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Specific Heat Capacity of Water
Heating up water takes a lot of energy. In fact it takes 4187 joules for every degree Celsius to heat up each litre.
The Specific Heat Capacity of Water is 4187 joules per kilogramme degree C.
Using this fact, it's possible to work out how much energy it would take to warm up a particular volume of water by however many degrees you want. Similarly, it's possible to check the power of a microwave oven by measuring how much heat it can put into a volume of water. This is a good way of testing a microwave oven.
In the page about the difference between power and energy, it's mentioned that a 650 watt microwave oven puts 650 joules of energy into water per second. If there's one litre of water heated for one minute, then it's warmed by 9 degrees C. So, why is 39,000 joules in 1 litre of water represented by 9 degrees C ? It's because in water, heat is represented by a temperature rise which is specific to the heat-absorption characteristics of water. The factor is 4187 joules of energy for each degree C for each litre.
That might sound a funny way of saying it, but it makes sense if you think about it. Supposing you've got one litre of water and you warm it up by one degree C, then that takes 4187 joules of energy. If instead, it was two litres, it would takes twice as much energy. Or, if it was heated up by 2 degrees C, it would be twice as much energy. You multiply the amount by the temperature rise and multiply by 4187 (the specific heat capacity of water) and you get how much energy would be required.
For example: Supposing you've got a swimming pool that's 5 metres by 10 metres and it's got an average depth of 2 metres, and it's at 10 degrees C as measured with a thermometer. You've decided that 10 degrees C is too cold to swim in, and you're going to heat the water up to 25 degrees C. How much would that cost?
First, you work out how much water there is. In this case there's 5 x 10 x 2 = 100 cubic metres (100 tonnes) of water. 100 tonnes of water is 100,000 litres. You want to heat up 100,000 litres by 15 degrees C (25 minus 10). Therefore the total heat required is: 100,000 x 15 x 4187 joules. When you work this out it comes to 6280.5 million joules. It sounds a lot, but a kilowatt-hour is 3.6 million joules, so to get the answer in KWh it's divided by 3.6M and comes to 1744.6 KWh. This is where some serious shopping-around at uSwitch comes in, because even if you got the electricity for 10p/unit it would still cost £174.46 to heat up the water. Obviously there are much cheaper ways to heat up water than to use electricity! Solar power, having your own CHP (combined heat and power) plant, or even having a bonfire with some old radiators in it!
Here's another example: Working out how much power a microwave oven produces:
You get a large cooking-grade plastic jug and fill it up with cold water. You have to know how much water there is, and what the temperature is to start with. Let's suppose there's 3 litres of water and it's at 15 degrees C.
Heat it in the microwave oven on full power for a known time, let's say four minutes (240 seconds).
Now give it a stir and measure the final temperature.
The result could be 29 degrees.
Now, using the "specific heat of water", you can work out how much energy has gone into the water. It's the volume of water x the temperature rise x 4187. In this case the temperature rise is 14 degrees, since it was 15 to start with and ended up being 14 degrees hotter at 29 degrees. The amount of energy is 3 x 14 x 4187 = 175854 joules.
The fact that it took 240 seconds to put 175854 joules into the water means that it put in 175854/240 = 732.725 joules per second, so the power is 732.725 watts. Let's call it 733 watts. That's not bad, especially if the microwave oven was rated at 720 watts. If it's within 5% either way, it shows the microwave oven is working properly. Also see How to Measure the Power of a Microwave Oven