I recall on more than one occasion when our kids were small, wewould come inside from playing in the snow and I would make hotchocolate by heating the milk in the microwave. Invariably, I wouldget the milk too hot for at least one of them. Rather than addingmore cold milk to the hot chocolate, I dropped a couple of smallice cubes into the mug. I figured they would never notice thedilution effect that the water had on the hot chocolate since theyalways enjoyed adding about thirty marshmallows to the mixtureafter I served it to them anyway.

Although I never actually took any temperatures of the hotchocolate, let’s assume that the initial temperature was 175⁰F andthe mug contained 350 g of hot chocolate.

a)   If the initial temperature of the hot chocolatewas is 175⁰F, what would the temperature of the drink be once theice cubes had completely melted and the mixture stirred? Expressyour answer in “degrees Celsius” to one decimal place. Since themarshmallows were added after I served the drinks, they would haveno impact on the temperature of the drink at the time I served it.Additional information is provided below.

b)   From a purely personal perspective (i.e., in yourown opinion), would you consider the chocolate milk still be toohot for you personally to drink or would it be a bit on the coolside? How can you support your answer?  

Additional information:

Weight of ice cubes   =   75 grams totalweight
Initial temperature of ice cubes   =   -18⁰C
Specific heat capacity of the hot chocolate  =   3.751 kJ/kg C⁰

Hints: While this problem can be a little bit “scary” at first,it should not really be that bad if you break the problem down intoa series of more manageable steps. Think about what is happeningbefore you charge ahead and start trying to actually solve thisproblem.  

You can simplify things greatly if you draw a diagram torepresent what is happening. Draw a vertical line to represent a“thermometer” and mark the important points on it (this diagramwill also assist you in determining the temperature changes etc.).These points will include the starting temperatures of thechocolate milk and the ice as well as the freezing / melting pointof the ice.  

Let the final temperature of the thawed ice and milk be X (thechocolate milk and the melted ice will both have the same finaltemperature since they are mixed together in the same mug). X willbe somewhere between the thawing / freezing point of the ice andthe initial temperature of the milk - the trick is to find out whatthe temperature actually is. Remember that the heat lost by themilk will be equal to the heat gained by the ice, if we assume thatno heat is lost to the surroundings (which we will do in thiscase).