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 vibrational motion, retaining only quantum mechanical, zero-point energy-induced particle motion. The theoretical temperature is determined by extrapolating the ideal gas law; by international agreement, absolute zero is taken as −273.15 degrees on the Celsius scale ( International System of Units), Note: The triple point of water is 0.01 °C, not 0 °C; thus 0 K is −2890.15 °C, not −273.16 °C. which equals −459.67 degrees on the Fahrenheit scale ( United States customary units or Imperial units). The corresponding Kelvin and Rankine temperature scales set their zero points at absolute zero by definition.

It is commonly thought of as the lowest temperature possible, but it is not the lowest ''enthalpy'' state possible, because all real substances begin to depart from the ideal gas when cooled as they approach the change of state to liquid, and then to solid; and the sum of the enthalpy of vaporization (gas to liquid) and enthalpy of fusion (liquid to solid) exceeds the ideal gas's change in enthalpy to absolute zero. In the quantum-mechanical description, matter (solid) at absolute zero is in its ground state, the point of lowest internal energy.

The laws of thermodynamics indicate that absolute zero cannot be reached using only thermodynamic means, because the temperature of the substance being cooled approaches the temperature of the cooling agent asymptotically. Even a system at absolute zero, if it could somehow be achieved, would still possess quantum mechanical zero-point energy, the energy of its ground state at absolute zero; the kinetic energy of the ground state cannot be removed.

Scientists and technologists routinely achieve temperatures close to absolute zero, where matter exhibits quantum effects such as Bose–Einstein condensate, superconductivity and superfluidity.

Thermodynamics near absolute zero

At temperatures near , nearly all molecular motion ceases and Δ''S'' = 0 for any adiabatic process, where ''S'' is the entropy. In such a circumstance, pure substances can (ideally) form perfect crystals with no structural imperfections as ''T'' → 0. Max Planck's strong form of the third law of thermodynamics states the entropy of a perfect crystal vanishes at absolute zero. The original Nernst '' heat theorem'' makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as ''T'' → 0:
:$\lim_ \Delta S = 0$

The implication is that the entropy of a perfect crystal approaches a constant value. An adiabat is a state with constant entropy, typically represented on a graph as a curve in a manner similar to isotherms and isobars.

The Nernst postulate identifies the isotherm T = 0 as coincident with the adiabat S = 0, although other isotherms and adiabats are distinct. As no two adiabats intersect, no other adiabat can intersect the T = 0 isotherm. Consequently no adiabatic process initiated at nonzero temperature can lead to zero temperature. (≈ Callen, pp. 189–190)

A perfect crystal is one in which the internal lattice structure extends uninterrupted in all directions. The perfect order can be represented by translational symmetry along three (not usually orthogonal) axes. Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping. For substances that exist in two (or more) stable crystalline forms, such as diamond and graphite for carbon, there is a kind of ''chemical degeneracy''. The question remains whether both can have zero entropy at ''T'' = 0 even though each is perfectly ordered.

Perfect crystals never occur in practice; imperfections, and even entire amorphous material inclusions, can and do get "frozen in" at low temperatures, so transitions to more stable states do not occur.

Using the Debye model, the specific heat and entropy of a pure crystal are proportional to ''T'' ³, while the enthalpy and chemical potential are proportional to ''T'' ⁴. (Guggenheim, p. 111) These quantities drop toward their ''T'' = 0 limiting values and approach with ''zero'' slopes. For the specific heats at least, the limiting value itself is definitely zero, as borne out by experiments to below 10 K. Even the less detailed Einstein model shows this curious drop in specific heats. In fact, all specific heats vanish at absolute zero, not just those of crystals. Likewise for the coefficient of thermal expansion. Maxwell's relations show that various other quantities also vanish. These phenomena were unanticipated.

Since the relation between changes in Gibbs free energy (''G''), the enthalpy (''H'') and the entropy is

:$\Delta G = \Delta H - T \Delta S \,$

thus, as ''T'' decreases, Δ''G'' and Δ''H'' approach each other (so long as Δ''S'' is bounded). Experimentally, it is found that all spontaneous processes (including chemical reactions) result in a decrease in ''G'' as they proceed toward equilibrium. If Δ''S'' and/or ''T'' are small, the condition Δ''G'' < 0 may imply that Δ''H'' < 0, which would indicate an exothermic reaction. However, this is not required; endothermic reactions can proceed spontaneously if the ''T''Δ''S'' term is large enough.

