# Fermilab

## Inquiring Minds

You Wrote:

Hi,

My name is Ian. I am a 12 year old student. I am in the 7th grade.

My question is: If a one pound block of ice (-2 degrees C) is placed in an insulated closed room that was 2 feet by 2 feet by 4 feet, set at 5 degrees C,

1.) How long would it take to melt the ice ?
2.) How many Btu's would it take to melt the ice ?
3.) Would the room temperature drop or stay at 5 deg. C ?
4.) If electricity was used to cool the room to 5 deg. C and cost 11 cents per kWh, how much would it cost to melt the ice ?

This may be part of a school project. Thanks.

Ian

Dear Ian,

To melt the ice you obviously need some heat (energy). Where does it come from? You say the box is perfectly insulated, that is, no heat will go in or out. Then the only heat available to melt the ice is the heat stored in the air inside the box. The air has a temperature of 5 degrees Celsius. As the air starts to warm up the ice, the air temperature drops. (Remember: perfectly insulated box. Nothing will reheat the air.) Once the air temperature throughout the box is at 0 degrees Celsius, all your heat for melting the ice is used up. (Ice melts at 0 degrees Celsius.) If the ice hasn't turned into water by then, you won't be able to do it at all. (The temperature of the air can, of course, drop below zero degrees throughout the box. For example, both ice and air could end up at -1 degrees Celsius. No ice would have melted.)

To melt the ice, you must keep in mind that it takes a certain amount of energy to warm up the ice from -2 degrees to 0 degrees, and then it takes some extra energy to MELT the ice of 0 degrees to liquid water of 0 degrees. (This is called melting energy, and it is the energy needed to break up the molecular bindings inside ice.) You can calculate both quantities with a simple formula that you can find in a physics textbook in your school library. (Look for the keywords "melting" or "heat" in the index.)

Using the "warming up" formula (not the melting formula), you can calculate how much heat (energy) you gain if air cools from 5 degrees to 0 degrees. The volume of the box minus the volume of the ice block gives you the volume and hence mass of the air.

All those energy values are usually calculated in Joules (J) or kilocalories (kcal, like the labels on food) and you can convert them into Btu. (Check a physics book.)

With these numbers you then know whether it is POSSIBLE to melt the ice just using the air inside the box. (My guess would be: no, it is not possible. You need a lot of warm air to melt ice. That's why ice stays around for a long time after it gets warm again. There won't be enough air inside the box to get the job done.)

Now if you use electricity to keep the air at 5 degrees C all the time, then the air can continuously transfer heat to the ice, and the ice will eventually melt. If no energy is lost, the electrical energy you need to put into the air of the box is the same amount as the sum of energy needed for warming up and melting the ice.

From that you can calculate the money you need to spend on electricity to keep the air at 5 degrees while the ice warms up and melts. (The price of electrical energy is always quoted in units of cents per kWh=cents per 1,000 Wh = cents per 1,000 Watthours, which equals cents per 3,600,000 Joules. [One Watt is one Joule/second and an hour has 3,600 seconds.]) Knowing the energy you need in units of Joules, you now can calculate the money you need.

Well, I hope I could help. Of course, I couldn't do your complete homework for you. Did you expect me to?

Best wishes, good luck,

Kurt

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