We were learning the Mishna Brura and the halacha regarding whether one can add cold water to a large bathtub of very hot water on Shabbos.

Specifically a ruling that one can add from the very hot water to a cold tub of water so long as there is more cold [water] than hot [water].

I find it hard to believe that we are talking about comparing fixed measures of water, and seek the help of someone familiar with heat energy calculations.

I remember from physics that there is a measure of heat energy, and that the Fahrenheit scale is less useful than others for a simple calculation.

Consider a tub of 212 degree water and a perfect insulated 1 cup ladel to take from it and add to a 1 cup measure of 80 degree water within a [more than 2 cup size] bowl. I say 212 degrees at the highest possible temperature for water in its liquid state. And I say 80 degrees as perhaps the highest ambient room temperature on a hot summer day.

What would be the resultant temperature of the 2-cup mixture of water, ignoring the expected cooling of pouring and touching the bowl and air?

Its not a simple average, but perhaps its close to the mean of 80 and 212, or 146 degrees, which is higher than the halachic boiling point?

And what would be the resultant temperature of 1 cup of 130 degree hot water coming from a hot water faucet mixed with 1 cup of 40 degree cold water as perhaps might come out during a middle of the winter day when pipes are prone to freezing?

Perhaps its close to the mean temperature of 130 and 40 which is 85 degrees, well under the halachic boiling point?

What is the right way of doing this sort of heat energy calculation for cups of hot and cold water (by definition at the same altitude since they are mixing)?

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Originally posted by rob:
We were learning the Mishna Brura and the halacha regarding whether one can add cold water to a large bathtub of very hot water on Shabbos.

Specifically a ruling that one can add from the very hot water to a cold tub of water so long as there is more cold [water] than hot [water].

The main point is whether or not you are going to be heating up the cold water to "yad soledes bo" [the point at which one would remove one's hand from it because of the temperature] which is around 40 to 45 degrees Celsius or 104 to 113 degrees Farenheit. Pouring a small amount of hot water into a large amount of cold water will probably not increase the temperature of the cold water to "yad soledes bo" and so would be permitted.

quote:

I find it hard to believe that we are talking about comparing fixed measures of water, and seek the help of someone familiar with heat energy calculations.

I think perhaps that you are making this more complicated than it actually is, based on what I have written above. Or perhaps I misunderstood your question, given that I was useless at physics in school!

rob,

Actually, the key is that the energy stays constant (the law of conservation fo energy). Since in both cases, you are mixing water, and liquid water's specific heat (amount of heat per unit mass per degree temperature) is constant, it will indeed be the simple average of the two, as long as the two cups carry the same mass of water:

m1 * T1 + m2 * T2 = (m1 + m2) * T, where m1 = mass of water in one cup; T1 = its temperature; m2 = mass of water in the other cup; T2 = its temperature; T = temperature of the mixture,

or m * (T1 + T2) = 2 * m * T, where m1 = m2 = m


And then T = (T1 + T2) / 2.

If we are talking different masses, then the first equation describes the situation.

If you need to know the mass of hot water, m1, to get temperature T of the mix,

you can derive it as:

m1 = m2 * (T - T2) / (T1 - T).


Now. If we are talking about water at 212 degrees Fahrenheit, technically, it is a water that is starting to boil (in fact, even at 210 degrees, bubbles of steam are already forming on the surfaces of the kettle), and then its heat capacity will be different than that of the liquid water. That becomes a more complicated calculation. But if you have just got the water to a boiling point and removed it from the source of heat (keeping the lid on, to prevent evaporation), then withoug cooling much it will all become liquid: boiling, condensation, and freezing are all isothermal (constant-temperature) processes, and you will have condensation occurring under the lid, - and then the original calculation will be true.