How Many Ice Cubes Melt in 10 Gallons of Water at 75 Degrees F?

[evotunedscc]

Homework Statement

A completely insulated tub contains 10 gallons of water at a temperature of 75 degrees F. Ice cubes are taken from a freezer at 15 degrees F and placed into the tub. How many 1-in^3 ice cubes can be completely melted by the water?

Homework Equations

The Attempt at a Solution

 

[arunma]

I've got a question for you. Say you've put in the maximum number of ice cubes that can be melted in the tub, and the next one you put in will not be melted. What is the temperature of the water at this point?

 

[evotunedscc]

15 degrees F

 

[arunma]

15 degrees F

But this wouldn't make physical sense. How can any amount of ice at 15 degrees be mixed with any amount of 75 degree water, and produce ice at 15 degrees? It turns out that this is not possible. Every time you drop an ice cube in the water, the ice melts, and the water reduces in temperature slightly. But the ice also warms. So there's no way that the water would ever have a temperature of 15 degrees, is there? The water always has to be at a sufficiently high temperature so that it can give away some heat to the ice cubes that you're dropping into it, and still be hot enough to give away more heat without freezing. Remember the difference between temperature and heat. The relationship is given by the following equation:

Q = mc\Delta T

Remember also that in order for an ice cube to melt, it needs to change phase, and that takes the following energy:

Q = mL

Here, c is specific heat, and L is latent heat of fusion. Since the relationships depend on mass, you'll need to figure out how much mass of water is in ten gallons, and how much mass of ice is in 1 cubic inch (for which you'll of course need the densities of water and ice).

Anyway, I don't want to give away too much more, since it's important that you understand the principle of heat transfer. But this should be enough for you to solve the problem.