Thermodynamic temperature

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Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an “absolute” scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the lowest possible temperature where nothing could be colder.

The Z machine at Sandia National Laboratories in Albuquerque, New Mexico, U.S.A., set a record man-made temperature for a bulk quantity of matter of greater than two billion kelvin. Courtesy, Sandia National Laboratories.

[edit] Overview

Fig. 1 The translational motion of fundamental particles of nature such as atoms and molecules gives a substance its temperature. Here, the size of helium atoms relative to their spacing is shown to scale under 136 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two trillion fold). At any given instant however, a particular helium atom may be moving much faster than average while another may be nearly motionless. Five atoms are colored red to facilitate following their motions.

Temperature arises from the random submicroscopic vibrations of the particle constituents of matter. These motions comprise the kinetic energy in a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy of a certain kind of vibrational motion of its constituent particles called translational motions. Translational motions are ordinary, whole-body movements in 3D space whereby particles move about and exchange energy in collisions. Fig. 1 at right shows translational motion in gases; Fig. 4 below shows translational motion in solids. Thermodynamic temperature’s null point, absolute zero, is the temperature at which the particle constituents of matter have minimal motion, retaining only quantum mechanical motion.^[1]

Throughout the scientific world where measurements are made in SI units, thermodynamic temperature is measured in kelvins (symbol: K). Many engineering fields in the U.S. measure thermodynamic temperature using the Rankine scale.

By international agreement, the unit “kelvin” and its scale are defined by two points: absolute zero, and the triple point of specially prepared (VSMOW) water. Absolute zero is defined as being precisely 0 K and −273.15 °C. The triple point of water is defined as being precisely 273.16 K and 0.01 °C. This definition does three things: 1) it fixes the magnitude of the kelvin unit as being precisely 1 part in 273.16 parts the difference between absolute zero and the triple point of water; 2) it establishes that one kelvin has precisely the same magnitude as a one-degree increment on the Celsius scale; and 3) it establishes the difference between the two scales’ null points as being precisely 273.15 kelvins (0 K = −273.15 °C and 273.16 K = 0.01 °C). Conversion from kelvins to degrees Rankine (°R) is accomplished as follows: T_K × 1.8 = T_°R.

[edit] Table of thermodynamic temperatures

The full range of the thermodynamic temperature scale and some notable points along it are shown in the table below.

^A For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated strictly per the two-point definition of thermodynamic temperature.
^B The 2500 K value is approximate. The 273.15 K difference between K and °C is rounded to 300 K to avoid false precision in the Celsius value.
^C For a true blackbody (which tungsten filaments are not). Tungsten filaments’ emissivity is greater at shorter wavelengths which makes them appear whiter.
^D Effective photosphere temperature. The 273.15 K difference between K and °C is rounded to 273 K to avoid false precision in the Celsius value.
^E The 273.15 K difference between K and °C is ignored to avoid false precision in the Celsius value.
^F For a true blackbody (which the plasma was not). The Z machine’s dominant emission originated from 40 MK electrons (soft x–ray emissions) within the plasma.

[edit] The relationship of temperature, motions, conduction, and heat energy

[edit] The nature of kinetic energy, translational motion, and temperature

At its simplest, “temperature” arises from the kinetic energy of the vibrational motions of matter’s particle constituents (molecules, atoms, and subatomic particles). The full variety of these kinetic motions contribute to the total heat energy in a substance. The relationship of kinetic energy, mass, and velocity is given by the formula E_k = ¹/₂m • v ².^[13] Accordingly, those particles with one unit of mass moving at one unit of velocity have the same kinetic energy—and the same temperature—as those with twice the mass but only 70.7% of the velocity.

Fig. 2 The translational motions of helium atoms occurs across a range of speeds. Compare the shape of this curve to that of a Planck curve in Fig. 5 below.

The thermodynamic temperature of any bulk quantity of a substance (a statistically significant quantity of particles) is directly proportional to the average—or “mean”—kinetic energy of a specific kind of particle motion known as translational motion. These simple movements in the three x, y, and z–axis dimensions of space means the particles move in the three spatial degrees of freedom.^[14] This particular form of kinetic energy is sometimes referred to as kinetic temperature. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the ideal gas law’s formula pV = nRT.

