Here is the best explanation I have found
http://boards.straightdope.com/sdmb/showthread.php?t=693803
"DLPoole DLPoole is offline
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Join Date: Aug 2013
Guys, Its not the weight, its the internal densities
Check out the Wikipedia entry on Alkaline Battery:
http://en.wikipedia.org/wiki/Alkaline_battery
And in particular, importantly, the cross section diagram:
http://en.wikipedia.org/wiki/File:Al...ry-english.svg
There's a cavity at the bottom with a pressure expansion seal in the middle. It's there because the battery manufacturers know the total volume of the anode and cathode above it are going to shrink during discharge and they can't allow that shrinkage to pull the cell apart. Also, because of the pressure seal, nothing enters the cell and nothing leaves, so the total mass and hence the weight don't necessarily change with discharge unless the seal bursts.
The half reactions show the electron flow electrochemically and through the discharge circuit. The overall reaction shows that after discharging, oxygen ions have moved from the Manganese Oxide cathode and into the Zinc Anode, which becomes Zinc Oxide.
No oxygen enters or leaves the cell, but the oxygen doesn't occupy the same volume when originally part of the Manganese Oxide and thereafter as part of the Zinc oxide. It depends on the "intercalation," or in ordinary language, how well the oxygen "fits" in between the other atoms in the cathode and then the anode.
In this case, the oxygen fits better and occupies less volume as part of the Zinc anode than it did as part of the Manganese Oxide cathode, so the total volume shrinks ever so slightly, the cavity at the bottom grows ever so slightly, and the bottom of the can becomes ever so slightly more elastic.
The difference is barely large enough to see in an iPhone video, but can be measured in its sound track. When new, at 1.60V, a cell bounces from a 5" fall about a half inch off a granite surface then falls on its side with the two peaks in the sound track about 150mS apart; when discharged to 1.40V, the same cell bounces about an inch and the time is closer to 200mS."