Specific Heat Capacity
Specific heat capacity (c) is the heat required to raise the temperature of a unit mass of a substance by one kelvin, c = (1/m)(δQ/dT), with SI unit J kg−1 K−1; the molar heat capacity is the same quantity per mole. Values differ for heating at constant pressure (cp) and constant volume (cv), and depend on temperature in ways that reveal the quantum structure of matter.
Curious
Put a metal pan and a pan of water on the same burner and the metal heats up much faster. The difference is specific heat capacity. Water's is unusually high, about 4,186 J per kilogram per kelvin, so oceans and lakes warm and cool slowly and moderate the climate of coastal regions, and the human body's high water content helps keep its temperature steady.
Heat capacity also explains why sand on a beach gets hot quickly in the sun while the sea stays cool. Together with latent heat it governs most practical heating and cooling calculations (see Heat).
Exploring
| Substance | c (J kg−1 K−1) | Molar C (J mol−1 K−1) |
|---|---|---|
| Water (liquid, 25 °C) | 4,181 | 75.3 |
| Aluminium | 897 | 24.2 |
| Copper | 385 | 24.4 |
| Lead | 129 | 26.6 |
| Dry air (cp) | 1,005 | 29.1 |
Deep Dive
Cp and Cv. At constant pressure, some heat goes into expansion work, so Cp > Cv. For an ideal gas, Cp − Cv = R per mole (Mayer's relation), and the ratio γ = Cp/Cv is 5/3 for monatomic and about 7/5 for diatomic gases near room temperature; γ sets the speed of sound and adiabatic behaviour. For solids and liquids the difference is small but nonzero, Cp − Cv = TVα²/κT.
Dulong–Petit and its failure. Classical equipartition predicts 3R = 24.94 J mol−1 K−1 for any solid (Dulong and Petit, 1819), which works for many elements at room temperature but fails for diamond and for all solids at low temperature. Einstein (1907) explained the fall by quantising lattice vibrations, and Debye (1912) refined the model to give the C ∝ T³ law (see Third Law). In metals, conduction electrons add a small term proportional to T (see Density of States).
Water. Liquid water's molar heat capacity (75.3 J mol−1 K−1, about 9R) is far above the equipartition value for a free molecule because hydrogen bonds store energy as they are broken and re-formed. Heat capacity peaks (anomalies) mark phase transitions, such as the lambda point of helium at 2.17 K, where liquid helium becomes superfluid.
Concept Relationship Map
Requires: heat, internal energy
Enables: calorimetry, third law tests, speed of sound
Commonly confused with: heat capacity; molar heat capacity
Sources
- Einstein, A. (1907). Annalen der Physik, 22, 180–190.
- Petit, A.T., Dulong, P.L. (1819). “Recherches sur quelques points importants de la théorie de la chaleur.” Annales de Chimie et de Physique, 10, 395–413.
- Schroeder, D.V. (2000). An Introduction to Thermal Physics. Addison-Wesley.