The extent to which the kinetic energy of translational motion of an individual atom or molecule (particle) in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the Boltzmann constant (symbol: k_B). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an individual particle’s translational motion as follows:

E_mean = 3/2k_BT

where…

E_mean = joules (symbol: J)

k_B = 1.380 6505(24) × 10⁻²³ J/K

T = thermodynamic temperature in kelvins

While the Boltzmann constant is useful for finding the mean kinetic energy of a particle, it’s important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in Fig. 1 above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in Fig. 2  shows the speed distribution of 5500 K helium atoms. They have a most probable speed of 4.780 km/s (0.2092 s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the x–axis to the right). This graph uses inverse speed for its x–axis so the shape of the curve can easily be compared to the curves in Fig. 5 below. In both graphs, zero on the x–axis represents infinite temperature. Additionally, the x and y–axis on both graphs are scaled proportionally.

[edit] The high speeds of translational motion

Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are very fast^[15] and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool cesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature.^[16]  Formulas for calculating the velocity and speed of translational motion are given in the following footnote.^[17]

[edit] The internal motions of molecules and specific heat

Fig. 3 Molecules have internal structure because they are composed of atoms that have different ways of moving within molecules. The heat energy stored in these internal degrees of freedom does not contribute to the temperature of a substance.

There are other forms of heat energy besides the kinetic energy of translational motion. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain its internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements. These are all types of internal degrees of freedom. This makes molecules distinct from monatomic substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom.^[14] Kinetic energy is stored in molecules’ internal degrees of freedom, which gives them an internal temperature.  Even though these motions are called “internal,” the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as heat is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a molecular-based substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles.

The kinetic energy stored internally in molecules does not contribute directly to the temperature of a substance (nor to the pressure or volume of gases). This is because any kinetic energy that is, at a given instant, bound in internal motions is not at that same instant contributing to the molecules’ translational motions. Since the internal temperature of the molecules in any bulk quantity of a substance in equilibrium is, on average, equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-equilibrium phenomena such as the sublimation of solids and the diffusion of hot gases in a partial vacuum.

Different molecules absorb different amounts of heat energy for each incremental increase in temperature. Water for instance, can absorb a large amount of heat energy per mole (a specific number of particles) with only a modest temperature change. This property is known as a substance’s specific heat capacity. High specific heat capacity arises, in part, because certain substance’s molecules possess more internal degrees of freedom than others. For instance, room-temperature nitrogen, which is a diatomic molecule, has five active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance to the equipartition theorem, nitrogen has five-thirds the molar heat capacity as do the monatomic gases.^[18] Larger, more complex molecules can have dozens of internal degrees of freedom.

[edit] The diffusion of heat energy: Entropy, phonons, and mobile conduction electrons

Fig. 4 The temperature-induced translational motion of particles in solids takes the form of phonons. Shown here are phonons with identical amplitudes but with wavelengths ranging from 2 to 12 molecules.

Heat conduction is the diffusion of heat energy from hot parts of a system to cold. A “system” can be either a single bulk entity or a plurality of discrete bulk entities. The term “bulk” in this context means a statistically significant quantity of particles (which can be a microscopic amount). Anytime heat energy diffuses within an isolated system, temperature differences within the system decrease (entropy increases).

One particular heat conduction mechanism occurs when translational motion—the particle motion underlying temperature—transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in Fig. 1. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can advance forward into new territory, bringing their kinetic energy with them. Consequently, heat diffuses through gases rather easily; especially for light atoms or molecules. Convection speeds this process even more.

Translational motion in solids however, takes the form of phonons (see Fig. 4 at right). Phonons are constrained, quantized wave packets traveling at the speed of sound for a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is usually inefficient^[19] and such solids are considered to be thermal insulators (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.

