Find a formula for the relation of temperature and time?

Question

You have 2 gallons of water under a 120 volt (15 watt) bulb. What is the formula used to represent the relation between water temperature and time? Plug into the equation what you know.

It is asking it like, "how long does it take the water to heat from 20 degrees to 30 degrees, or how many degrees will the temperature change in 2 hours." That type of relation. I have no idea how to do this. Please Help! I wasnt in class yesterday.

Answer 1

If 100% of the energy produced by the light bulb actually went into the water, you could determine it like this:

It takes one Btu of heat to warm one pound of water by 1° Fahrenheit. 2 gallons of water is approximately 16 pounds of water. Therefore it takes 16 BTU of heat to warm 2 gallons of water 1° Fahrenheit

A typical incandescent bulb produces 90% of its energy as heat, so 15 watts * .90 = 13.5 watts of power as heat.

The number of BTU produced per hour = 3.413 x watts, so
13.5 * 3.413 = 46.0755 BTU per hour

It follows that the 15-watt incandescent light bulb will heat two gallons of water 56.0755/16 = 2.8797 desgrees Fahrenheit per hour.

Given i= initial temperature of water, and t=time in hours elapsed, the final temperature f can be calculated as

f = i+2.8797t degrees Fahrenheit

Of course, all of the heat from that light bulb will not make it into the water - an appreciable percentage of it will radiate out into the atmosphere. So consider the above as a theoretical maximum. Your actual temperature increase will actually be less.

Answer 2

If you only missed one day. No problem.
If you've been goofing off all quarter - you're screwed.

Assuming you've been truthful
And assuming this is a 'perfectly insulated' container
- here's the deal.

This is a question about SPECIFIC HEAT, i.e. how much energy does it take to change the temperature of something.

Water is used as a reference. It takes 1 calorie, to raise 1 gram (1ml) of water 1 degree Centigrade.

A watt is a joule per second. There are ~4.187 joules per calorie. (calories, joules,BTUs, kWh etc. are all units of energy.)

So if you know you have 15 joules per second going into 2 gallons of water, convert the joules to calories, covert the gallons to ml or grams and the rest is basic algebra, because you know how many calories per second are entering the water, and you know how many calories it takes to change the temperature of each ml(or gram) 1 degree C.

Beware your units. The easy to remember SPECIFIC HEAT of water ( 1 calorie per gram degree Centigrade) is METRIC, so don't forget to convert from/to Fahrenheit.

Good Luck