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Thermo Problem

A second pass at Mr. K's famous cocoa problem. Take 250 mL of cocoa at 98 degrees C and add 25 grams of ice at 0 C. What is the final temperature of the cocoa?

Background knowledge:

    Energy is not created nor destroyed in Thermo (or anywhere else).

    Since Energy is not created nor destroyed then the heat lost by cooling the cocoa will do two things: melt the ice
    cube, and then warm the resulting water up to Tfinal

Important equations for calorimetry:

 

Adding ice to 98 C liquid causes ice to melt. This cools cocoa (which we will assume behaves like pure water)

Melting: ice travels from point 4 to point 5.

Enthalpy = m Hf   (mass of ice x heat of fusion of ice)

                 25 g x 333.5 J/g = + 8337.5 J   this must come from cocoa so it travels from point 1 to point 2

Enthalpy = m Cp DeltaT  (mass of cocoa x heat capacity of cocoa ( use water) x Delta T)

    -8337.5 = 250 g x 4.18 J/gºC x DeltaT

    Delta t = -7.97 ºC so at this step our cocoa is 91ºC.  Now the melted ice is at 5 and the cocoa is at 2. They must meet at
    point 3. But where is this. Again the heat lost by the cocoa will be gained by the melted ice.

    m Cp delta T (of melted ice) = m Cp delta T (of cooling cocoa)

    25 g x 4.18 J/gºC  x (Tf - 0) = 250 g x 4.18 J/gºC  x ( 91-Tf)

    104.5 Tf = -1045 Tf - 95095

   1149.5 Tf= 95095

    Tf = 83 ºC  and hopefully ready to drink.

Try it at home and tell us how close Thermodynamics actually matches the real world.