Metals however, are not restricted to only phonon-based heat conduction. Heat energy conducts through metals extraordinarily quickly because instead of direct molecule-to-molecule collisions, the vast majority of heat energy is mediated via very light, mobile conduction electrons. This is why there is a near-perfect correlation between metals’ thermal conductivity and their electrical conductivity.^[20] Conduction electrons imbue metals with their extraordinary conductivity because they are delocalized, i.e. not tied to a specific atom, and behave rather like a sort of “quantum gas” due to the effects of zero-point energy (for more on ZPE, see Note 1 below). Furthermore, electrons are relatively light with a rest mass only ¹/₁₈₃₆th that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Sir Isaac Newton once wrote with his third law of motion:

“Law #3: All forces occur in pairs, and these two forces

 are equal in magnitude and opposite in direction.”

However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner—because they are much less massive—heat energy is readily borne by mobile conduction electrons. Too, because they’re delocalized and very fast, kinetic heat energy conducts extremely quickly through metals with abundant conduction electrons.

[edit] The diffusion of heat energy: Black-body radiation

Fig. 5 The spectrum of black-body radiation has the form of a Planck curve. A 5500 K black body has a peak emittance wavelength of 527 nm. Compare the shape of this curve to that of a Maxwell distribution in Fig. 2 above.

Thermal radiation is a byproduct of the collisions arising from atoms’ various vibrational and rotational motions. These collisions cause atoms to emit thermal photons (known as black-body radiation). Photons are emitted anytime an electric charge is accelerated (as happens when two atoms’ electron clouds collide). Even individual molecules with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. In any bulk quantity of a substance at equilibrium, black-body photons are emitted across a range of wavelengths in a spectrum that has a bell curve-like shape called a Planck curve (see graph in Fig. 5 at right). The top of a Planck curve—the peak emittance wavelength—is located in particular part of the electromagnetic spectrum depending on the temperature of the black body. Substances at extreme cryogenic temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short gamma rays (see Table of thermodynamic temperatures, above).

Black-body radiation diffuses heat energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.

As established by the Stefan–Boltzmann law, the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black body at 824 K (just short of glowing dull red) emits 60 times the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, black-body radiation can be the principal mechanism by which heat energy escapes a system.

Fig. 6  Ice and water: two phases of the same substance

[edit] The heat of phase changes

The kinetic energy of particle motion is just one contributor to the total heat energy in a substance. The other is the potential energy of molecular bonds that can yet form in a substance as it cools (such as during condensing and freezing). This concept may be more easily grasped by visualizing it in the reverse direction: as the heat energy required to break molecular bonds (such as during evaporation and melting). These processes are known as phase transitions. The heat energy required for a phase transition is called latent heat. Anyone who has compared the 100 °C air from a hair dryer to 100 °C steam knows that the steam can quickly cause severe burns whereas the air can not. The burn occurs because a large amount of heat energy is liberated as steam condenses into liquid water on the skin. Even though heat energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic alloys exhibit no temperature change whatsoever while they undergo them (see Fig. 7, below right).

This phenomenon can be readily understood by examining one particular type of phase transition: the melting of a solid. When a solid melts, crystal lattice chemical bonds break apart; the substance has gone from what is known as a more ordered state to a less ordered state (see Topological order). In Fig. 7, the melting of ice is shown within the lower left box heading from blue to green. At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are—on average—at the maximum energy threshold the lattice bonds can withstand without breaking and jumping to a higher quantum energy state.

Fig. 7 Water’s temperature does not change during phase transitions as heat flows into our out of it. The total heat capacity of a mole of water in its liquid phase (the green line) is 7.5507 kJ.

Quantum transitions are a complete jump from one energy level to another; no intermediate values are possible. Consequently, when a substance is at its melting point, every joule of heat energy that is added to it only breaks bonds of a specific quantity of its atoms or molecules, releasing them from the crystal lattice and converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional heat energy can’t make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), heat energy must be removed from a substance.

