Possible to upgrade the CPU in the intel based Mini (late 2011)

I cannot even turn on my mac anymore after it endured countless hours of rendering and compiling projects this little machine has served me well. Now, it wont even turn on anymore, as soon as the screen turns white, it shuts off, and repeats the whole process again and again. I've brought it into a shop, they said the OS was fine (i had thought it was a glitch). They explained the CPU had burnt out after extensive use. I asked to have it replaced but they said that's not what they do. So i was wondering, instead of putting money into a new one (which i dont have) is there any way to put a new CPU into the thing? I've scoured the web with no real answer. If someone has one, or at least has an explanation as to why you cannot. Please reply. Thank you.