Quest for absolute zero, Nernst heat theorem.

Absolute zero is the lowest possible temperature on the thermodynamic temperature scale. It corresponds to

\[ \boxed{ T=0\,\text{K}=-273.15^\circ\text{C} } \]

At absolute zero, a system is in its lowest accessible energy state, called the ground state. However, quantum-mechanical zero-point motion may remain present.

The production of extremely low temperatures has been an important objective of experimental physics because many quantum phenomena, including superconductivity, superfluidity, and magnetic ordering, become observable at cryogenic temperatures.

Cooling by Evaporation of Liquid Helium

Before the development of magnetic refrigeration, one of the principal methods of producing temperatures below 4.2 K was the evaporation of liquid helium at reduced pressure.

When the pressure above liquid helium is reduced by pumping, its boiling temperature decreases.

The evaporation of the more energetic atoms removes energy from the remaining liquid, thereby lowering its temperature.

This method is called evaporative cooling.

By pumping on liquid helium-4, temperatures of the order of 1 K can be obtained. Further cooling using helium-3 evaporation can reach substantially lower temperatures.

Adiabatic Demagnetization of Paramagnetic Salts

In 1926, Debye and Giauque independently proposed the principle of magnetic refrigeration. The technique was subsequently developed experimentally during the early 1930s.

The method is based on the magnetic properties of paramagnetic materials and is known as adiabatic demagnetization refrigeration.

Principle

A paramagnetic material contains magnetic moments that tend to align with an applied magnetic field.

When a magnetic field is applied at constant temperature, the magnetic moments become more ordered, reducing their magnetic entropy.

If the magnetic field is subsequently reduced under approximately adiabatic conditions, the magnetic entropy tends to increase. To conserve the total entropy, the temperature of the material decreases.

Working

1. The paramagnetic salt is first cooled to a low temperature using a suitable cryogenic refrigeration system.
2. A strong magnetic field is applied while the material remains in thermal contact with a cold reservoir. The magnetic moments become partially aligned.
3. The material is thermally isolated from the surroundings.
4. The applied magnetic field is gradually reduced under approximately adiabatic conditions.
5. The magnetic moments become less ordered, and the temperature of the material decreases.

For an ideal non-interacting paramagnet in the appropriate temperature and field range, the entropy depends primarily on the ratio \(B/T\).

Therefore, during an ideal isentropic demagnetization process,

\[ \frac{B_i}{T_i} = \frac{B_f}{T_f} \]

Hence,

\[ \boxed{ T_f=T_i\frac{B_f}{B_i} } \]

where

\[ \begin{aligned} T_i &= \text{Initial temperature},\\ T_f &= \text{Final temperature},\\ B_i &= \text{Initial magnetic field},\\ B_f &= \text{Final magnetic field}. \end{aligned} \]

This simplified relation neglects internal magnetic interactions, heat leaks, and other non-ideal effects. In practice, these factors limit the lowest attainable temperature.

Adiabatic demagnetization of suitable paramagnetic salts has been used to produce temperatures in the millikelvin range.

Nuclear Adiabatic Demagnetization

To achieve temperatures much lower than those obtained by electronic paramagnetic demagnetization, nuclear magnetic moments can be used as the refrigerant.

This method is called nuclear adiabatic demagnetization.

Nuclear magnetic moments are generally much smaller than electronic magnetic moments, typically by approximately three orders of magnitude.

Consequently, nuclear spin systems can retain significant magnetic entropy at temperatures far below those characteristic of electronic magnetic refrigeration.

In a typical experiment, a nuclear refrigerant such as copper is first precooled to a very low temperature using another refrigeration method.

A strong magnetic field is then applied to polarize the nuclear spins. After thermal isolation, the magnetic field is reduced slowly, causing the nuclear spin temperature to decrease.