As stated above, the heat energy required for a phase transition is called latent heat. In the specific cases of melting and freezing, it’s called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements.^[21] If the substance is one of the monatomic gases, (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole.^[22] Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the heat energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals’ ratios are even greater, typically in the range of 400 to 1200 times.^[23] And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a one-degree increase.^[24] Water’s sizable enthalpy of vaporization is why one’s skin can be burned so quickly as steam condenses on it (heading from red to green in Fig. 7 above). In the opposite direction, this is why one’s skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient wet-bulb temperature that is dependent on relative humidity).

[edit] Internal energy

The total kinetic energy of all particle motion — including that of conduction electrons — plus the potential energy of phase changes, plus zero-point energy comprise the internal energy of a substance, which is its total heat energy. The term internal energy mustn’t be confused with internal degrees of freedom. Whereas the internal degrees of freedom of molecules refers to one particular place where kinetic energy is bound, the internal energy of a substance is composed of heat energy in all its various forms.

Fig. 8 When many of the chemical elements, such as the noble gases and platinum-group metals, freeze to a solid — the most ordered state of matter — their crystal structures have a closest-packed arrangement. This yields the greatest possible packing density and the lowest energy state.

[edit] Heat energy at absolute zero

As a substance cools, many forms of heat energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions are liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic temperature decreases); the internal motions of molecules diminish (their internal temperature decreases); conduction electrons (if the substance is an electrical conductor) travel somewhat slower;^[25] and black-body radiation’s peak emittance wavelength increases (the photons’ energy decreases). When the particles of a substance are as close as possible to complete rest and retain only quantum mechanical motion, the substance is at the temperature of absolute zero (T=0).

Note that whereas absolute zero is the point of zero temperature, absolute zero is not necessarily the point at which a substance contains zero heat energy; one must be very precise with what one means by “heat energy.” Often, all the phase changes that can occur in a substance, will have occurred by the time it reaches absolute zero. However, this is not always the case. For instance, T=0 helium remains liquid at room pressure and must be under a pressure of at least 25 bar to crystallize. This is because helium’s heat of fusion — think of it as the energy required to melt helium ice — is so low (only 21 J mol⁻¹) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures.^[1] If helium is cooled while under at least 25 bar of pressure, this latent heat energy is liberated as the helium freezes while approaching absolute zero.

Further complicating matters is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars). These are known as solid-solid phase transitions wherein heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one. These complexities make for rather cumbersome blanket statements regarding the internal energy in T=0 substances. Regardless of pressure though, what can be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a closest-packed arrangement (see Fig. 8, above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy.^[26] One can also say that for all substances at any fixed pressure, absolute zero is the point of minimal enthalpy (a measure of heat content that takes pressure into consideration).^[27] Lastly, it is always true to say that all T=0 substances have zero kinetic heat energy.^[1] [17]

[edit] The origin of heat energy

Earth’s proximity to the Sun is why most everything near Earth’s surface is warm with a temperature substantially above absolute zero.^[28] The Sun constantly replenishes heat energy the Earth loses into space. Because matter is everywhere in the natural world, and because of the wide variety of heat diffusion mechanisms (one of which is black-body radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287 to 288 K (14 to 15 °C). The more an object’s or system’s temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.

[edit] History of thermodynamic temperature

• 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. He made the discovery while endeavoring to improve upon the air-thermometers in use at the time. 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.

• 1742:
  

  

  Anders Celsius
  Anders Celsius (1701 – 1744) created a “backwards” version of the modern Celsius temperature scale whereby zero represented the boiling point of water and 100 represented the melting point of ice. In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that ice’s melting point was effectively unaffected by pressure. He also determined with remarkable precision how water’s boiling point varied as a function of atmospheric pressure. He proposed that zero on his temperature scale (water’s boiling point) would be calibrated at the mean barometric pressure at mean sea level.

