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Our observations have already shown us that the temperature change is double for half as much water. We can repeat these observations for many different masses of water, and we also find that the temperature change is inversely proportional to the mass of the water. This means that the heat capacity C itself is proportional to the mass of the substance heated. (Look back at Equation 1 to convince yourself that this is true. For a fixed amount of heat, what happens to the temperature change and the heat capacity if we double the mass of water heated?) So now we rewrite Equation 1 with this new information:
Here, m is the mass of the material being heated, and the proportional constant is now called the “heat capacity per gram” or more commonly the “specific heat.” Experiments show that, for any particular material, C _{s} is a relatively constant property of the material. (C _{s} actually varies slowly with the temperature, so it is about constant unless we make very large temperature changes.)
This equation so far is not very helpful, though, because we do not know values for the heat q or for the specific heat C _{s} . If we knew one, we would know the other from Equation 2, so somehow we have to devise an experiment to measure one or the other.
Here’s one way to do the experiment. Since heat is a form of energy and energy is the capacity to do work, we just need to measure how much work can be done for a specific amount of heat, e.g. for burning a specific amount of methane. This is tricky, but we’ve already seen that we can use the heat generated by a reaction to push a piston back in a cylinder. If we burn 1.0 g of methane, we can measure how much work is done on the piston by measuring how much force is generated and for what distance. From these measurements and the rules of physics, we find that burning 1.0 g of methane can produce a maximum amount of work equal to 55.65 kJ.
(A second way to do the experiment is to use work to increase the temperature of water and to measure how much work is required to increase the temperature of water by 1 °C. We’ll leave it as an exercise to devise a way to elevate temperature by doing work.)
What do the data tell us? If 55.65 kJ of work can be done by burning 1.0 g of methane, then burning 1.0 g of methane must produce 55.65 kJ of heat. This is q in Equation 2. But we have already measured that, for 1.0 kg of water, the temperature change is 13.3°C. This is ΔT in Equation 2, and m is 1000 g. From these data, we can directly calculate that, for water, C _{s} = 4.184 J/g·°C. This is called the specific heat of water, or somewhat more loosely, the heat capacity of water. Pay attention to the units of this quantity, as they are unusual.
In similar ways, it is possible to find the specific heat or heat capacity of any material of interest. A set of specific heats for different substances is shown in [link] . This is very valuable for predicting temperature changes in different materials. For our purposes, it has an even greater value. We can use this to determine the energy change in a chemical reaction.
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