thermodynamics - Best way to chill a cup of coffee with cold water and 5 minutes

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

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.