• 1744:
  

  

  Carolus Linnaeus
  Coincident with the death of Anders Celsius, the famous botanist Carolus Linnaeus (1707 – 1778) effectively reversed ^[29] Celsius’s scale upon receipt of his first thermometer featuring a scale where zero represented the melting point of ice and 100 represented water’s boiling point. His custom-made “linnaeus-thermometer,” for use in his greenhouses, was made by Daniel Ekström, Sweden’s leading maker of scientific instruments at the time. For the next 204 years, the scientific and thermometry communities world-wide referred to this scale as the “centigrade scale.” Temperatures on the centigrade scale were often reported simply as “degrees” or, when greater specificity was desired, “degrees centigrade.” The symbol for temperature values on this scale was °C (in several formats over the years). Because the term “centigrade” was also the French-language name for a unit of angular measurement (one-hundredth of a right angle) and had a similar connotation in other languages, the term “centesimal degree” was used when very precise, unambiguous language was required by international standards bodies such as the Bureau international des poids et mesures (BIPM). The 9th CGPM (Conférence générale des poids et mesures) and the CIPM (Comité international des poids et mesures) formally adopted “degree Celsius” (symbol: °C) in 1948.^[30]

• 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 75 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.

• Circa 1787: Notwithstanding the work of Guillaume Amontons 85 years earlier, Jacques Alexandre César Charles (1746 – 1823) is often credited with “discovering”, but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was V₁/T₁ = V₂/T₂.

• 1802: Joseph Louis Gay-Lussac (1778 – 1850) published work (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called Charles’s Law and is one of the gas laws. His are the first known formulas to used 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:
  

  

  Lord Kelvin
  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.

• 1859: William John Macquorn Rankine (1820 – 1872) proposed a thermodynamic temperature scale similar to William Thomson’s but which used the degree Fahrenheit for its unit increment. This absolute scale is known today as the Rankine thermodynamic temperature scale.

• 1877 - 1884:
  

  

  Ludwig Boltzmann
  Ludwig Boltzmann (1844 – 1906) made major contributions to thermodynamics through an understanding of the role that particle kinetics and black-body radiation played. His name is now attached to several of the formulas used today in thermodynamics.

• Circa 1930s: Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed that absolute zero was equivalent to −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 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. Also, both the CIPM (Comité international des poids et mesures, also known as the International Committee for Weights and Measures) and the CGPM formally adopted the name “Celsius” for the “degree Celsius” and the “Celsius temperature scale.” ^[30]

• 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, in combination with Resolution 3 of the 9th CGPM, had the effect of defined absolute zero as being precisely zero kelvin and −273.15 °C.

• 1967/1968: Resolution 3 of the 13th CGPM created the unit increment of thermodynamic temperature (as distinct from the scale) and gave it the name “kelvin”, symbol K, instead of “degree Kelvin”, symbol °K. In so doing, feeling it useful to explicitly define the magnitude of this new unit increment, 13th CGPM also decided in Resolution 4 that “The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.”

• 2005: The CIPM (Comité International des Poids et Mesures, also known as the International Committee for Weights and Measures) affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the Kelvin thermodynamic temperature scale would refer to water having an isotopic composition defined as being precisely equal to the nominal specification of VSMOW water.

[edit] Derivations of thermodynamic temperature

Strictly speaking, the temperature of a system is well-defined only if its particles (atoms, molecules, electrons, photons) are at equilibrium, and so obey a Boltzmann distribution (or its quantum mechanical counterpart). There are many possible scales of temperature, derived from a variety of observations of physical phenomena. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratios of temperatures, T₂/T₁, are the same in all absolute scales.

Loosely stated, temperature controls the flow of heat between two systems and the Universe, as we would expect any natural system, tends to progress so as to maximize entropy. Thus, we would expect there to be some relationship between temperature and entropy. In order to find this relationship let's first consider the relationship between heat, work and temperature. A heat engine is a device for converting heat into mechanical work and analysis of the Carnot heat engine provides the necessary relationships we seek. The work from a heat engine corresponds to the difference between the heat put into the system at the high temperature, q_H and the heat ejected at the low temperature, q_C. The efficiency is the work divided by the heat put into the system or:

(1)

where w_cy is the work done per cycle. We see that the efficiency depends only on q_C/q_H. Because q_C and q_H correspond to heat transfer at the temperatures T_C and T_H, respectively, q_C/q_H should be some function of these temperatures:

(2)

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T₁ and T₃ must have the same efficiency as one consisting of two cycles, one between T₁ and T₂, and the second between T₂ and T₃. This can only be the case if:

Consequently, we have:

where T₁ is the temperature of the triple point of water. So we can define the thermodynamic temperature as:

This temperature scale has the property that:

(3)

Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:

(4)

Notice that for T_C=0 K the efficiency is 100% and that efficiency becomes greater than 100% below 0 K. Since an efficiency greater than 100% violates the first law of thermodynamics, this requires that 0 K must be the minimum possible temperature. This makes intuitive sense; since temperature is the motion of particles, no system can, on average, have less motion than the minimum permitted by quantum physics. In fact, as of June 2006, the coldest man-made temperature was 450 pK.^[3] Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives:

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by:

(5)

where the subscript indicates a reversible process. The change of this state function around any cycle is zero, as is necessary for any state function. This function corresponds to the entropy of the system, which we described previously. We can rearranging Equation 5 to get a new definition for temperature in terms of entropy and heat:

For a system, where entropy S may be a function S(E) of its energy E, the thermodynamic temperature T is given by:

The reciprocal of the thermodynamic temperature is the rate of increase of entropy with energy.

[edit] See also

[edit] Notes

In the following notes, wherever numeric equalities are shown in concise form — such as 1.854 87(14) × 10⁴³ — the two digits between the parentheses denotes the uncertainty (the standard deviation at 68.27% confidence level) in the two least significant digits of the mantissa.

1. ^ ^{a b c d}
   

   

   Absolute zero’s relationship to zero-point energy
   While scientists are obtaining temperatures ever closer to absolute zero, they can not fully achieve a state of “zero” temperature. Even if scientists could remove all kinetic heat energy from matter, quantum mechanical zero-point energy (ZPE) causes ZPE-induced particle motion that can never be eliminated. ZPE is the product of all-pervasive energy fields in the vacuum between the fundamental particles of nature and is responsible for the Casimir effect and other phenomena. Click here for an excellent explanation of ZPE and here for an account of its effect on Bose–Einstein condensates of helium. Encyclopedia Britannica Online defines zero-point energy as the “vibrational energy that molecules retain even at the absolute zero of temperature.”
   
   Although absolute zero (T=0) is not a state of zero molecular motion, it is the point of zero temperature and, in accordance with the Boltzmann constant, is also the point of zero particle kinetic energy and zero kinetic velocity. To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two T=0 helium atoms in zero gravity are carefully positioned and observed to have an average separation of 620 pm between them (a gap of ten atomic diameters). It’s an “average” separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83 yoctokelvin (1 yK = 1 × 10^–24 K). This is done in a way that directs this atom’s velocity vector at the other atom. With 83 yK of kinetic energy between them, the 620-pm gap through their common barycenter would close at a rate of 719 pm/s and they would collide after 0.862 second. This is the same speed as shown in the Fig. 1 animation above. Before being given the kinetic kick, both T=0 atoms had zero kinetic energy and zero kinetic velocity because they could persist indefinitely in that state and relative orientation even though both were being jostled by ZPE. At T=0, no kinetic energy is available for transfer to other systems. The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of chaos, i.e., unpredictability, to rebound kinetics; a given collision is as likely to have less ZPE-induced kinetic energy after the collision as it is to have more. Its random nature is why ZPE has no net effect upon either the pressure or volume of any bulk quantity (a statistically significant quantity of particles) of T>4.2 K gases. In T=0 liquids and solids however, ZPE causes inter-atomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the real-world effects that ZPE has on substances can vary as one alters a thermodynamic system (helium for instance, won’t freeze due to ZPE unless under a pressure of at least 25 bar), zero-point energy is very much a form of heat energy and is properly included when tallying a substance’s internal energy.
   
