Dulong–Petit law
The Dulong - Petit law states that the molar heat capacity of a composite of single atoms solid have a universal and constant value, which is three times the universal gas constant R.
Pierre Louis Dulong and Alexis Thérèse Petit discovered experimentally that many of them examined substances had practically the same molar heat capacity, and published in 1819, the presumption that it was this to be a general law. Classical statistical thermodynamics (which were unknown quantum effects ), monatomic solids later found for the molar heat capacity actually the constant value 3R. This statement of thermodynamics is in honor of the two experimenters called the Dulong - Petit law. On wider temperature ranges extensive measurements and theoretical investigations taking into account quantum mechanical principles, however, show that this law is only approximately valid.
Derivation
The particles in a solid are bound to their sites in the crystal lattice vibrations and lead to these resources positions. The vibration of each particle can be described in a first approximation as a harmonic oscillator. After equipartition classical statistical thermodynamics each of the three degrees of freedom of lattice vibration contributes each particle (one in the x-, y -and z- direction) at the temperature T in the middle, the kinetic energy 1/2 kT. The potential energy of the harmonic oscillator is a homogeneous function of degree 2 in the deflection. Thus follows the virial that the average potential energy is equal to the average kinetic energy. Therefore, a degree of freedom vibration means is omitted in the energy kT and a particle having three degrees of freedom for the lattice vibration energy 3kT. One mole of such particles thus carries the energy E = 3NAkT is = 3RT, and the molar heat capacity
Where R is the universal gas constant, T the absolute temperature, k is Boltzmann's constant, NA is the Avogadro constant. Furthermore: molar heat capacity at constant pressure (p) and: molar heat capacity at constant volume (V).
Confines
Despite its simplicity makes the Dulong - Petit law relatively good predictions for the specific heat capacity of solids with simple crystal structure at sufficiently high temperatures (eg, at room temperature).
In areas of low temperatures it deviates increasingly from the experimental findings. Since the lattice vibrations are quantized, they can only absorb energy per degree of freedom quantum size hv (h: Planck's constant, ν: frequency of oscillation ). In particular at least 1 × energy hv per degree of freedom necessary to excite the vibration at all. Is the available thermal energy kT is too low, some degrees of freedom are not excited and can not contribute to the heat capacity by energy intake. The heat capacity of solids therefore increases at very low temperatures down considerably and is aiming for T → 0 towards zero (third law of thermodynamics ). At low temperatures the Debye model gives better predictions.
Is a solid built not from individual atoms but of more complicated molecules (eg, CaSO4 ), so come to the 3 degrees of freedom of lattice vibration for each particle additional degrees of freedom of the molecular vibration added ( the particles of the molecule vibrate against each other ). The molar heat capacity of such a solid can be significantly higher than predicted by the Dulong - Petit law.
Metals as monoatomic solids are usually in good agreement with the Dulong - Petit law. Because of the metal binding one would initially expect anything else, when the atoms leave in their bonding electrons from their outer electron shell, which can move freely through the crystal. Each electron takes three translational degrees of freedom contribute so that when each emits one electron atom, the molar heat capacity 3R 3 * R / 2 = 9/ 2 R would be. However, since all states are already occupied below the Fermi distribution in the electron gas, most of the electrons can not go into a state of higher energy and therefore do not contribute to the heat capacity.