Best way to chill a cup of coffee with cold water and 5 minutes [duplicate]

Question

Initial data

• 1 x 3/4 full cup of hot coffee / tea / your favorite morning beverage
• cold water
• 5 minutes

Considering that it's starting to get hot outside, and we all want to drink reasonably cold coffee / tea (well, in any case, not scalding hot), I have the following options:

1. Wait for 5 minutes

So the hot coffee releases as much energy as possible in the 5 minutes, then add cold water to fill the coffee cup

2. Add cold water

So the temperature is lowered as fast as possible in the start, then let the air do the rest.

Question

Which option cools the coffee more ? Are there other, better options ?

From what I can figure out, waiting 5 minutes first is the most obvious choice, but I noticed that physics isn't always as intuitive as we may think.

PS: Would a spoon to stir the beverage have any impact on the difference between the two ?

Answer 1

Lets take each case and make some calculation. So, the first case, waiting for 5 minutes than adding some cold water. Assume the following values:

The initial "hot" temperature of the coffee $T_H=80^{\circ}C$

The temperature of the surrounding medium $T_m=23^{\circ}C$

Using Newtons cooling law

$$\frac{dT}{dt}=-k(T-T_m)$$

and after a simple integration we get

$$T_c=T_m+(T_H-T_m)\mathrm{e}^{-kt}$$

Taking $k=0.05$ we find $T_c$ to be $T_c\simeq63^{\circ}C$. Now we have to add some cold water. Lets say we add $1/4^{th}$ of a cup at $T_{cold}=10^{\circ}C$.

$$-Q_{coffee}=Q_{water}$$ $$-cm_{c}\Delta T_{coffe}=cm_{w}\Delta T_{water}$$ $$-\frac{3}{4}(T_f-T_{c})=\frac{1}{4}(T_f-T_{cold})$$

And we find $T_f$ to be $T_f\simeq49^{\circ}C$

Now lets look at the second method, mixing them from the start. We just have to replace the numerical value for $T_c$ in the above formula with the initial temperature of the coffee $T_H$. Doing this, we find that the temperature after mixing is $T_{f}'\simeq 62^{\circ}C$.

Thus, as Neuneck already said, the first method is the best one. These are at best some approximate calculations, but even so, the difference is clearly visible.

Edit:

As a response to the comments, if you add hotter and hotter water to your coffee the final temperature after mixing will be higher (it will grow linearly with the temperature of the water you add.) Here is a plot for it.

Answer 2

Option 1 is the way to go. To a good approximation the cooling process has an exponentially decaying temperature, so the rate of cooling is proportional to the temperature itself. If you add the cool water first, you will reduce the cooling effect of waiting 5 minutes.

We did this experiment back in high school. The assumption that enters here is that the temperature drop caused by the addition of cold water is approximately independent of the beverage's temperature. This is justified as long as the temperature difference between cold water and hot beverage is large. I guess one could put more detail into the simulation by calculating the mixing temperature explicitly, though...

Answer 3

If you have an additional mug, then the most effective way to cool the coffee will be to run the empty mug under cold water to chill it, dry it, then pour your coffee into that mug. Then repeat with cooling the empty mug with water and pouring your coffee back into it.

Or you could just pour the coffee between the two cups continuously to aid cooling by exposing it to their air.

Stirring will help too, as it will keep the coffee at a constant temperature rather than the top layer being a colder temperature than the rest (which will reduce the rate of cooling.)

For actually mixing cold water with your drink, this should be done last for the reasons mentioned in Nijankowski V's answer; you will lose more heat in the same time by being at a higher temperature.