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.
Best Answer
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!!