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A look at the technical specifications of the Core i3 processors automatically puts them into a negative light when it comes to power consumption. The desktop Core i3 parts at listed as having a 73 Watt TDP, while most Core 2 Duo desktop parts have a 65 Watt TDP. In laptops the Core i3 has a 35 watt TDP, while Core 2 Duo
mobile processors usually have a 25 Watt TDP.
These differences pan out about how you'd expect them to when it comes to absolute power consumption. The Core i3 processors do consume just slightly more power than Core 2 Duo processors at load and at idle. We're talking a difference of around 10 Watts on desktops and a few on laptops - nothing huge, but a difference none the less.
However, when it comes to power efficiency the answer becomes less clear. In order for a processor to be power efficient, it needs to not only have low power consumption but also the ability to complete tasks quickly. This lowers the overall "task energy" because a faster processor will be done with a task before a slower processor, and once done it will slip back into an idle state.
When viewed from this perspective, the Core i3 is much more efficient than the Core 2 Duo on both the desktop and the laptop. This means that the Core i3 will probably not use any more power than a Core 2 Duo - and may actually use less - unless your usage patterns place a constant load on your processor.