SPECIFIC HEAT CAPACITY
Heat is energy. You know that when a mass is attached to a spring and set into motion, kinetic energy and potential energy are traded back and forth, and their average values are equal. The total energy is half kinetic and half potential, on the average. So what's this got to do with heat? Molecules in a solid or liquid are masses in a springy environment, and their energy is half kinetic and half potential if it acts monatomic (elements, alloys). The kinetic energy is derived for gases, and it turns out to be (3/2)nRT for a monatomic gas, where n is the number of moles (n= m/M, mass divided by "molecular mass"), R is the universal gas law constant, and T is the absolute temperature. (Find this under "kinetic theory of gases" in your typical text.) Molecular kinetic energy is the same for a given temperature whether we are talking about solids, liquids or gases, so all you need to do is calculate the energy for a monatomic gas and double it if you are dealing with solids or liquids.
From the above discussion it is evident that when you add heat energy D Q to a solid or liquid, this will cause a temperature increase D T, where D Q = 3nRD T. If you add heat to a monatomic gas (under conditions of constant volume, otherwise some of the energy will go elsewhere due to the work of expansion), use 3/2 instead of 3 in the equation (and for a diatomic gas, use 5/2). As usual, certain conditions are necessary for these things to be true. See the last paragraph and the table at the bottom of this document.
The equation in the preceding paragraph has 3 constants in it: 3, R, and M (because n = m/M). So a more efficient way of writing it is D Q = CmD T, where C=3R/M and m is the mass. C is known as the specific heat capacity, and the conventional units for it are calories/(gramoC) or use kcal and kg, and the numerical value is the same. D T is the same for Kelvin and Celsius: they are the same scales with different zero points. To get Kelvins, add 273 to the Celsius temperature.
Typical text problem: put two things of different temperature together; one loses heat and the other gains the same amount (assuming no heat exchanged with the environment). Set these quantities equal to each other and calculate the unknown, whatever it is. If a change of phase is involved, when hell freezes over, for example, you need to know about heat of fusion or vaporization.
The above is oversimplified, but the expression for C is valid for most monatomic materials at everyday temperatures. It turns out that quantum mechanics must be used at low temperatures, and that is a long story. Below are some values of C found experimentally for some elements at 25o C and the values calculated by 3R/M. Note that diamond and silicon disagree the most. They have the same crystalline structure, and both are semiconductors, so I thought this was a clue until I added germanium to the list.
Note: Except for carbon and silicon, for the elements listed here we could do quite well with one single quantity, 6 cal/mole oC. This is called the molar specific heat, and if you are familiar with moles, and you don't require high precision, this is nice. The last column shows the experimental molar specific heat capacity. Most of them round off to 6 if you just require one digit precision. It is interesting that lithium is a little low, but chemically similar sodium, potassium and rubidium round off to 7.
Element
M, g/mole
Cexp, cal/g oC or kcal/kgoC
..3R/M, (same units)
.. Cmolar,exp
Lithium ..6.94 0.834 .0.860 .. ..5.79
Carbon (diamond) 12
0.124
.
.0.497
1.49
Sodium ..23.0 ...0.293 .0.260 .6.74
Magnesium..24.3 .0.243 .0.246 .5.90
Aluminum 27.0 0.215 0.221 .5.81
Silicon
.28.1
0.168
.0.212
.4.72
Sulfur (yellow)32.1 ..0.175 . .0.186 .5.62
Potassium ..39.1 0.180 0.153 7.04
Calcium 40.1 . . 0.156 ..0.149 . 6.26
Iron ..55.8 0.106 ..0.107 .5.91
Copper .63.5 .0.092 .0.094 .5.84
Germanium 72.6 0.0764 0.0822 .5.56
Rubidium 85.5 .0.086 0.0698 . 7.35
Tin (a ).. 119 ..0.051 .0.0502 ...6.07
Lead ..207 ..0.0305 ...0.0288 6.31
Water has a specific heat capacity of 1 cal/(goC), but 3R/M for water is 0.33. Maybe water has rotational and/or vibrational modes with a lot of energy. About semiconductors (diamond, silicon and germanium): When heated, more electrons in semiconductors become free from their bonds. This takes energy, so it would seem that C should be higher than predicted, but it is lower. Go figure.
Main pages: mechanics,
fluid/heat/e&m,
vibrations/waves,
quantum
alphabetical stuff: index
Q&A: fredrick.gram @ tri-c.edu (remove spaces)