temperature diffusion

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Does temperature diffuse between two objects at a constant rate or does the difference in temperature determine the rate?

For example, you place two cans of coke in a 35 degree refrigerator. One can is 95 degrees and the other is 40 degrees. Will their temperatures lower at the same rate or will the hotter can diffuse more rapidly?

In: Physics
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This actually depends how technical you want to be.

The rate of energy transfer between the objects is the same (thats conservation of energy) and is done by the equation Q (energy transfer/change) = m (mass) * cp (heat capacity) * dT (difference in temperature between the two objects.

So if the two have the same mass and are made of the same thing (m and cp) then they change temperature at the same rate. And if not they don’t.

If you want to be super super picky technically cp is dependent on temperature, so they go at slightly different rates, though the scale difference has to be pretty huge for the number the change enough to change the answer. The an equation is used to figure out a new CP at a given temperature for a specific compound.

This also changes if one undergoes a phase change (like ice to water) in which case it will absorb the energy but rather than use that energy to change T it will use it to break the structure down, so it will stay at one temperature while the other drops.

Edit: and if you want to be super super super picky then the above is assuming its not losing heat to the surrounding air, in reality the hot one would lose heat more quickly to the air (with the same equation) so would drop slightly faster because it is losing heat to both places.

Edit2: I’ve forgotten the name of the equation and I thought it was the Arrhenius equation but thats slightly different, sorry its been a while since phys chem

EDIT3: it has been correctly pointed out that the Arrhenius equation is actually the right one for calculating the thermal conductivity of a substance, thats me being sleepy and I apologize. I’m leaving the above as is though because I think it hits on the key points around what determines how much energy is lost from a thing when next to another thing of a different temperature and probably addresses OPs concerns and follow up questions.

There are two parts in this: Energy and Temperature.

The rate of cooling depends on the rate of energy flow, the rate of energy flow depends on the temperature difference and the temperature difference depends on the temperature.

The rate at which the energy flows is roughly linearly propotional to the temperature difference between the object and its surrounding.

So doubling the temperature difference doubles the flow of energy. (this doesn’t hold if the temperature is very high. Blackbody radiation grows at T^4 but that is negligble at normal temperatures).

Note that this all needs to be done with kelvins. 35 F ≈ 275 K ≈ 2 C, 40 F ≈ 278 K ≈ 5 C, 96 F ≈ 308 K ≈35 C.

Energy would flow out of 90 F coke about 11 times faster than out of 40 F coke. ( 308 – 275 ) / ( 278 – 275 ) = 11

But the tricky thing is that is only the single moment. The hotter coke would cool down faster than the cooler coke. And so the temperature difference between them would go smaller. And the difference in rate of cooling would also go smaller. I am not going to go into details on math of this. Wikipedia has more details https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling

But I happen to have some code for plotting the results for this kind of thing. The results aren’t exact but they should be decently close to reality.

https://i.imgur.com/INcCuNN.png

Yellow line is the temperature of the 35 C (96 F) coke and purple line is the temperature of 5 C (40 F) coke, enviroment is at 2 C (35 F). (numbers are rounded. The time axis may be scaled wrong)

You can see that the hot coke cools down greatly while the cooler coke barely cools down at all at the beginning. But as the hot coke cools down its rate of cooling down also slows down.