You are planning a birthday party for your niece and need to make at least 4 gal

Question

You are planning a birthday party for your niece and need to make at least 4 gallons of Kool-Aid, which you would like to cool down to 32 oF (0 oC) before the party begins. Unfortunately, your refrigerator is already so full of treats that you know there will be no room for the Kool-Aid. So, with a sudden flash of insight, you decide to start with 4 gallons of the coldest tap water you can get, which you determine is 50 oF (10 oC), and then cool it down with a 1-quart chunk of ice you already have in your freezer. The owner's manual for your refrigerator states that when the freezer setting is on high, the temperature is -20 oC. Will your plan work? You assume that the density of the Kool-Aid is about the same as the density of water. You look in your physics book and find that the density of water is 1.0 g/cm3, the density of ice is 0.9 g/cm3, the heat capacity of water is 4200 J / (kg oC), the heat capacity of ice is 2100 J / (kg oC), the heat of fusion of water is 3.4 x 10^5 J/kg, and its heat of vaporization is 2.3 x 10^6 J/kg.

Explanation / Answer

heat required by ice to melt down to 0 oc Qgain = m*Cice*dt1 + m*Lf + m*Cw*dt2

mass of ice = 4 gallon = 4*0.003785411784*1000 = 15.142 kg

Qgain = (15.142*2100*20) + (15.142*3.4*10^5) +(15.142*4200*10) = 6420208 J

mass of water = that can be colled to 0 degrees = m = 6420208/(4200*10) = 152.86

volume of water = Vwater = 5.78 gallons

1quart of ice can cool 5.78 gallons of water

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