# Hypothetical Science Experiment

I have two 100 ml jars with lids. I’m in a room where the air temperature is 70 degrees F. Into one jar, I pour 100 ml of water that is 60 degrees F and secure the lid. Into the second jar, I pour 100 ml of water that is 80 degrees F and secure the lid.

Does the water in each jar reach 70 degrees at the same time?

It’s very unlikely both will equalize to 70 degrees at the same moment. They might seem to if you don’t have a small enough increment of resolution on your thermal sensing or timing device. Taking the problems individually and assuming perfection of other variables gives this.

1. The 60 degree water is poured first and capped off. This gives it a head start over the 80 degree water in equalizing to air temperature.

2. You didn’t say what temperature the jars were before you poured in the water.

3. The thickness of the jar wall material will vary throughout the jar and so the transfer of heat will vary over the surfaces.

4. The total mass of the jars will vary so it will affect the transfer of heat.

5. Water has a slight decrease in density as it warms in it’s liquid state, so the same volume of water in both jars means the 60 degree water is slightly denser than the 80 degree water. This means there are slightly more water molecules in the colder water for a given volume.

Given that this is a thought experiment, it seems to me that #5 is the only one that actually matters.

For the theoretical outcome, I believe that Fourier’s Law says that the heat flux will be proportional to the temperature difference.
Hence, all things being equal, they should come in pretty close.

And the density difference (however small) probably can’t be whisked away by a wave of the hand since it would be present in even a perfect scenario.

Depends on how many factors your thought experiment takes into account. In general, heat should flow from 70degrees to 60degrees just as fast as from 80degrees to 70degrees.

With water, heat capacity should be about the same at 60 degrees and 80 degrees. As mentioned there is a tiny density difference between water at 60 and 80, so if you fill the two containers by volume, there’s a tiny mass difference, making the cold water heat up a tiny bit faster.

More important, though, is that the two jars are transferring heat with the surrounding air, and so air currents (convection) are the big factor in heat transfer. Heated air will move a little faster, so the hot container should cool down a little faster than the cold one warm up.

Strictly speaking, it’s impossible to pinpoint the exact time either jar reaches room temperature. The temperature approaches the room temperature asymptotically. The closer it gets to room temperature, the slower the rate of change. Mathematically it never actually reaches room temperature.

Practically though, the thermometer in each jar will read “70” at pretty much the same time. There may be a slight difference depending on how the air currents work. Though I’m not sure I agree with Quercus’s assertion that heated air would always move faster - I think it would depend on the geometry (e.g. shape of the jar, whether there is something above and below to block airflow, etc).

Thanks for the replies, and the discussion of the variables. I had thought of quite a few variables, but didn’t want to appear any more ignorant about physics than I already do.

I wondered, too, if it mattered with high temps (in the high 100s F) versus relatively low temps (just above freezing), and with higher differences between the three temps (20 or 30 between instead of just 10).

For some reason, it wouldn’t have surprised me if the hotter water would have lost heat faster than the cooler water would have gained it, but I didn’t know why.

It sounds like you’re talking about the Mpemba effect, Runs With Scissors. In some circumstances, hot water CAN cool off faster than cool water.

The first order theoretical approach to this would observe that the jars would have the same heat transfer coefficient, maybe 8 or so watts per square meter kelvin, and so they would relax toward 70 with the same relaxation time but would never reach 70.

Minor7flat5 says something like this, but it is wrong to use Fourier’s law to justify it. Fourier was talking about conduction. In the case of these jars, convection would probably be more important.

There are several details that would make a difference between the jars, though. If they are sitting on a table, the convection caused by the hot one would send a plume upward, drawing more air along the table toward the jar. The cold jar would not cause a similar downward plume from above. This is because fluid flow with an inertial (nonviscous) component is not symmetric with respect to time - it’s why your vacuum cleaner exhaust can move things 10 or 20 feet away, while the suction only reaches out a few inches or maybe a foot or so. Also, you could have radiative heat transfer. Generally the infrared flux would be higher between 70 and 80 than between 70 and 60, because radiation is a strongly increasing function of temperature. Of course, thermal conductivity and heat capacity are both roughly constant but in fact have a little sensitivity to temperature. The viscosity of air goes up as it gets warmer, while the viscosity of liquid water goes down as it gets warmer. You’d like both of the viscosities to be low if you want the jar to equilibrate quickly. We get to wonder which one would be more important (I dibs the air viscosity).