How much more productive for writing essays and reading papers is a faster computer going to be? Unless the OP is doing a course requiring more sophisticated computer work I can't imagine much of a productivity change.
I asked the same question of my late 2006 macbook about a year back and ended up replacing the battery and fitting an SSD. Replacing the battery made it possible to work on the sofa without shutting down then moving and plugging in which did improve productivity somewhat by letting me vary where I was working making it easier to work for longer, the SSD just made it open word in a sensible amount of time.
I'd vote for replace or just bin the battery and keep going till after you finish university.
My old MacBook gives me the spinning wheel a good bit when switching apps when I've got lots of tabs open, though that's been improved with an SSD. Office and related apps have also gotten more resource hungry in newer versions, not to mention a Mac that old also probably won't run Mavericks or some newer apps.
I also didn't say it would justify all the cost, I think most folks probably need to upgrade far less than they do. That being said, this is a 7 year old machine that's certainly been eclipsed in performance and it's not like the OP won't have a use for whatever Mac they have after they're done with school.
A new machine is going to be a lot snappier and worth the investment if it's in the budget. If it's not then a new SSD and the RAM bump will keep it usable for a bit longer.