Room temperature related question

If two objects are +10C and -10C from room temperature, would they converge to the same temperature at the same time? Say, room temperature is 20C and object a) is at 30C and object b) is at 10C would they get at the same time to 20C?

• There are way too many variables that you have left out to even begin to answer the question. What are the masses of the two objects?. What are their shapes? (surface area to mass ratio can effect heat transfer). What are their specific heats. What is the size of the room (how much air surrounds the objects). You are wording this like the room is a thermal reservoir whose temperature will not change when it exchanges heat with the objects. Are the masses suspended in the air only or in thermal contact with other objects? These are just to name a few. – Bob D Sep 4 '18 at 15:35
• Beware of the Mpemba effect. :) – aventurin Sep 4 '18 at 18:56
• You are correct Bob, the problem is that since I'm noob at physics I didn't even know that I didn't know about those details. Thanks for the suggestions, I will work them out asap. – matt_zarro Sep 5 '18 at 17:55

The rate of heat transfer is approximately proportional to the difference in temperature between them. The heat transfer equation is an exponential one: $$T(t) = T_0 + (T_0 -T_{room})e^{-kt}$$ So, setting $T = T_{room}$, we see that they would reach room temperature at about the same time. However, this is a rough approximation that assumes that both objects and their environments are identical and that the temperature difference isn't too large.

• They would never reach room temperature, but the absolute difference from room temperature will be equal for all $t$. – Jasper Sep 4 '18 at 15:38
• @Jasper: You're right, theoretically, they never reach equilibrium. But practically, yes. – user7777777 Sep 4 '18 at 15:44
• Small remark: Also known as "Newton's law of cooling". – R.G.J Sep 4 '18 at 16:47

If two objects are +10C and -10C from room temperature, would they converge to the same temperature at the same time?

Not necessarily.

For instance, if the objects are placed on a carpeted floor (presumably, with a low thermal conductivity), we may expect that the dominant cooling mechanism will be air convection.

Air convection should be more robust for the warm object, since warm air will rise and promote circulation, while cold air will tend to stay down and cause stagnation.

So, under these conditions, everything else the same, the warm object is likely to get to the room temperature sooner than the cold object.

The answer is that it depends. As worded, there are too many unknowns to assert that one will reach equilibrium (or near equilibrium) temperature quicker than the other. For instance, the $C_V$ dependence on temperature, which usually is an increasing function of $T$, will have an impact on the solution. Usually, it takes less energy to heat up a metal from $10°C$ to $20°C$ than the energy "released to the environment" for letting cooling an object from $30°C$ to $20°C$. But the difference is small. This difference might not be small at all at a few Celsius degrees above the absolute zero though.

Then there is another effect that will have an impact: the hotter object will lose more heat to the environment than the cooler one, due to Stefan-Boltzmann law. The larger the surface, the larger this effect will have an impact, making the cooling of the hotter object faster than the heating of the cooler object. Here, the higher the temperature, the stronger this effect will be. At room temperature the effect might be small, but at higher temperatures (near $1000°C$), it might be quite significant.

Lastly, as V.F. claims, there could be also an impact from convection, which may favor the hotter object to cool down faster than the colder one to heat up. But it depends on whether the objects are in vacuum, or air, or anything else, and their position in the box/room too. This could involve fluid mechanics and make the problem quite messy.