Early nuclear demagnetization experiments achieved temperatures in the microkelvin range.

\[ \boxed{ 1\,\mu\text{K}=10^{-6}\,\text{K} } \]

Modern low-temperature techniques can produce even lower temperatures in selected nuclear-spin and atomic systems. However, the temperatures of different degrees of freedom must be distinguished: a nuclear-spin temperature need not be identical to the lattice or electronic temperature.

Limitations in Reaching Absolute Zero

Although increasingly sophisticated cooling methods have made it possible to approach absolute zero extremely closely, reaching exactly 0 K is prohibited by the unattainability formulation of the third law of thermodynamics.

No finite sequence of physically realizable cooling operations can reduce a system to absolute zero.

Thus, absolute zero represents a limiting temperature that can be approached asymptotically but cannot be attained by a finite cooling process.

Nernst's Heat Theorem and Third Law of Thermodynamics

The second law of thermodynamics introduces entropy as a thermodynamic state function. For a reversible process, the change in entropy is defined by

\[ \boxed{ dS=\frac{\delta Q_{\mathrm{rev}}}{T} } \]

where

\[ \begin{aligned} dS &= \text{Infinitesimal change in entropy},\\ \delta Q_{\mathrm{rev}} &= \text{Reversible heat transfer},\\ T &= \text{Absolute temperature}. \end{aligned} \]

The second law enables the calculation of entropy differences between equilibrium states, but it does not independently establish the absolute value of entropy.

To determine absolute entropy, a reference state with a known entropy is required.

In 1906, Walther Nernst introduced a principle based on the behaviour of substances at very low temperatures.

This principle is known as Nernst's heat theorem and forms the foundation of the third law of thermodynamics.

Nernst's Heat Theorem

According to Nernst's heat theorem, the entropy change associated with a reversible isothermal process between equilibrium states approaches zero as the temperature approaches absolute zero, under the conditions in which the theorem applies.

Mathematically,

\[ \boxed{ \lim_{T\to0}\Delta S=0 } \]

For two equilibrium states A and B,

\[ \Delta S=S_B(T)-S_A(T). \]

Therefore,

\[ \boxed{ \lim_{T\to0} \left[S_B(T)-S_A(T)\right]=0 } \]

This implies that the entropy differences between the relevant equilibrium states vanish in the zero-temperature limit.

Third Law of Thermodynamics

The third law of thermodynamics establishes the limiting behaviour of entropy as the temperature approaches absolute zero.

(i) Planck's Statement

According to Planck's formulation, the entropy of a pure, perfectly crystalline substance approaches zero as its temperature approaches absolute zero, provided it has a unique ground state.

Mathematically,

\[ \boxed{ \lim_{T\to0}S(T)=0 } \]

or

\[ \boxed{S(0)=0} \]

for a perfect crystal with a non-degenerate ground state.

(ii) Lewis and Randall's Statement

According to the conventional formulation associated with Lewis and Randall, the entropy of a pure, perfectly crystalline substance in thermodynamic equilibrium is zero at absolute zero, provided the ground state is unique.

The entropy of a real material may retain a nonzero residual value if configurational disorder or ground-state degeneracy persists.

(iii) Principle of Unattainability

The third law may also be expressed through the principle of unattainability.

Unattainability Principle: It is impossible to reduce the temperature of a physical system to absolute zero through a finite number of thermodynamic operations.

Thus, absolute zero can only be approached as a limiting condition.

Mathematical Expression of the Third Law

Consider a pure crystalline substance at constant pressure.

For a reversible process, the entropy change is

\[ dS=\frac{\delta Q_{\mathrm{rev}}}{T}. \]

At constant pressure, for a closed system performing only pressure–volume work,

\[ \delta Q_{\mathrm{rev}}=C_P\,dT. \]

Substituting,

\[ \boxed{ dS=\frac{C_P}{T}\,dT } \]

where \(C_P\) is the heat capacity at constant pressure.

Suppose the substance is heated reversibly from a temperature approaching 0 K to a temperature \(T\), without crossing a phase transition.

