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The third law of thermodynamics states, regarding the properties of closed systems in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic field. At absolute zero (zero
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
s) the system must be in a state with the minimum possible energy. Entropy is related to the number of accessible
microstate A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
s, and there is typically one unique state (called the ground state) with minimum energy. In such a case, the entropy at absolute zero will be exactly zero. If the system does not have a well-defined order (if its order is
glassy Glassy is an adjective meaning, of or resembling glass. Glassy may also mean: * Glassy, amorphous metal * ''Glassy'', a 2021 single album, or the title song by Jo Yu-ri * Glassy phase, amorphous solid * Glassy water, amorphous ice * Glassy carbon, ...
, for example), then there may remain some finite entropy as the system is brought to very low temperatures, either because the system becomes locked into a configuration with non-minimal energy or because the minimum energy state is non-unique. The constant value is called the
residual entropy Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crys ...
of the system. The entropy is essentially a state-function meaning the inherent value of different atoms, molecules, and other configurations of particles including subatomic or atomic material is defined by entropy, which can be discovered near 0 K. The Nernst–Simon statement of the third law of thermodynamics concerns thermodynamic processes at a fixed, low temperature: Here a condensed system refers to liquids and solids. A classical formulation by Nernst (actually a consequence of the Third Law) is: There also exists a formulation of the third law which approaches the subject by postulating a specific energy behavior:


History

The third law was developed by chemist Walther Nernst during the years 1906–12, and is therefore often referred to as Nernst's theorem or Nernst's postulate. The third law of thermodynamics states that the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
of a system at
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...
is a well-defined constant. This is because a system at zero temperature exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state. In 1912 Nernst stated the law thus: "It is impossible for any procedure to lead to the isotherm in a finite number of steps." An alternative version of the third law of thermodynamics as stated by
Gilbert N. Lewis Gilbert Newton Lewis (October 23 or October 25, 1875 – March 23, 1946) was an American physical chemist and a Dean of the College of Chemistry at University of California, Berkeley. Lewis was best known for his discovery of the covalent bond a ...
and
Merle Randall Merle Randall (January 29, 1888 – March 17, 1950) was an American physical chemist famous for his work with Gilbert N. Lewis, over a period of 25 years, in measuring reaction heat of chemical compounds and determining their corresponding free ...
in 1923: :If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances. This version states not only \Delta S will reach zero at 0 K, but S itself will also reach zero as long as the crystal has a ground state with only one configuration. Some crystals form defects which cause a residual entropy. This residual entropy disappears when the kinetic barriers to transitioning to one ground state are overcome. With the development of statistical mechanics, the third law of thermodynamics (like the other laws) changed from a ''fundamental'' law (justified by experiments) to a ''derived'' law (derived from even more basic laws). The basic law from which it is primarily derived is the statistical-mechanics definition of entropy for a large system: : S - S_0 = k_\text \ln \, \Omega where S is entropy, k_\mathrm is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, and \Omega is the number of
microstate A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
s consistent with the macroscopic configuration. The counting of states is from the reference state of absolute zero, which corresponds to the entropy of S_0.


Explanation

In simple terms, the third law states that the entropy of a perfect crystal of a pure substance approaches zero as the temperature approaches zero. The alignment of a perfect crystal leaves no ambiguity as to the location and orientation of each part of the crystal. As the energy of the crystal is reduced, the vibrations of the individual atoms are reduced to nothing, and the crystal becomes the same everywhere. The third law provides an absolute reference point for the determination of entropy at any other temperature. The entropy of a closed system, determined relative to this zero point, is then the ''absolute'' entropy of that system. Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
. The entropy of a ''perfect'' crystal lattice as defined by Nernst's theorem is zero provided that its ground state is unique, because . If the system is composed of one-billion atoms, all alike, and lie within the matrix of a perfect crystal, the number of combinations of one-billion identical things taken one-billion at a time is . Hence: S - S_0 = k_\text \ln\Omega = k_\text\ln = 0 The difference is zero, hence the initial entropy can be any selected value so long as all other such calculations include that as the initial entropy. As a result, the initial entropy value of zero is selected is used for convenience. S - S_0 = S - 0 = 0 S = 0


Example: Entropy change of a crystal lattice heated by an incoming photon

Suppose a system consisting of a crystal lattice with volume of identical atoms at , and an incoming photon of wavelength and energy . Initially, there is only one accessible microstate: S_0 = k_\text \ln\Omega = k_\text\ln = 0. Let's assume the crystal lattice absorbs the incoming photon. There is a unique atom in the lattice that interacts and absorbs this photon. So after absorption, there is possible microstates accessible by the system, each of the microstates corresponding to one excited atom, and the other atoms remaining at ground state. The entropy, energy, and temperature of the closed system rises and can be calculated. The entropy change is: \Delta S = S - S_ = k_\text \ln From the second law of thermodynamics: \Delta S = S - S_0 = \frac Hence: \Delta S = S - S_0 = k_\text \ln(\Omega) = \frac Calculating entropy change: S - 0 = k_\text \ln = 1.38 \times 10^ \times \ln = 70 \times 10^ \,\mathrm We assume and . The energy change of the system as a result of absorbing the single photon whose energy is : \delta Q = \varepsilon = \frac =\frac=2 \times 10^ \,\mathrm The temperature of the closed system rises by: T = \frac = \frac = 0.02857 \,\mathrm This can be interpreted as the average temperature of the system over the range from 0 < S < 70 \times 10^\,\mathrm. A single atom was assumed to absorb the photon but the temperature and entropy change characterizes the entire system.


