Ok, so I got bored at work and did a
little research. Turns out that "volumes of CO2" is a pretty nebulous measure and is only valid at 32 degrees F. According to this
website, a "volume of CO2" can be approximated by 2 grams of CO2 per liter of water.
Some quick conversion says 5 gals is 18.9 liters, so you'd need 94.5 grams (3.33 oz) of CO2 to carbonate a standard homebrew batch of beer at 2.5 volumes.
Now, if you just measure 3.33 oz of dry ice into your keg and seal it, you technically have all of the CO2 you need, but you will probably not successfully dissolve 100% of the CO2 into your beer. Therefore you'd need an additional amount to create the pressure required in the "headspace," and I'm not quite sure how much extra you'd need.
If I were G. Cretin, I might start with 5 gallons of COLD water in a Corny keg, attach a pressure gauge to the gas ball lock, add 1 oz of dry ice, seal it, and monitor the results. If the pressure gets much above 30 psi, use the relief valve to avert disaster. Otherwise, when the pressure stops rising, shake the keg A LOT, and watch the pressure drop as CO2 goes into solution. When the pressure stops dropping, pour yourself a glass and taste the result. Repeat with increasing 1 oz increments of dry ice until you are either satisfied that this works, or decide to abandon it because shaking 5 gallons of water is a lot of work.
You could also scale the experiment down to a 1 or 2 liter plastic bottle...
just be careful!
If I decide to give this a shot, I'll certainly post the results! If you try this,
please use caution, but definitely share your results.
On a different note, the
website above is an excellent resource, and I carbonate homemade soda quite often with a similar system to the one shown there.