I didn't notice you double posted, even though that's kinda bad, I figured I'd make sure you saw my response.
You need to find out what Socket the system board in that laptop has, then find a dual core that will fit it.
I've heard that the Acer Aspire 5532 has an S1 socket.
( You can find out for sure by running CPU Z )
So just go there and look at the processors that use S1, and find a dual core that suits you.
One thing you HAVE TO WATCH OUT FOR, is that you need to find out what your current CPU model is ( CPU Z can tell you ) And how much voltage it uses. (Listed on the wikipedia link there)
You should only replace it with a CPU that uses the same or less voltage to avoid heat issues.
Also make sure the TDPs are close.
As far as the BEST dual core CPU you can install.
Turion 64 X2 TL-68 2400 MHz 2 x 512 KB 800 MHz 12x 1.075/1.10/1.125 V 35 W Socket S1 December 19, 2007 TMDTL68HAX5DM
I think the Turion 64 X2 TL-68 would probably be the one, as it's clocked at 2.4Ghz and is built on 65nm tech.
Athlon Neo TF-20 1600 MHz 512 KB 800 MHz 8x 1.0 V 15 W Socket S1 Q1 2009 AMGTF20HAX4DN
I found your processor.
The voltage matches up sort of with the Turions, BUT the TDP on your CPU is a LOT lower.
If you install a dual core CPU you may run into heat issues.
So unless you also have a way to deal with the heat it might be a bad idea.