Systems with non-zero entropy at absolute zero

An example of a system which does not have a unique ground state is one whose net spin is a half-integer, for which time-reversal symmetry gives two degenerate ground states. For such systems, the entropy at zero temperature is at least (which is negligible on a macroscopic scale). Some crystalline systems exhibit geometrical frustration, where the structure of the crystal lattice prevents the emergence of a unique ground state. Ground-state helium (unless under pressure) remains liquid. In addition, glasses and solid solutions retain large entropy at 0 K, because they are large collections of nearly degenerate states, in which they become trapped out of equilibrium. Another example of a solid with many nearly-degenerate ground states, trapped out of equilibrium, is
ice Ih Photograph showing details of an ice cube under magnification. Ice Ih is the form of ice commonly seen on Earth. Phase space of ice Ih with respect to other ice phases. Ice Ih (hexagonal ice crystal) (pronounced: ice one h, also known as ice-p ...
, which has "proton disorder". For the entropy at absolute zero to be zero, the magnetic moments of a perfectly ordered crystal must themselves be perfectly ordered; from an entropic perspective, this can be considered to be part of the definition of a "perfect crystal". Only
ferromagnetic Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
, antiferromagnetic, and diamagnetic materials can satisfy this condition. However, ferromagnetic materials do not, in fact, have zero entropy at zero temperature, because the spins of the unpaired electrons are all aligned and this gives a ground-state spin degeneracy. Materials that remain paramagnetic at 0 K, by contrast, may have many nearly-degenerate ground states (for example, in a spin glass), or may retain dynamic disorder (a quantum spin liquid).


Consequences


Absolute zero

The third law is equivalent to the statement that :It is impossible by any procedure, no matter how idealized, to reduce the temperature of any closed system to zero temperature in a finite number of finite operations. The reason that cannot be reached according to the third law is explained as follows: Suppose that the temperature of a substance can be reduced in an isentropic process by changing the parameter ''X'' from ''X''2 to ''X''1. One can think of a multistage nuclear demagnetization setup where a magnetic field is switched on and off in a controlled way. If there were an entropy difference at absolute zero, could be reached in a finite number of steps. However, at ''T'' = 0 there is no entropy difference so an infinite number of steps would be needed. The process is illustrated in Fig. 1.


Specific heat

A non-quantitative description of his third law that Nernst gave at the very beginning was simply that the specific heat can always be made zero by cooling the material down far enough.''Einstein and the Quantum'', A. Douglas Stone, Princeton University Press, 2013. A modern, quantitative analysis follows. Supposed that the heat capacity of a sample in the low temperature region has the form of a power law asymptotically as , and we wish to find which values of are compatible with the third law. We have By the discussion of third law (above), this integral must be bounded as , which is only possible if . So the heat capacity must go to zero at absolute zero if it has the form of a power law. The same argument shows that it cannot be bounded below by a positive constant, even if we drop the power-law assumption. On the other hand, the molar specific heat at constant volume of a monatomic classical ideal gas, such as helium at room temperature, is given by with the molar ideal gas constant. But clearly a constant heat capacity does not satisfy Eq. (). That is, a gas with a constant heat capacity all the way to absolute zero violates the third law of thermodynamics. We can verify this more fundamentally by substituting in Eq. (), which yields In the limit this expression diverges, again contradicting the third law of thermodynamics. The conflict is resolved as follows: At a certain temperature the quantum nature of matter starts to dominate the behavior. Fermi particles follow
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac d ...
and Bose particles follow Bose–Einstein statistics. In both cases the heat capacity at low temperatures is no longer temperature independent, even for ideal gases. For Fermi gases with the Fermi temperature ''T''F given by Here is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
, the molar volume, and the molar mass. For Bose gases with ''T''B given by The specific heats given by Eq. () and () both satisfy Eq. (). Indeed, they are power laws with ''α''=1 and ''α''=3/2 respectively. Even within a purely classical setting, the density of a classical ideal gas at fixed particle number becomes arbitrarily high as goes to zero, so the interparticle spacing goes to zero. The assumption of non-interacting particles presumably breaks down when they are sufficiently close together, so the value of gets modified away from its ideal constant value.


Vapor pressure

The only liquids near absolute zero are 3He and 4He. Their heat of evaporation has a limiting value given by with ''L''0 and ''C''p constant. If we consider a container, partly filled with liquid and partly gas, the entropy of the liquid–gas mixture is where ''S''l(''T'') is the entropy of the liquid and is the gas fraction. Clearly the entropy change during the liquid–gas transition ( from 0 to 1) diverges in the limit of ''T''→0. This violates Eq.(). Nature solves this paradox as follows: at temperatures below about 50 mK the vapor pressure is so low that the gas density is lower than the best vacuum in the universe. In other words: below 50 mK there is simply no gas above the liquid.


Latent heat of melting

The melting curves of 3He and 4He both extend down to absolute zero at finite pressure. At the melting pressure, liquid and solid are in equilibrium. The third law demands that the entropies of the solid and liquid are equal at . As a result, the latent heat of melting is zero and the slope of the melting curve extrapolates to zero as a result of the Clausius–Clapeyron equation.


Thermal expansion coefficient

The thermal expansion coefficient is defined as With the
Maxwell relation file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...
and Eq. () with it is shown that So the thermal expansion coefficient of all materials must go to zero at zero kelvin.


See also

*
Adiabatic process In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
* Ground state * Laws of thermodynamics * Quantum thermodynamics *
Residual entropy Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crys ...
* Thermodynamic entropy * Timeline of thermodynamics, statistical mechanics, and random processes * Quantum heat engines and refrigerators


References


Further reading

*Goldstein, Martin & Inge F. (1993) ''The Refrigerator and the Universe''. Cambridge MA: Harvard University Press. . Chpt. 14 is a nontechnical discussion of the Third Law, one including the requisite elementary
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. * ** * {{Use dmy dates, date=October 2014 3