# How make the perfect Hot Chocolate? Mixing liquids and temperature

I sometimes like to make hot chocolate, though when I do usually use powdered hot chocolate that you mix with hot water. However I think it taste better with some actual milk added in from the fridge. But since the milk is cold (4 ºC) I can't add too much or the Hot Chocolate won't be hot anymore. The water kettle that I use to warm up the water goes up to 90 ºC, and I don't think I can/should warm milk with it.

So my question is how much milk can I add to the drink while having the temperature stay hot (≥60 ºC). Let's assume the I have a container that can hold 0.5 L and doesn't dissipate heat to the outside environment. Also I don't know if this is necessary information but this is done at sea-level, so 1 atm. Not doing this in space or something crazy like that.

I also want to say I don't care about how long it takes the mixture to become a single even temperature. Nor how the heat distribute itself while the liquid is being mixed. I only care about the final temperature after a equilibrium has been reached and how the amount of milk affects that number.

I would guess energy is preserved in this system so an energy equation could be used. Though I don't know how the temperature of a liquid relates to it's energy, neither of water, milk or the resulting mix between the two. Hopefully someone here with more knowledge can help me out.

• @Countto10 Well it's hard not to make the title too long and give people a good idea about that the question is about. I did add "Mixing liquids and temperature" to the title. Though this is of course not a debate on what exactly is perfect hot chocolate, but given a definition of it, how do you make it. What's the physics behind it. Anyway, I've trying searching around like "energy in a liquid" or things of that nature though I didn't find something helpful I could understand. If it is very elementary then hopefully it should be easy then for you help me out and answer my question. Commented Feb 2, 2017 at 17:15
• @Countto10 I can tell that you are a hot chocolate connoisseur and have taken great offense from my question. However your comments are not helpful and I would like to keep discussions focused on the physics. Commented Feb 2, 2017 at 17:55

For mixing any two liquids, in any quantity, and at any temperature (in Kelvin)

the equation you need is:

$$T_f=\frac {(M_1\cdot C_1 \cdot T_1) + (M_2\cdot C_2 \cdot T_2)}{(M_1\cdot C_1)+(M_2\cdot C_2)}$$

$T_f$ is final temperature

$M_1$ is water and chocolate mass

$C_1$ is the specific heat capacity of water

$T_1$ is initial temperature of water

$M_2$ is mass of milk

$C_2$ is the specific heat capacity of milk

$T_2$ is initial temperature of milk

You can take the mass of water and milk as equal to their volume.

The heat capacity of water is 4.19 kJ/kg.K

The heat capacity of milk is 3.93 kJ/kg.K

Enjoy!!

• Thank you so much for the answer! It seems to be exactly what I was looking for. Think you made a small mistake though. Milk's heat capacity should be 3.93 kJ/kg.K, but other than that it seems to give me sensible numbers.Also try using <br> tags. Just place them where you want to start a new line. It doesn't create a new paragraph so you won't have the gap between the lines. Commented Feb 2, 2017 at 20:22