MacBook – Changing the Logic Board on the Macbook, does it changes the serial number

I think you're out of luck. From iFixit, here's the logic board from the original MacBook Air:

And here's the logic board from a 2012 MacBook Air:

Unlike desktops, which typically use standardized parts, laptops use fairly specialized components, and on top of that, Apple tends to iterate its designs fairly aggressively even within models with similar outwards appearance to achieve better cost or performance (e.g. smaller logic boards means more space for the battery). Because of this, most Apple components beyond hard drives and RAM are only interchangeable with those of the exact same generation.

I have received my new logic board today and installed it successfully. I also upgraded the Macbook with a 240GB SSD and 4GBs of RAM.

I only encountered 2 problems which can be worked around easily. The first one was the different SuperDrive connector. I simply didn't connect the SuperDrive, but you can easily order the right SuperDrive, since they're not that expensive. The second and most important problem I encountered was the different layout of the heat sink. This needs replacement as well in order to get it working properly. I'm still waiting for this part to be delivered. For now, my Macbook shuts down after a few minutes, because it gets too hot.

After I've received the right heat sink and installed Yosemite, I'll give you guys another update.