you can also qualify by taking a certain number of science classes in designated areas. Chemistry isn't a listed major, but I would guess you could qualify based on the courses you have taken. Check out this website for info about the patent bar. http://www.uspto.gov/ip/boards/oed/grb.pdf
. Passing the Fundamentals of Engineering exam is one way of qualifying to take the patent bar, but since you have a science major there is no reason to take it because you would likely qualify anyway. It doesn't make sense for a Chemistry major to try to take an engineering exam anyway.
You do not need to go to law school to take the patent bar. If you pass it without going to law school, you will not be a patent attorney, you will be a patent agent. This means you can prosecute (write) patents, but cannot litigate patents (go to court to invalidate a patent or uphold a patent). Therefore, if you only want to write patents and have no desire to go to court, then taking the patent bar might be a better idea for you than law school. If you go to law school but don't take the patent bar, you can litigate patents but you cannot write patents. The only way to be qualified to write patents is passing the patent bar, whether you go to law school or not.