Since heat flows spontateneously from object with high T to an object with low T, the temperature of a body changes when heat is added or removed. The heat Q is proportional to the size of the object [mass m ] and also depends on the type of material,

**Q = mc ΔT**

where c is the specific heat Calorimetry. Since heat is a form of energy [just like kinetic and potential energy], the unit of the specific heat is c = J/(kg °C). Heat must be added or removed during a phase change [gas-liquid (evporation or condensation), liquid-solid (melting or freezing), or gas-solid (sublimation)]. Because the temperature is constant during a phase change,

**Q = mL, **

where the latent heat L has the unit [LI = J/kg . The specific and latent heat for materials are measured experimentally. The specific heats are different for the different phases of the same substance and likewise, the latent heats are different for different phase transformations. Furthermore, the specific and latent heats are different for different substances.

**Example:** The best way to make hot chocolate is by putting steam into chilled milk. An Espresso machine produces vapor at 100°C and pressure P = 9.0 x 10^{5} Pa [or 9 atm]. The barista puts msteam = 50.4 g [ V = 9.6 x 10^{-3} m^{3} or 9.6 L] of steam into the chilled milk. A 12-ounce cup of milk contains 0.36 kg of milk at the temperature 6°C. Find the temperature of the Hot Chocolate! Ignore the specific heat of the cup. Useful data: Specific heat of water C_{water} = 4, 186 J/(kg •° C), Specific heat of milk C_{milk} = 3, 890 J/(kg •° C) , and latent heat of vaporization of water L_{water} = 22.6 x 10^{5} J/kg.

**Solution:**We have for the heat given off by the vapor [and hot water] and the absorbed heat:

That’s about 190^{0} F.

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