This should be an easy one.
Let's say you are swimming in a pool that has 64,000 gallons of water at a temperature of 80 degrees Fahrenheit. How many calories will you have to burn off to raise the temperature of the water by one degree Fahrenheit? And approximately how long will it take?
Additional assumptions:
Gallons are U.S. gallons, since that's how American pools are measured, and that's what I've been swimming in lately.
Calories are the dietary calories as listed on American nutritional labels.
The pool is a common American fiberglass lined, inground, cement swimming pool.
Assume for the sake of this puzzle that the pool is filled with pure (or distilled) water.
Assume that the only heat transferred to the water is from your body and comes from burning calories.
Assume the temperature of the water is not affected in any way by any other source, such as the Sun, atmosphere, ground, other swimmers, dogs, pool heaters, etc.
Bonus points:
If you add eight molecules of a homeopathic active ingredient (such as arsenic) to the above pool, what is the homeopathic dilution factor stated in terms of X or C?
More bonus points:
According to the U.S. Environmental Protection Agency, how many molecules of arsenic can you have in the above pool and still be in compliance with the current Safe Drinking Water Act? And what is that dilution factor stated in terms of X or C?
How many Calories?

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Re: How many Calories?
Well, a gallon is 3.785 liters and there's one kilogram to a liter of water. So your 64,000 gallons have a mass of 242240 kg.
A diet Calorie is also known as a kilocalorie, where 1 calorie is the heat required to raise one gram of water by 1C. So a diet Calorie will raise 1kg by 1C. There are actually different versions of the definition of the calorie which define the amount of heat at different temperatures, and these days the modern calorie is defined as a chosen number of Joules. Do you care about the thermal coefficient of water at exactly 80F vs, say, 58F (close to the socalled 15 degree calorie)?
You're only wanting to heat the pool by 1F instead of 1C. 1F is 5/9's of 1C, so the Calories required are around 135,000C.
This assumes, as you suggested, that the water transfers no heat to or from its surroundings.
I have no idea how long it would take, though  it depends on your rate of exercise. If you assumed you were expending 500 C/day exercising (a moderate amount going by the 1lb/week guideline for weight loss), this would take 270 days of swimming, but that's probably only an hour a day.
Now, as to the homeopathic dilution. H20 is pretty close to 18 grams/mole, so going from the mass above, we're looking at 1.35 E 7 moles which works out to 8.10 E 31 molecules. Ah, your choice of 8 arsenic atoms was providential: this would be a 30X (15C) dilution.
Unfortunately, as of this writing the EPA website appears to be offline, but according to the Idaho Department of Environmental Quality, the EPA's standard as of 2000 was 10 ppb, down from the standard of 50ppb established in 1975 (according to WebElements, the typical human body contains arsenic at about a 50ppb concentration). 10ppb is 10 parts per billion, or 10/1E9 which works out to a paltry 8X, assuming you started with pure arsenic. (In homeopathy, the concentration of the socalled "mother tincture" is likely unclear, at least to me.)
In any case, the number of acceptable arsenic atoms would be 8.10E30/1E8 or 8.10 E 22. This is 0.135 moles, and if you get it all out, it would weigh about 10 grams. According to a casual search that landed on a PDF at the CDC (following links from WebElements), a potential fatal dose of arsenic is on the order of 100mg.
(Edit to fix an arithmetic error. It's 4:00 in the morning here.)
(Edit2: remove unnecessary quoting.)
A diet Calorie is also known as a kilocalorie, where 1 calorie is the heat required to raise one gram of water by 1C. So a diet Calorie will raise 1kg by 1C. There are actually different versions of the definition of the calorie which define the amount of heat at different temperatures, and these days the modern calorie is defined as a chosen number of Joules. Do you care about the thermal coefficient of water at exactly 80F vs, say, 58F (close to the socalled 15 degree calorie)?
You're only wanting to heat the pool by 1F instead of 1C. 1F is 5/9's of 1C, so the Calories required are around 135,000C.
This assumes, as you suggested, that the water transfers no heat to or from its surroundings.
I have no idea how long it would take, though  it depends on your rate of exercise. If you assumed you were expending 500 C/day exercising (a moderate amount going by the 1lb/week guideline for weight loss), this would take 270 days of swimming, but that's probably only an hour a day.
Now, as to the homeopathic dilution. H20 is pretty close to 18 grams/mole, so going from the mass above, we're looking at 1.35 E 7 moles which works out to 8.10 E 31 molecules. Ah, your choice of 8 arsenic atoms was providential: this would be a 30X (15C) dilution.
Unfortunately, as of this writing the EPA website appears to be offline, but according to the Idaho Department of Environmental Quality, the EPA's standard as of 2000 was 10 ppb, down from the standard of 50ppb established in 1975 (according to WebElements, the typical human body contains arsenic at about a 50ppb concentration). 10ppb is 10 parts per billion, or 10/1E9 which works out to a paltry 8X, assuming you started with pure arsenic. (In homeopathy, the concentration of the socalled "mother tincture" is likely unclear, at least to me.)
In any case, the number of acceptable arsenic atoms would be 8.10E30/1E8 or 8.10 E 22. This is 0.135 moles, and if you get it all out, it would weigh about 10 grams. According to a casual search that landed on a PDF at the CDC (following links from WebElements), a potential fatal dose of arsenic is on the order of 100mg.
(Edit to fix an arithmetic error. It's 4:00 in the morning here.)
(Edit2: remove unnecessary quoting.)