I also have an i5-2400. And my gfx card also was too shabby to run Doom (Radeon HD 6870). I talked to people and they said the processor should be fine, just get a new gfx card. My new 970 arrives tomorrow, I will let you know how Doom works with it and the i5-2400 if you are interested.
(It also turned out that my PSU has too much power, even with the new card. It is a 650W unit -- I'm actually considering to downgrade it to a quiet 450W unit.)