You are using the same cup, making the same amount of coffee out of the same coffee machine. Why does your coffee cool down faster than it does in the summer? Of course you’d say that is because winter is much colder, but shouldn’t coffee cool down at some constant rate? like maybe 4°F every minute or something?

I don’t know if Isaac Newton used to drink coffee or not, but his law of cooling does tell us the reason to the problem above.

The equation for Newton’s law of cooling is given above, in which:

- t means time
- T represents T(t), the temperature function in terms of time
- k is a positive proportionality constant
- Te indicates the temperature of the environment

It is a first-order linear differential equation. All it says is that the rate of change in temperature is proportional to the difference between the temperature of the environment and the temperature of the object. [1][2] So your coffee cools faster in the winter is indeed because there is a greater temperature difference, but the rate at which your coffee cools is not a constant.

To actually get the equation for the temperature of your coffee, we need to solve the equation above. Without going into too much maths, I’ll just give the solution(s) for the temperature function T(t).

Now, this equation is useless for engineers because they almost always want a predicted temperature, instead of a temperature function.[3] Therefore, a definite integral is always used instead of the indefinite form, to project what the temperature will be at a certain point in time. Here is the definite integral form:

As time goes to positive infinity, C*e^(-kt), the part on the right of the addition sign, goes to zero. That is why it is called a transient. It does not last for long and as time goes by, we can simply ignore it. That said, the steady state temperature should just be the equation above without the transient. [4]

Of course, you probably would never use this complicated formula to predict how long you could leave your hot coffee on the table. Coffee is there for you to drink, anyways. Evaluating such integral(s) above would just be another exercise to recall high school calculus.

Works Cited:

[1] https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling

[2][3][4] Prof. Arthur Mattuck, MIT OpenCourseWare. Course number 18.03 Differential Equations (Spring 2010).

Image Credits:

Coffee: http://wtop.com/wp-content/uploads/2015/03/getty_030315_coffee.jpg

Newton, Science Rocks: https://s-media-cache-ak0.pinimg.com/564x/78/c6/bd/78c6bd96f97ac662f3827e7b4c6aebe5.jpg

*Equations are all created via http://www.hostmath.com/

kevinwang11This is a very interesting blog post, Max! Your title is intriguing and attracts readers like me to open your blog post. You also did a amazing job in explaining how the greater temperature difference causes faster temperature change.