Integrating both sides,

\[ \int_{S(0)}^{S(T)}dS = \int_0^T\frac{C_P(T')}{T'}\,dT'. \]

Therefore,

\[ S(T)-S(0) = \int_0^T\frac{C_P(T')}{T'}\,dT'. \]

Rearranging,

\[ \boxed{ S(T)=S(0)+ \int_0^T\frac{C_P(T')}{T'}\,dT' } \]

According to the third law, for a perfect crystal with a unique ground state,

\[ S(0)=0. \]

Hence,

\[ \boxed{ S(T)= \int_0^T\frac{C_P(T')}{T'}\,dT' } \]

This expression gives the absolute entropy of the substance at temperature \(T\), provided no phase transition occurs within the integration interval.

If one or more phase transitions occur between 0 K and \(T\), their entropy changes must also be included.

For a first-order transition at temperature \(T_{\mathrm{tr}}\),

\[ \boxed{ \Delta S_{\mathrm{tr}} = \frac{L_{\mathrm{tr}}}{T_{\mathrm{tr}}} } \]

where \(L_{\mathrm{tr}}\) is the latent heat of transition.

Thus, the absolute entropy is calculated by adding the heat-capacity integrals over the stable phases and the entropy changes associated with phase transitions.

Behaviour of Heat Capacity Near Absolute Zero

The third law imposes important restrictions on the low-temperature behaviour of thermodynamic properties.

For a typical non-metallic crystalline solid, the Debye model predicts that the lattice heat capacity at sufficiently low temperatures varies approximately as

\[ \boxed{ C_V\propto T^3 } \]

Therefore, writing

\[ C_V=aT^3, \]

the entropy change at constant volume is

\[ S(T)-S(0) = \int_0^T\frac{aT'^3}{T'}\,dT'. \]

Simplifying,

\[ S(T)-S(0) = a\int_0^T T'^2\,dT'. \]

On integration,

\[ S(T)-S(0)=\frac{aT^3}{3}. \]

For a perfect crystal,

\[ \boxed{ S(T)=\frac{aT^3}{3} } \]

Thus,

\[ \boxed{ \lim_{T\to0}S(T)=0 } \]

This result is consistent with the third law of thermodynamics.

Residual Entropy and Limitations

The zero-entropy formulation applies to a pure, perfectly ordered crystal with a unique ground state.

Some materials retain configurational disorder even at very low temperatures. Such materials may possess residual entropy.

Residual entropy is commonly described using Boltzmann's relation

\[ \boxed{ S=k_B\ln\Omega } \]

where

\[ \begin{aligned} k_B &= \text{Boltzmann constant},\\ \Omega &= \text{Number of accessible microstates}. \end{aligned} \]

For a unique ground state,

\[ \Omega=1. \]

Therefore,

\[ S(0)=k_B\ln1=0. \]

However, if the ground state has multiple configurations,

\[ \Omega>1. \]

Hence,

\[ \boxed{ S(0)=k_B\ln\Omega>0 } \]

Examples of materials exhibiting residual entropy under appropriate conditions include orientationally disordered crystals, certain forms of ice, and some frustrated magnetic systems.

Glasses and other nonequilibrium materials may also retain configurational entropy associated with frozen-in disorder. Such behaviour does not invalidate the third law for equilibrium perfect crystals.

Importance of the Third Law of Thermodynamics

The third law of thermodynamics has several important applications:

1. It provides a reference for calculating the absolute entropy of pure substances.
2. It explains the limiting behaviour of thermodynamic systems near absolute zero.
3. It establishes restrictions on the temperature dependence of heat capacities and other thermodynamic properties.
4. It is useful in determining entropy changes associated with chemical reactions and phase transitions.
5. It contributes to the calculation of Helmholtz free energy and Gibbs free energy.
6. It establishes the unattainability of absolute zero through finite thermodynamic operations.