This is probably a really easy question and I'm sure I already know the answer, but here goes:
I just finished building my first computer from scratch with the help of some more knowledgeable friends, and before I managed to boot it up for the first time, set up the BIOS, and install Windows, the heat sink fell off the processor. I reattached it but did not apply a new layer of thermal grease, and started the system. A dumb move, I'm sure, but I'm learning as I go.
The computer is now up and running, but the CPU is constantly idling somewhere between 40-50 degrees Celsius, spiking massively when I run programs, play video files, load DVDs, etc. When I run minor programs, it usually hovers around 50-60 degrees, but it sometimes rises above 80 with no sign of stopping. When this happens, I immediately exit all programs and the temperature drops fairly quickly.
Is this overheating issue most likely caused by uneven thermal, and will removing the existing grease and applying a new layer fix the issue? It seems to me that if it were, my CPU would be idling even hotter, but I'm not an expert, which brings me here.