Residual entropy

Nullpunktsentropie is the entropy of a substance at absolute zero.

Initial situation

The state of a substance is detected macroscopically numerically by its state variables. While the atoms / molecules can be distributed in very different and changing ways over the space and the energy levels of microscopic, has the substance for many of these distribution patterns or micro-states the same macroscopic properties, they form a particular macrostate. At higher temperature and free-moving particles the substance tends very quickly to the macrostate that corresponds to the largest number W of microstates. It has the largest statistical weight (W ) and the highest entropy S = kB · LNW ( kB = Boltzmann constant ). In the solid state and when approaching absolute zero several micro-states could theoretically still be formed, but due to the highly restricted mobility of the particles, the change are in a different configuration outside of the observation period. In the low available energy levels barely above the ground state can be occupied.

Ideal cases

In the crystal of the heat capacity is cp ≈ cv when approaching absolute zero according to the Debye T3 law against zero. The reaction entropy (the difference between the entropies of products and reactants ) is always smaller ( Nernst theorem) and by Planck is the entropy of a perfect crystalline, pure solid at absolute zero is zero.

Thermodynamically, we can calculate the entropy over the course of the specific heat cp with temperature. The entropy change between the temperatures T1 and T2 is:

( There are a further amounts for phase transitions. )

When entropy is zero according to the microscopic idea just a microstate is possible. It would have to be a rigid structure, while a crystal at absolute zero still possesses vibrational energy, in accordance with the uncertainty principle, giving a margin in the structure is possible. Quantum mechanically, however, one can consider the ground state as a state.

Special cases

If a non-crystalline system as glass is considered, the entropy at the neutral point is not zero, but reaches a minimum there. This residual entropy is extrapolated or calculated.

Examples

After Wedler can at absolute zero for other reasons exist several equivalent microstates and lead greater than zero to an entropy, such as when the ground state is energetically degenerate (more than one configurations with the same energy has; configurational eg, spin ice and spin- glass).

• In several ice crystal configurations exist with the same energy and its Nullpunktsentropie is given as 3.41 J · mol -1 · K -1.

• As another example, CO - crystal is known in which the molecules can be oriented parallel or anti-parallel to each other, in a mole, W = 2NA (NA = Avogadro's number ), different arrangements of the molecules are possible and of S0 = kB LNW = X ln2 obtained a Nullpunktsentropie of 5.76 J · mol -1 · K -1. CO has a very low dipole moment and formed on cooling is not fast enough to the perfect crystal.

• The Nullpunktsentropie of H2 and D2 based on configurations of the nuclear spins. Crystals, which consist of several types of molecules, we may ascribe an entropy of mixing.

• For glasses, as undercooled melts, a large number of different configurations are possible. Also the case is conceivable that at T → 0, the specific heat cp ≠ 0.

Calculation

By Siebert can obtain a value for the entropy of glasses when approaching zero in two ways:

1 from the number of possible configurations. Glass has a lower density than the chemically same crystal modification. It is concluded cavities in the glass, giving the molecules mobility and so allow many different configurations could. In silica glass the cavity accounts for a share of 21 % compared to quartz. with

(N = number of molecules added = Nz cavity cells)

One obtains the number of possible configurations / micro-states of the glass molecules in total. Then the entropy difference compared to quartz? S is for 1 mol of silica glass (0) = kB · LNW = 4.6 J · mol -1 · K -1. However, a relaxation between the different configurations takes place so slowly that in a realistic observation period there is only one configuration and it does not come to that a certain state because of its greater number of equivalent microstates in the time average preferred setting. Therefore, the opinion is expressed that the configuration portion of the entropy in glasses - as in the ideal crystal - at T → 0K go to zero.

2 From the difference of the specific heats between glass and crystal, Δcp = ( CPGL - cpkrist ), where one starts from the melting temperature Ts and the molar enthalpy of fusion? H (Ts ). The entropy difference between glass and crystal is obtained as one approaches absolute zero

1K below follows the specific heat of glass is not the Debye T3 law. Even at very low temperatures remains a difference in the molar entropy Δ ( sgl - skrist ) of ≈ 4 J · mol -1 · K -1.

Quantum mechanical model

For the observed heat capacity of glasses at the lowest temperatures the following statement is proposed: In the frozen melt just one of many possible configurations of the glass molecules is present. This does not correspond to the absolute but only a relative energy minimum which is separated from other minima by potential barriers. Tunneling splits the vibrational state, and transitions between these levels are possible. Suffice to excite very small amounts of energy, as they can be applied by phonons ( phononeninduziertes tunnels ).

Complex cases

A calculation is not possible, if the system under study is a mixture of molecules.