Absolute zero

Absolute zero is the lowest possible temperature where nothing could be colder and no heat energy remains in a substance. Absolute zero is the point at which molecules stop and they have minimal movement vibrations or none, retaining only quantum mechanical, zero-point energy-induced particle motion.

By international agreement, absolute zero is defined as precisely…

• 0 K on the Kelvin scale, which is a thermodynamic (absolute) temperature scale, and
• –273.15 °C on the Celsius scale.

Absolute zero is also precisely equivalent to…

• 0 °R on the Rankine scale (also a thermodynamic temperature scale), and
• –459.67 °F on the Fahrenheit scale.

While scientists cannot fully achieve a state of “zero” heat energy in a substance, they have made great advancements in achieving temperatures ever closer to absolute zero (where matter exhibits odd quantum effects). In 1994, the NIST achieved a record cold temperature of 700 nK (billionths of a kelvin). In 2003, researchers at MIT eclipsed this with a new record of 450 pK (0.45 nK).

History of absolute zero

• 1702–1703: Guillaume Amontons (1663 – 1705) published two papers that credit him with being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation lead him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240 degrees on today’s Celsius scale—only 33.15 degrees short of the true value of −273.15 °C.

• 1777: In his book Pyrometrie (Berlin: Haude & Spener, 1779) completed four months before his death, Johann Heinrich Lambert (1728 – 1777)—sometimes incorrectly referred to as Joseph Lambert—proposed an absolute temperature scale based on the pressure / temperature relationship of a fixed volume of gas. This is distinct from the volume / temperature relationship of gas under constant pressure that Guillaume Amontons discovered 77 years earlier. Lambert stated that absolute zero was the point where a simple straight-line extrapolation reached zero gas pressure and was equal to −270 °C.

• 1802: Joseph Louis Gay-Lussac (1778 – 1850) published papers with the first known formulas to use the number “273” for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C).

• 1848: William Thomson, (1824 – 1907) also known as “Lord Kelvin,” wrote in his paper, On an Absolute Thermometric Scale, of the need for a scale whereby “infinite cold” (absolute zero) was the scale’s null point, and which used the degree Celsius for its unit increment. As did Gay-Lussac, Thomson calculated that absolute zero was equivalent to –273 °C on the air–thermometers of the time. This absolute scale is known today as the Kelvin thermodynamic temperature scale. It’s noteworthy that Thomson’s value of “–273” was actually derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of –0.00366 expressed to four significant digits is –273.2 °C which is remarkably close to the true value of –273.15 °C.

• Circa 1930s: Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed that absolute zero was –273.15 °C.

• 1948: Resolution 3 of the 9th CGPM (Conférence Générale des Poids et Mesures, also known as the General Conference on Weights and Measures) fixed the triple point of water at precisely 0.01 °C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared “will be fixed at a later date.” The implication at this date is that if the value of absolute zero measured in the 1930s was truly –273.15 °C, then the triple point of water (0.01 °C) was equivalent to 273.16 K.

• 1954: Resolution 3 of the 10th CGPM gave the Kelvin scale its modern definition by choosing the triple point of water as its second defining point and assigned it a temperature of precisely 273.16 kelvin (what was actually written 273.16 “degrees Kelvin” at the time). This defined absolute zero as being precisely zero kelvin and –273.15 °C.

Kinetic theory and motion

According to kinetic theory, there should be no movement of individual molecules at absolute zero, so any material at this temperature would be solid. In a monatomic gas, most of the energy is in the form of translational motion, and the temperature can be measured in terms of the distribution of this motion, with slower speeds corresponding to lower temperatures, perhaps even down to absolute zero. But this is contrary to experimental evidence, as helium will never solidify at normal pressures, regardless of temperature...

Because of quantum-mechanical effects, the speed at absolute zero is larger than zero and depends, along with the energy, on the volume within which a particle is confined. At absolute zero, the molecules and atoms in a system are all in their ground state, the state of lowest possible energy, and a system has the least amount of kinetic energy allowed by the laws of physics. But the lowest possible zero-point energy for a confined particle in a box is not zero. Rather than being fixed and non-moving, the equation for the energy levels shows that no matter how low the temperature gets, even when the quantum number takes its minimum value of one, a particle still has some translational kinetic energy and motion. This is a reflection of Heisenberg's uncertainty principle, which states that the position and the momentum of a particle cannot both be known precisely at any given time.

Similarly, using the harmonic approximation for the vibrations of a diatomic molecule, the quantum harmonic oscillator yields a positive zero-point energy even when the vibrational quantum number takes its minimum value of zero. For polyatomic molecules, and for bodies such as crystals, whose normal mode motions can not be assigned to individual atoms or chemical bonds, the lowest-energy state is that of the system as a whole.

