Zero point entropy

Zero point entropy is the entropy of a substance at absolute zero .

initial situation

While the atoms / molecules of a substance can be distributed microscopically in very different and changing ways over space and over the energy levels , the substance has the same macroscopic properties for many of these distribution patterns or microstates , they form a certain macrostate . This is recorded numerically by its state variables .

At higher temperatures and freely movable particles, the substance tends very quickly to the macrostate, which corresponds to the largest number W of microstates. It has the greatest statistical weight W and the highest entropy S:

{\ displaystyle S = k _ {\ mathrm {B}} \ cdot \ ln \, W}

with the Boltzmann constant k _B .

In the solid state and when approaching absolute zero, several microstates could theoretically still be formed, but due to the severely restricted mobility of the particles, the change to a different configuration may be outside the observation period. With the low energy available, levels above the ground state can hardly be occupied.

Ideal cases

In pure crystal , the heat capacity c _p ≈ c _v approaches zero when approaching absolute zero according to Debye's T ³ law . The reaction entropy (the difference in entropies between products and educts) becomes smaller and smaller ( Nernst theorem ), and according to Planck , the entropy of an ideally crystallized, pure solid is zero at absolute zero.

Thermodynamically, one calculates the entropy over the course of the specific heat c _p with the temperature. The change in entropy between temperatures T ₁ and T ₂ is:

{\ displaystyle \ Delta S = \ int _ {T_ {1}} ^ {T_ {2}} {\ frac {c_ {p} (T)} {T}} dT}

(There are also amounts for phase transitions .)

With the entropy value zero, according to the microscopic idea, only a microstate is possible. There should therefore be a rigid structure, while a crystal still has oscillation energy even at absolute zero , in accordance with the uncertainty principle , so that a margin in the structure is conceivable. In terms of quantum mechanics , however, the ground state can be viewed as a state.

special cases

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

Examples

According to Wedler, several equivalent micro-states can exist at absolute zero for other reasons and lead to an entropy greater than zero, for example if the ground state is energetically degenerate (more than one configuration with the same energy; configuration entropy ; e.g. spin-ice or spin Glass ).

In ice there are several crystal configurations with the same energy, its zero point entropy is given as 3.41 J · mol ⁻¹ · K ⁻¹ .

Another example is the CO crystal, in which the molecules can be oriented parallel or antiparallel to one another: In one mole , W = 2 ^{N _A} (N _A is Avogadro's constant ) different arrangements of the molecules are possible, and from S ₀ = k _B lnW = R · ln2 we get a zero point entropy of 5.76 J · mol ⁻¹ · K ⁻¹ ( is the universal gas constant ). CO has a very low electrical dipole moment and does not form into a perfect crystal fast enough when it cools down. ${\ displaystyle R = N _ {\ mathrm {A}} \ cdot k _ {\ mathrm {B}}}$

The zero point entropy of H ₂ and D ₂ is based on configurations of the nuclear spins . An entropy of mixing can be ascribed to crystals that consist of several types of molecules .

With glasses as supercooled melts , a large number of different configurations are possible. The case is also conceivable that at T → 0 the specific heat is c _p ≠ 0.

calculation

According to Siebert, a value for the entropy of glasses when approaching the zero point can be obtained in two ways:

1. From the number of possible configurations. Glass has a lower density than the chemically identical crystal modification. This leads to the conclusion that there are cavities in the glass, which give the molecules mobility and thus enable many different configurations. In the case of silica glass, the cavity accounts for 21% compared to quartz. With

${\ displaystyle W = {\ frac {(N + N_ {z})!} {N! N_ {z}!}}}$ (N = number of molecules, N _z = additional cavity cells)

the number of possible configurations / microstates of the glass molecules in the total volume is obtained. For 1 mol of silica glass the difference in entropy compared to quartz is Δs (0) = k _B · InW = 4.6 J · Mol ⁻¹ · K ⁻¹ . However, relaxation between the individual configurations takes place so slowly that only one configuration is present in a realistic observation period and a certain state does not occur at all because of its larger number of equivalent micro-states on average over time. This is why the opinion is held that the configuration part of the entropy in glasses - as in the ideal crystal - approaches zero at T → 0K.

2. From the difference in the specific heats between glass and crystal, Δc _p = (c _p^gl - c _p^crystal ), where one ^starts from the melting temperature T _s and the molar melting enthalpy Δh (T _s ). The difference in entropy between glass and crystal is obtained when approaching absolute zero

{\ displaystyle \ Delta s (T \ rightarrow 0) = {\ frac {\ Delta h (T_ {s})} {T_ {s}}} - \ int _ {T \ rightarrow 0} ^ {T_ {s} } \ Delta c_ {p} \ cdot d (\ ln T)}

Below 1K, the specific heat of glasses does not follow Debye's T ³ law. Even at very low temperatures there remains a difference in the molar entropy Δ (s ^gl - s ^crystal ) of ≈ 4 J · Mol ⁻¹ · K ⁻¹ .

Quantum mechanical model

The following explanation is proposed for the observed heat capacity of glasses at the lowest temperatures: In the frozen melt there is only one of very many possible configurations of the glass molecules. This does not correspond to the absolute, but only to a relative energy minimum, which is separated from other minima by potential barriers. The tunnel effect splits the vibrational ground state, and transitions between these levels are possible. Very small amounts of energy, such as those applied by phonons , are sufficient for excitation (phonon-induced tunneling).

Complex cases

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

Individual evidence

↑ Wedler, Textbook of Physical Chemistry, Verlag Chemie 1982, chap. 2.4, p. 268.
^ Christian Gerthsen , Helmut Vogel: Physics . 17th edition. Springer, Berlin 1993, p. 727.
↑ Adam, Läuger, Stark. Physical chemistry and biophysics. Springer, chap. 2.2.3.3, p. 75.
^ Andreas Heintz (2010). Statistical Thermodynamics. Basics and treatment of simple chemical systems. Cape. 4.4, p. 138.
↑ Lars Siebert: Magnetic field dependence of the thermal properties of multi-component glasses at low temperatures , dissertation, University of Heidelberg, 2001, chap. 2.1, pp. 6-8.
↑ Lars Siebert: Magnetic field dependence of the thermal properties of multi-component glasses at low temperatures , dissertation, University of Heidelberg, 2001, chap. 2.3.1, p. 13.

literature

Lars Siebert: Magnetic field dependence of the thermal properties of multicomponent glasses at low temperatures , dissertation, University of Heidelberg, 2001.
University of Rostock: Lecture notes Thermodynamics
Welf A. Kreiner: Entropy - what is it? (Overview for students and teachers)

[1] Wedler, Textbook of Physical Chemistry, Verlag Chemie 1982, chap. 2.4, p. 268.

[2] Christian Gerthsen , Helmut Vogel: Physics . 17th edition. Springer, Berlin 1993, p. 727.

[3] Adam, Läuger, Stark. Physical chemistry and biophysics. Springer, chap. 2.2.3.3, p. 75.

[4] Andreas Heintz (2010). Statistical Thermodynamics. Basics and treatment of simple chemical systems. Cape. 4.4, p. 138.

[5] Lars Siebert: Magnetic field dependence of the thermal properties of multi-component glasses at low temperatures , dissertation, University of Heidelberg, 2001, chap. 2.1, pp. 6-8.

[6] Lars Siebert: Magnetic field dependence of the thermal properties of multi-component glasses at low temperatures , dissertation, University of Heidelberg, 2001, chap. 2.3.1, p. 13.