   Note too that absolute zero serves as the baseline atop which thermodynamics and its equations are founded because they deal with the exchange of heat energy between “systems” (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPE-induced particle motion within a system that is at absolute zero but there can never be a net outflow of heat energy from such a system. Also, the peak emittance wavelength of black-body radiation shifts to infinity at absolute zero; indeed, a peak no longer exists and black-body photons can no longer escape. Due to the influence of zero-point energy however, virtual photons are still emitted at T=0. Such photons are called “virtual” because they can’t be intercepted and observed. Furthermore, this ZP radiation has a unique zero-point spectrum. However, even though a T=0 system emits ZP radiation, no net heat flow Q out of such a system can occur because if the surrounding environment is at a temperature greater than T=0, heat will flow inward, and if the surrounding environment is at T=0, there will be an equal flux of ZP radiation both inward and outward. A similar Q equilibrium exists at T=0 with the ZPE-induced “spontaneous” emission of photons (which is more properly called a stimulated emission in this context). The graph at upper right illustrates the relationship of absolute zero to zero-point energy. The graph also helps in the understanding of how zero-point energy got its name: it’s the vibrational energy matter retains at the “zero kelvin point.” Citation: Derivation of the classical electromagnetic zero-point radiation spectrum via a classical thermodynamic operation involving van der Waals forces, Daniel C. Cole, Physical Review A, Third Series 42, Number 4, 15 August 1990, Pg. 1847–1862.
2. ^ The cited emission wavelengths are for true black bodies in equilibrium. In this table, only the sun so qualifies. CODATA 2002 recommended value of 2.897 7685(51) × 10⁻³ m K used for Wien displacement law constant b.
3. ^ ^{a b} A record cold temperature of 450 ±80 pK in a Bose–Einstein condensate (BEC) of sodium atoms was achieved in 2003 by researchers at MIT. Citation: Cooling Bose–Einstein Condensates Below 500 Picokelvin, A. E. Leanhardt et al., Science 301, 12 Sept. 2003, Pg. 1515. It’s noteworthy that this record’s peak emittance black-body wavelength of 6,400 kilometers is roughly the radius of Earth.
4. ^ The peak emittance wavelength of 2.897 77 m is a frequency of 103.456 MHz
5. ^ Measurement was made in 2002 and has an uncertainty of ±3 kelvins. A 1989 measurement produced a value of 5777 ±2.5 K. Citation: Overview of the Sun (Chapter 1 lecture notes on Solar Physics by Division of Theoretical Physics, Dept. of Physical Sciences, University of Helsinki). Download paper (252 kB PDF)
6. ^ The 350 MK value is the maximum peak fusion fuel temperature in a thermonuclear weapon of the Teller–Ulam configuration (commonly known as a “hydrogen bomb”). Peak temperatures in Gadget-style fission bomb cores (commonly known as an “atomic bomb”) are in the range of 50 to 100 MK. Citation: Nuclear Weapons Frequently Asked Questions, 3.2.5 Matter At High Temperatures. Link to relevant Web page. All referenced data was compiled from publicly available sources.
7. ^ Peak temperature for a bulk quantity of matter was achieved by a pulsed-power machine used in fusion physics experiments. The term “bulk quantity” draws a distinction from collisions in particle accelerators wherein high “temperature” applies only to the debris from two subatomic particles or nuclei at any given instant. The >2 GK temperature was achieved over a period of about ten nanoseconds during “shot Z1137.” In fact, the iron and manganese ions in the plasma averaged 3.58 ±0.41 GK (309 ±35 keV) for 3 ns (ns 112 through 115). Citation: Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over 2 × 10⁹ Kelvin, M. G. Haines et al., Physical Review Letters 96, Issue 7, id. 075003. Link to Sandia’s news release.
8. ^ Core temperature of a high–mass (>8–11 solar masses) star after it leaves the main sequence on the Hertzsprung–Russell diagram and begins the alpha process (which lasts one day) of fusing silicon–28 into heavier elements in the following steps: sulfur–32 → argon–36 → calcium–40 → titanium–44 → chromium–48 → iron–52 → nickel–56. Within minutes of finishing the sequence, the star explodes as a Type II supernova. Citation: Stellar Evolution: The Life and Death of Our Luminous Neighbors (by Arthur Holland and Mark Williams of the University of Michigan). Link to Web site. More informative links can be found here, and here, and a concise treatise on stars by NASA is here.
