You have two ice square cubes. Each is 100 grams. They are standard water ice composition.
You have a lab freezer that can take things down to -100 degrees Celsius.
You put two ice cubes taken straight out of the freezer on two separate plates on a table in a room temp environment of 80F or 26.7C . One cube has been cooled to 0 deg celsius the other to -100 deg celsius.
How much longer does the -100 C cube take to melt?
I ask because I’m wonder how much of a real difference the -100 C temp v the 0 deg C temp is really going to make.
It takes 54.5 kJ of heat to melt the -100C cube and 33.4kJ to melt the 0C cube. The colder cube will transfer heat faster than the warmer cube so the upper limit is it will take 63% longer.
We aren’t really given enough details to fully model the heat transfer process (in particular, the relative importance of convective and conductive losses; and whether radiation is an issue), and modeling all of these are potentially complicated as well.
But heat transfer by convection is relatively simple to model reasonably well. If you can assume that radiative and convective heat transfer are negligible (a reasonable assumption in some cases, though convection may be an issue with the setup described in the OP), then you can come up with a better estimate.
The simplest model of conductive heat transfer is that conduction through a region is proportional to the temperature gradient, with the proportionality constant (thermal conductivity) assumed constant. Because of this, the ice cube at -100°C will initially have a much higher rate of heat transfer than the one at 0°C, and so even though the colder ice cube requires 63% more heat transfer, the heat is initially conducted to it about five times faster than to the melting ice cube.
So for a somewhat better estimate you could guess that “on average” the cold ice cube has temperature -50°C, for a temperature difference (relative to the environment) of 76.7K, about three times the 26.7K difference for the warm ice cube. So you can estimate that the cold ice cube warms to the melting point at roughly three times the rate that the ice cubes melt; and so instead of an additional 63% the colder ice cube only takes an additional 63%/3=21% as long to melt.
This model for the warming ice cube isn’t too hard to solve “exactly,” though (“exact” within the assumptions made here). It’s the simplest kind of differential equation, a first-order linear DE; the “exact” answer turns out to be that the cold ice cube takes about 26% longer to melt.
I’ve been thinking about this more and the initial 63% figure may be overwhelmed by a couple of other factors:
The cube at 0C will rapidly form a film of water and so heat transfer comes from convection as well as evaporation which is going to be a more efficient process than convection and sublimation.
Unless the bottom of the cube and the top of the plate are mirror flat, there’s going to be imperfect contact. The 0C cube will rapidly form a liquid layer which will increase the efficiency of the conduction.
Absolutely! It’s true that boiling longer won’t make the water hotter - but once it has changed to steam the steam can continue to get hotter. Likewise with ice, the temperature won’t drop until the ice has formed, but once it’s ice it will drop to the temperature of its environment.
Right, going from a solid to a gas takes the same amount of energy no matter what route you go through to get there. Since you have to add energy to a liquid to evaporate it, it follows that solid to gas takes more energy than solid to liquid.
Boiling water won’t make the liquid water in the pot hotter because once water reaches its boiling point, it turns to gas (steam).
You can heat that gas to any temperature you want, however.
Likewise, 0C is the maximum temperature at which water will freeze under normal conditions. That doesn’t stop you from getting it much, much colder, however.