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Measuring HEAT CAPACITY
While electrical batteries store electricity, all matter is capable of storing heat. All you have to do is heat it to get the atoms and molecules vibrating (note 1), and it will stay hot for awhile. During that time you can use that hot thing to heat other things (note 2) - such as that old camping trick of dropping hot rocks from the campfire into a pot of water and soon the water boils! However, different kinds of matter can store different amounts of heat. And it is just that which you will determine on this page. You probably have heard the child's riddle: "Which is heavier - an ounce of silver or an ounce of feathers?"* Well, here we can ask the question: "Which has the higher heat capacity - a pound of silver or a pound of feathers?"
If you are a glutton for equations, well-, here is one devised by - of all persons - Albert Einstein, who posited that each sustance has a characteristic frequency of vibration "v", and that the absorption of energy by oscillators does not obey continuous classical mechanics, but rather follows discontinuous quantum theory a la Planck. Here is an Einstein equation that is a bit more difficult to remember than E = mc2:
where:
T =absolute temperature; v = characteristic frequency; N = Avogadro's number
k = R/N = 1.380x10-15 erg/degree; and h = Planck's constant = 6.625 x 10-27 erg seconds**
Happily - no - Joyously! we can escape Einstein because there is another way to define heat capacity! That is to measure how much heat (calories) it takes to heat up a gram of the specified material. By definition, a calorie is almost precisely the amount of heat needed to increase the temperature of a gram of water by one degree centigrade. ('Almost precisely' because it differs slightly depending on the temperature of the water.)
How one can do this is usually done in an easily made calorimeter - a well insulated container of low heat capacity (note 3) holding some water. You place a known volume of water into the container, and then measure its temperature. Into that water you introduce something of known mass that has a far different temperature. Then you look to see how much the whole system changes in its temperature by measuring the water after thermal equilibrium is attained.
Let's take an obvious example. Were you to add 100 gm of water at 10°C to 100 gm of water at 30°C, you would expect to end up with 200 gm of water at 20°C. To discuss this ad nauseum would be to explain that every gram of 30°C water would give off the same amount of its stored heat per degree drop that it would take to warm a gram of 10°C water one degree - an example of the "conservation of energy" equation.
It would be less obvious if we were to take 100 gm of water at 30°C and add to it a 100 gm piece of iron at 10°C. Here, however, things are different. Perhaps the amount of stored heat in the water (1 cal/gm/degree) is sufficient to raise the iron's temperature more than a degree/gram. Let's pretend that water's can store twice as much heat as can iron. Then water's temperature would drop on half as much as the iron's temperature would rise. The final temperature would be 23.33°C (the water's temperature dropped 6.67°C, while the iron's went up 13.33°C. Qualitatively: water's heat capacity is greater than that of iron (in this pretend case); and quantitatively, water's heat capacity would be 2.00 times that of iron - in this pretend case.
PROTOCOL
Preparations:
The First "Run"
RELATIVE heat capacity = |(Δtempwater x masswater)/(Δtempx x massx)|
The SPECIFIC heat capacity, Cs, is the heat capacity for a specified standard mass of material versus the heat capacity of that same mass of water. This will work for grams, tons, kilograms, or even those peculiar British weights called "stones" or jeweler's pennyweights - just so long as the units are the same for the thing added and for the water. It should thus be happily noted that the relative heat capacity is unit-less.
Subsequent "Runs"
* An ounce of silver, which being a precious metal, is commonly weighed in Troy ounces, which are 1/12 of a pound, while feathers are commonly weighed in ounces Avoirdupois (1/16 of a pound).
Note 1: A corrollary of the 3rd Law of Thermodynamics (Return to reading place.)
Note 2: Another Law of Thermodynamics! Which one? (Return to reading place.)
Note 3: Textbook authors generally persist in reproducing diagrams of antique devices designed before the invention of styrofoam. So we shall simply be using a styrofoam cup!(Return to reading place.)
** Interestingly, this heat capacity equation is so filled with constants that if they are all combined into a term we'll call "K", then Cp=Kv2. Since Newton's equation F=½mv2, v at any given T is thus defined by the constant mass of the atom in question. Thus, at the very fundamental level, Einstein's heat capacity equation is little more than a corrollary of Newton's equation. Indeed, Einstein's most famous equation, E=mc2, is merely Isaac Newton's equation taken to the extreme velocity of the speed of light! Like Einstein once said: "I am merely standing on the shoulders of a giant [Newton], so that I can reach higher still." In any event, this all means that heat capacity ought to be inversely related to the atomic mass - which you might check out for yourself by looking at the Table of Specific Heat Capacities of the Elements, in the links below.
Two interesting points about this heat capacity equation: (1) Einstein considered it his life's best piece of work, and (2) the reason Einstein could come up with his famous E=mc2 formula and no one before him, was that previously it was not known that the speed of light was constant.
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