Moreover, the slopes of the derivatives of Δ''G'' and Δ''H'' converge and are equal to zero at ''T'' = 0. This ensures that Δ''G'' and Δ''H'' are nearly the same over a considerable range of temperatures and justifies the approximate empirical Principle of Thomsen and Berthelot, which states that ''the equilibrium state to which a system proceeds is the one that evolves the greatest amount of heat'', i.e., an actual process is the ''most exothermic one''. (Callen, pp. 186–187)

One model that estimates the properties of an electron gas at absolute zero in metals is the Fermi gas. The electrons, being fermions, must be in different quantum states, which leads the electrons to get very high typical velocities, even at absolute zero. The maximum energy that electrons can have at absolute zero is called the Fermi energy. The Fermi temperature is defined as this maximum energy divided by the Boltzmann constant, and is on the order of 80,000 K for typical electron densities found in metals. For temperatures significantly below the Fermi temperature, the electrons behave in almost the same way as at absolute zero. This explains the failure of the classical equipartition theorem for metals that eluded classical physicists in the late 19th century.

Relation with Bose–Einstein condensate

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero. Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale.

This state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper from English to German and submitted it for Bose to the '' Zeitschrift für Physik'', which published it. Einstein then extended Bose's ideas to material particles (or matter) in two other papers.

Seventy years later, in 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST-JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) ().

A record cold temperature of 450 ± 80 picokelvin (pK) () in a BEC of sodium atoms was achieved in 2003 by researchers at the Massachusetts Institute of Technology (MIT). The associated black-body (peak emittance) wavelength of 6,400 kilometers is roughly the radius of Earth.

Absolute temperature scales

Absolute, or thermodynamic, temperature is conventionally measured in kelvin ( Celsius-scaled increments) and in the Rankine scale (Fahrenheit-scaled increments) with increasing rarity. Absolute temperature measurement is uniquely determined by a multiplicative constant which specifies the size of the ''degree'', so the ''ratios'' of two absolute temperatures, ''T''₂/''T''₁, are the same in all scales. The most transparent definition of this standard comes from the Maxwell–Boltzmann distribution. It can also be found in Fermi–Dirac statistics (for particles of half-integer spin) and Bose–Einstein statistics (for particles of integer spin). All of these define the relative numbers of particles in a system as decreasing exponential functions of energy (at the particle level) over ''kT'', with ''k'' representing the Boltzmann constant and ''T'' representing the temperature observed at the macroscopic level.

Negative temperatures

Temperatures that are expressed as negative numbers on the familiar Celsius or Fahrenheit scales are simply colder than the zero points of those scales. Certain systems can achieve truly negative temperatures; that is, their thermodynamic temperature (expressed in kelvins) can be of a negative quantity. A system with a truly negative temperature is not colder than absolute zero. Rather, a system with a negative temperature is hotter than ''any'' system with a positive temperature, in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat flows from the negative to the positive-temperature system.

Most familiar systems cannot achieve negative temperatures because adding energy always increases their entropy. However, some systems have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease. Because temperature is defined by the relationship between energy and entropy, such a system's temperature becomes negative, even though energy is being added. As a result, the Boltzmann factor for states of systems at negative temperature increases rather than decreases with increasing state energy. Therefore, no complete system, i.e. including the electromagnetic modes, can have negative temperatures, since there is no highest energy state, so that the sum of the probabilities of the states would diverge for negative temperatures. However, for quasi-equilibrium systems (e.g. spins out of equilibrium with the electromagnetic field) this argument does not apply, and negative effective temperatures are attainable.

On 3 January 2013, physicists announced that for the first time they had created a quantum gas made up of potassium atoms with a negative temperature in motional degrees of freedom.

History

One of the first to discuss the possibility of an absolute minimal temperature was Robert Boyle. His 1665 ''New Experiments and Observations touching Cold'', articulated the dispute known as the ''primum frigidum''. The concept was well known among naturalists of the time. Some contended an absolute minimum temperature occurred within earth (as one of the four classical elements), others within water, others air, and some more recently within nitre. But all of them seemed to agree that, "There is some body or other that is of its own nature supremely cold and by participation of which all other bodies obtain that quality."

Limit to the "degree of cold"

The question whether there is a limit to the degree of coldness possible, and, if so, where the zero must be placed, was first addressed by the French physicist Guillaume Amontons in 1702, in connection with his improvements in the air thermometer. His instrument indicated temperatures by the height at which a certain mass of air sustained a column of mercury—the volume, or "spring" of the air varying with temperature. Amontons therefore argued that the zero of his thermometer would be that temperature at which the spring of the air was reduced to nothing. He used a scale that marked the boiling point of water at +73 and the melting point of ice at +, so that the zero was equivalent to about −240 on the Celsius scale. Amontons held that the absolute zero cannot be reached, so never attempted to compute it explicitly.
The value of −240 °C, or "431 divisions n Fahrenheit's thermometerbelow the cold of freezing water" was published by George Martine in 1740.

This close approximation to the modern value of −273.15 °C for the zero of the air thermometer was further improved upon in 1779 by Johann Heinrich Lambert, who observed that might be regarded as absolute cold.

Values of this order for the absolute zero were not, however, universally accepted about this period. Pierre-Simon Laplace and Antoine Lavoisier