Okay, I’m not a computer expert. However I have some thoughts about this particular type of computer CPU. First, there should be no doubt that it is a 64 bit chip. Well, then if there are two programs in use, each using 32 bits, it follows that the program resources will be called upon more often and without reason for processing one program or another in the CPU. And so I believe, if the CPU is functioning there will be problems relatively. I recall that the Intel information about this processor as being energy-efficient is perhaps descriptive of this type of concern. See link to Intel Core 2.
In my opinion the basic platform for computing power is satisfactory with a 32 bit processor. I do not recommend a Duo Core 2.
45 PERCENT 
I am a computer professional – 12 years of experience. I STRONGLY do recommend a dual core processor of some sort for any user. The reason is this: a dual core processor can do two things at once. A computer must do many tasks, not just the program you have open at the time. It must also run software to 1) manage the network connection, 2) manage writing and reading to storage, 3) antivirus, 4) anti-spyware, and many system level items that most users are unaware of. A dual core processor allows the system to do that and minimize the impact on the user experience. You won’t have as many of those moments of waiting for a task to complete while you move on to something else. Your comment on two programs each using 32 bits isn’t exactly technically the way it works either.