Classically, the absolute temperature T of a system of molecules at thermodynamic equilibrium assigns an average of ½ kT to each quadratic kinetic and/or potential energy term in each mechanical degree of freedom, where k is Boltzmann's constant. (See equipartition of energy and the role of the Boltzmann distribution in relating temperature to energy.) But quantum mechanics shows that this is obeyed only for temperatures such that kT > hν, where h is Planck's constant and ν is a characteristic frequency. As T decreases, the assumption that energy is continuously variable fails whenever hν exceeds kT. For vibrational modes in crystals, this happens at room temperature, which explains the deviation of the calculated specific heats of atomic crystals from the experimental Dulong-Petit law value of 3R /mole, a fact which puzzled late 19th century physicists and physical chemists. (Rushbrooke, p. 33)

Record cold temperatures approaching absolute zero

It can be shown from the laws of thermodynamics that absolute zero can never be achieved, though it is possible to reach temperatures arbitrarily close to it through the use of cryocoolers. This is the same principle that ensures no machine can be 100% efficient.

At very low temperatures in the vicinity of absolute zero, matter exhibits many unusual properties including superconductivity, superfluidity, and Bose-Einstein condensation. In order to study such phenomena, scientists have worked to obtain ever lower temperatures.

• In September 2003, MIT announced a record cold temperature of 450 pK, or 4.5 × 10^-10  K in a Bose-Einstein condensate of sodium atoms. This was performed by Wolfgang Ketterle and colleagues at MIT.^[1]

• As of February 2003, the Boomerang Nebula, with a temperature of 1.15  K, is the coldest place known outside a laboratory. The nebula is 5000 light-years from Earth and is in the constellation Centaurus.^[2]

• As of November 2000, nuclear spin temperatures below 100 pK were reported for an experiment at the Helsinki University of Technology's Low Temperature Lab. However, this was the temperature of one particular type of motion— a quantum property called nuclear spin — not the overall average thermodynamic temperature for all possible degrees of freedom.^[3]

Thermodynamics near absolute zero

At 0 K, (nearly) all molecular motion ceases and <math>\Delta</math>S = 0 for any adiabatic process. Pure substances can (ideally) form perfect crystals as T <math>\rightarrow</math>0. Planck's strong form of the third law of thermodynamics states that the entropy of a perfect crystal vanishes at absolute zero. However, if the lowest energy state is degenerate (more than one microstate), this cannot be true. 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

<math> \lim_{T \to 0} \Delta S = 0 </math>

which implies that the entropy of a perfect crystal simply approaches a constant value.

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)

An even stronger assertion is that It is impossible by any procedure to reduce the temperature of a system to zero in a finite number of operations. (≈ Guggenheim, p. 157)

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 which have 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 materials, simply 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 as 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 the Gibbs free energy, the enthalpy and the entropy is

<math> \Delta G = \Delta H - T \Delta S \,</math>

it follows that as T decreases, ΔG and ΔH approach each other (so long as ΔS is bounded). Experimentally, it is found that most chemical reactions are exothermic and release heat in the direction they are found to be going, toward equilbrium. That is, even at room temperature T is low enough so that the fact that (ΔG)_T,P < 0 (usually) implies that ΔH < 0. (In the opposite direction, each such reaction would of course absorb heat.)

More than that, the slopes of the temperature derivatives of ΔG and ΔH converge and are equal to zero at T = 0, which ensures that ΔG and ΔH are nearly the same over a considerable range of temperatures, justifying the approximate empirical Principle of Thomsen and Berthelot, which says that the equilibrium state to which a system proceeds is the one which evolves the greatest amount of heat, i.e., an actual process is the most exothermic one. (Callen, pp. 186-187)

Absolute temperature scales

As mentioned, absolute or thermodynamic temperature is conventionally measured in kelvins (Celsius-size degrees), and increasingly rarely in the Rankine scale (Fahrenheit-size degrees). Absolute temperature is uniquely determined up to a multiplicative constant which specifies the size of the "degree", so the ratios of two absolute temperatures, T₂/T₁, are the same in all scales. The most transparent definition comes from the classical Maxwell-Boltzmann distribution over energies, or from the quantum analogs: Fermi-Dirac statistics (particles of half-integer spin) and Bose-Einstein statistics (particles of integer spin), all of which give the relative numbers of particles as (decreasing) exponential functions of energy over kT. On a macroscopic level, a definition can be given in terms of the efficiencies of "reversible" heat engines operating between hotter and colder thermal reservoirs.

Negative temperatures

Main article: Negative temperature

Certain semi-isolated systems (for example a system of non-interacting spins in a magnetic field) can achieve negative temperatures; however, they are not actually colder than absolute zero. They can be however thought of as "hotter than T=∞", as energy will flow from a negative temperature system to any other system with positive temperature upon contact.

See also

References

• Herbert B. Callen (1960). Thermodynamics, Chapter 10. John Wiley & Sons, Inc.. Library of Congress Catalog Card No. 60-5597. The clearest presentation of the logical foundations of the subject.
• E.A. Guggenheim (1967). Thermodynamics: An Advanced Treatment for Chemists and Physicists, 5th ed.. North Holland; John Wiley & Sons, Inc.. Library of Congress Catalog Card No. 60-20003. A remarkably astute and comprehensive treatise.
• G. S. Rushbrooke (1949). Introduction to Statistical Mechanics. Oxford Univ. Press. The classic, compact introduction to the subject.

Notes

1. ^ Leanhardt, A. et al. (2003) Science 301 1513. Physicsweb news report
2. ^ Press report February 21 2003
3. ^ The experimental methods and results are presented in detail in T.A. Knuuttila’s Ph.D. thesis which can be accessed from this site. Also the university’s pressure release on its achievement is here

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