Can Coffee Cool Below Room Temperature?

[homology]

Hey Folks,


I was showing my students Newton's Law of cooling (precalc class) and one of my students pointed out that this doesn't really model the cooling of an object, he said that he had recently learned in chemistry that the temperature of the object that is cooling actually dips down below the "room temperature" and then asymptotically approaches it from below.

I admitted ignorance, its not my area but am curious to learn about this, if this is the chase. So my question: Given a cup of coffee at 110 degrees (for example) in a room at 65 degrees will the coffee ever have a temperature lower than 65 degrees.

Thanks,

Kevin

 

[Tide]

That's a little tricky. The problem is that the temperature of the liquid isn't really uniform until thermal equilibrium is achieved. What the student left out of his/her explanation is that evaporative cooling is taking place at the exposed surface of the coffee thereby cooling SOME of the liquid to below room temperature (i.e. the hottest molecules leave by evaporation leaving cooler molecules behind). But the remaining molecules are interacting with the hotter ones below it.

The details of how a liquid undergoing evaporation and with nonuniform temperature throughout its volume approaches equilibrium is complicated. Newton's Law of cooling doesn't account for those details but often provides a decent approximation.

 

[charng]

Dear Kevin,

Thank you for sharing your experience with your students and for bringing up an interesting question. I can provide some insight into Newton's Law of Cooling model and the cooling of an object in general.

Firstly, it is important to understand that Newton's Law of Cooling is a simplified model that assumes certain conditions, such as constant room temperature and uniform heat transfer. In reality, the cooling of an object is a complex process that is affected by various factors such as air movement, humidity, and the specific heat capacity of the object.

In the case of your student's observation about the temperature of the cooling object dipping below the room temperature, this is possible and can occur due to evaporative cooling. When an object is hotter than its surrounding environment, it will transfer heat to the environment through convection and radiation. However, if the surrounding air is dry, the object can also lose heat through evaporation of water molecules on its surface, causing its temperature to decrease below the room temperature.

In the specific scenario you mentioned, the coffee at 110 degrees may experience a slight decrease in temperature below 65 degrees due to evaporation, but it will eventually reach an equilibrium with the room temperature of 65 degrees. This is because the rate of heat loss through evaporation will decrease as the temperature of the coffee gets closer to the room temperature, and eventually it will stop altogether.

Overall, while Newton's Law of Cooling provides a useful model for understanding the cooling of objects, it is important to keep in mind that it is a simplified representation of a complex process. I hope this helps clarify any confusion and encourages further exploration into the fascinating world of thermodynamics.