Looking at this d/s indicates 600 mA-hr for 25 mA load, so a regular Arduino without any loading will run
for maybe 24 hours. Poo on that, says I. Rechargeable 9V are only rated at 200 mA-hr, so 3X worse.
http://www.jameco.com/Jameco/Products/ProdDS/198731.pdf
I have used the 9V in dataloggers with 32-Khz xtal. At that rate, it will run for about 30,000 hours or
about 3 years. That makes more sense.