There should be no problem placing this processor into an eMachines et1831-05. If everything in your question is provided correctly, the et1831 uses the required motherboard socket, the LGA775. You really should be able to slip it right in and go without any issues whatsoever.
Keep in mind you need to apply thermal paste thinly and evenly on your processor before putting the fan back on. Clean the heatsink + fan all around (Including the thermal paste from the previous processor) before applying. other than that, you should be great!