# Which Cup Will Be Hottest After 15 Minutes?

• liz
In summary, the answer to the question is that cup 1, which has the milk added at the beginning, would be the hottest. This is because the mixture will have a higher average temperature for the duration of the 15 minute period, resulting in more heat loss and a cooler final temperature when the milk is eventually added. This is supported by the equations showing that in both cases, the temperature after 15 minutes will be higher for cup 1. However, it is possible for cup 2 to also be hotter depending on the thermal conductivity and other factors.
liz
this was in a quiz i did ages ago and i never found out the answer.

someone makes 2 cups of coffee. Cup 1 has coffee then water then milk added and is left to stand for 15mins. Cup 2 has coffee then water added then is left to stand for 15 mins, then milk is added. which one would be hottest?

i think it would be cup 2 but I am not sure and i can't really explain it properly either.

thank to anyone who can help, I've been trying to work it out for ages

i think it would be cup 2
Wrong guess.

The greater is the temperature then the greater is the rate of heat loss. This means that "cup 2" which is not initially cooled by the addition of milk will be at a higher average temperature for the duration of the 15 minute period. It will therefore have lost more heat during that time and after the milk is eventually added it will surely be cooler than cup 1.

let's see what is going on here
Let's $$T_1$$ to be the hot water temperature, $$T_2$$ to be
milk (room) temperature, $$\kappa$$ to be the thermal conductivity, (which we assume to be constant) $$\tau$$ =15 min, $$C_1$$ and $$C_2$$ to be the heat capacity of water and milk.

1) if we mix the hot water with milk at the beginning

then the mixture temperature is
$$T_m =[tex](C_1T_1 +C_2 T_2)/(C_1+C_2$$

The temperature after 15 minutes will be

$$T_{f1} =(T_m-T_2)exp^{-\kappa \tau/C_1+C_2}+T_2=(C_1/C_1+C_2)(T_1-T_2)exp^{-\kappa \tau/C_1+C_2)}+T_2$$

2)if we mix afterwards
the water temperature after 15 minutes will be
$$T_3=(T_1-T_2)exp^{-\kappa \tau/C_1}+T_2$$

The temperature after mixing will be

$$T_{f2}=(C_1T_3 +C_2 T_2)/(C_1+C_2)$$=
=$$(C_1/C_1+C_2)(T_1-T_2)exp^{-\kappa \tau/C_1}+T_2$$

Now because $$exp^{-\kappa \tau/C_1$$<$$exp^{-\kappa\tau/C_1+C_2$$, then $$T_{f1}>T_{f2}$$

the answer: the if we mix at the beginning, the cup will be slightly hotter, because it will take mor time for cup to cool.

We assumed that the thermal conductivity does not depend on the volueme of the liquid. Actually, for the metal cup the thermal loss will be higher for the cup with milk, and both cups could have the same temperature at the end. It is also possible for cup 2 to be hotter as well. As to validity of summation of the specific heat of water and milk, it is justified because milk is actually a mixture itself, and no chemical reaction occurs if we mix milk with water.

## 1. How long does it take for a cup of coffee to cool down?

The rate at which a cup of coffee cools down depends on several factors such as the initial temperature of the coffee, the ambient temperature, and the material of the cup. In general, a cup of coffee takes about 10-15 minutes to cool down to a comfortable drinking temperature.

## 2. Can I speed up the cooling process of my coffee?

Yes, there are a few methods you can use to cool down your coffee faster. One way is to pour the coffee into a wider, shallow container which increases the surface area and allows for quicker heat dissipation. Another method is to add a few ice cubes to your coffee, but be careful not to dilute it too much. You can also place the cup of coffee in the refrigerator for a few minutes to cool it down.

## 3. Does the type of cup affect how quickly my coffee cools down?

Yes, the type of cup can have a significant impact on the rate of cooling. For example, a ceramic mug will retain heat longer than a paper cup, so your coffee will stay warm for a longer period. On the other hand, a metal cup will conduct heat more quickly, causing your coffee to cool down faster.

## 4. Can I reheat my coffee after it has cooled down?

Yes, you can reheat your coffee after it has cooled down. However, reheating coffee can affect the taste and flavor, so it is best to reheat it in small increments and stir it well to distribute the heat evenly. Additionally, avoid reheating coffee multiple times as it can make it taste stale.

## 5. Why does my coffee taste different when it has cooled down?

Coffee has a complex chemical composition, and when it cools down, the chemical reactions that occur can change the taste and flavor of the coffee. As the coffee cools, the aromatic compounds become less volatile, resulting in a less intense flavor. This is why coffee may taste different when it has cooled down compared to when it was freshly brewed.

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