Cost of Boiling Water

So in today's exciting math adventure I'm trying to figure out the cost to boil a pot of water. I was reading something saving money, and it said to save money instead of wasting soap and sponge life on scrubbing a pot with gunk in it, boil a vinegar solution in it for 5 minutes, then clean normally. Note I think it will be cheaper, but that's not the point I want to do the math.
http://www.frugalurbanite.com/?p=14
So it breaks down like this:
savings = (wear + extra soap) - (water + vinegar + energy), where wear is the extra wear on the sponge, soap is extra soap you use, keep in mind you still need to use soap and sponge to clean it, this should just use less. Then you subtract the cost of the water, the energy to heat to water, and the vinegar. I suppose the value of your time should be figured in, but I'm going to ignore it, both because it's hard to peg a value, and because I'm not sure which would take more time, and how much more.

Finding the average cost of tap water is kind of hard, but I know water is very cheap, and everything I've found list prices well under a penny/gallon. Vinegar is also cheap ($1.60/gallon), and it says a weak solution. At first I assumed a weak solution would just be a couple tablespoons, but I am seeing 1 part vinegar to 3 parts water called a weak solution online, so we'll go with that (25% vinegar). I'm also using 2 quarts as our pot size, meaning we'll need 1pint or 1/8th gallon of vinegar = $0.20.

Now come the cost of energy, this is the fun part. This is going to be complicated. First I'll need to find the amount of energy to boil water for 5 minute, then I need to find out how efficient common stoves are (gas and electric), lastly I'll need to find to find the cost of the gas or electric. I decided to start with efficiency, as I knew this would be the hardest to find. It turns out I was right, because I've been reading for about an hour and I've yet to come up with a reliable figure. One link I found list 40% and that seems reasonable. Next the cost of energy, since I'm growing tired of this, I'm going to use $0.10/kWh for electricity, and $10/1000 cubic feet of gas (1ft^3 = 1030 BTUs). Now to convert all this nonsense into joules:
1 kWh = 3,600,000 joules
1 MJ = $0.027

1000 ft^3 = 1,030,000 BTUs = 1,086 MJ
1 MJ = $0.009

Now how much energy does it take to boil water:
50F (10C) tap water, 2L (2.11 quarts)
90C * 4.184 * 2000 = 753,120 J, 90 degrees to 100C, 4.184 is the specific heat of liquid water, 2000 grams.
However, now it again get complicated. Because it takes a lot of energy to actually vaporize the water (2260J/g, compare with 418.4J/g to heat it 100C). The problem is that not all the water is vaporized when you boil it for 5 minutes, since you still have liquid water left. In fact not much water is lost at all, but since the energy to vaporize is so high it can't be ignored. I'm going to assume you lose 100g (5% of the total) of water. I'm guessing that since I've never cooked anything in my life a lot of this is harder, but that is what we are using.
2260 * 100 = 226,000 J
226,000 + 753,120 = 979,120 J ~ 1 MJ
Using 40% efficiency that means we'll use 2.5 MJ to put 1 MJ into the water. This will cost $0.0675 electric, or $0.0225 gas.
Putting it all together, $0.20 (vinegar and water) + $0.07 (energy) = $0.27 ($0.23 gas).

Just for fun, let's boil all the water, and assume 10% efficiency.
2260 * 2000 = 4,520,000 J
4,520,000 J + 226,000 J = 4,746,000 J ~ 5 MJ
5 MJ * 10 (10% efficiency) = 50 MJ
$1.35 electric, $0.45 gas (plus $0.20 vinegar/water)

What did we learn from this? Don't try to figure things out with so many unknowns.

http://www.newton.dep.anl.gov/askasci/eng99/eng99196.htm

http://www.homebrewtalk.com/showthread.php?t=56371

http://www.portlandonline.com/water/index.cfm?&a=126197&c=ecica