9. ^ Based on a computer model that predicted a peak internal temperature of 30 MeV (350 GK) during the merger of a binary neutron star system (which produces a gamma–ray burst). The neutron stars in the model were 1.2 and 1.6 solar masses respectively, were roughly 20 km in diameter, and were orbiting around their barycenter (common center of mass) at about 390 Hz during the last several milliseconds before they completely merged. The 350 GK portion was a small volume located at the pair’s developing common core and varied from roughly 1 to 7 km across over a time span of around 5 ms. Imagine two city-sized objects of unimaginable density orbiting each other at the same frequency as the G4 musical note (the 28th white key on a piano). It’s also noteworthy that at 350 GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass (m) 5% greater than its rest mass (m₀).  Citation: Torus Formation in Neutron Star Mergers and Well-Localized Short Gamma-Ray Bursts, R. Oechslin et al. of Max Planck Institute for Astrophysics., arXiv:astro-ph/0507099 v2, 22 Feb. 2006.  Download paper (725 kB PDF) (from Cornell University Library’s arXiv.org server). To view a browser-based summary of the research, click here.
10. ^ Results of research by Stefan Bathe using the PHENIX detector on the Relativistic Heavy Ion Collider at Brookhaven National Laboratory in Upton, New York, U.S.A.  Bathe has studied gold-gold, deuteron-gold, and proton-proton collisions to test the theory of quantum chromodynamics, the theory of the strong force that holds atomic nuclei together.  Link to news release.
11. ^ Citation: How do physicists study particles? by CERN.
12. ^ The Planck frequency equals 1.854 87(14) × 10⁴³ Hz (which is the reciprocal of one Planck time). Photons at the Planck frequency have a wavelength of one Planck length. The Planck temperature of 1.416 79(11) × 10³² K equates to a calculated b /T = λ_max wavelength of 2.045 31(16) × 10⁻²⁶ nm. However, the actual peak emittance wavelength quantizes to the Planck length of 1.616 24(12) × 10⁻²⁶ nm.
13. ^ At non-relativistic temperatures of less than about 30 GK, classical mechanics are sufficient to calculate the velocity of particles. At 30 GK, individual neutrons (the constituent of neutron stars and one of the few materials in the universe with temperatures in this range) have a 1.0042 γ (gamma or Lorentz factor). Thus, the classic Newtonian formula for kinetic energy is in error less than half a percent for temperatures less than 30 GK.
14. ^ ^{a b} Two conventions are used in science when counting degrees of freedom. The one used here counts each dimension or mode of movement as a single degree of freedom. The other convention counts each of these as two degrees (back and forth, up and down, etc.). Thus, the monatomic gases such as helium have three degrees of freedom according to the convention used here and six in the other.
15. ^ Even room–temperature air has an average molecular translational speed (not vector-isolated velocity) of 1822 km/hour. This is relatively fast for something the size of a molecule considering there are roughly 2.42 × 10¹⁶ of them crowded into a single cubic millimeter. Assumptions: Average molecular weight of wet air = 28.838 g/mol and T = 296.15 K. Assumption’s primary variables: An altitude of 194 meters above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C (40.85% relative humidity), and 760 mmHg (101.325 kPa) sea level–corrected barometric pressure.
16. ^ Citation: Adiabatic Cooling of Cesium to 700 nK in an Optical Lattice, A. Kastberg et al., Physical Review Letters 74, No. 9, 27 Feb. 1995, Pg. 1542. It’s noteworthy that a record cold temperature of 450 pK in a Bose–Einstein condensate of sodium atoms (achieved by A. E. Leanhardt et al. of MIT) equates to an average vector-isolated atom velocity of 0.4 mm/s and an average atom speed of 0.7 mm/s.
17. ^ ^{a b} The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature as follows:
    
    

    

    where…

    = vector-isolated mean velocity of translational particle motion in m/s

    k_B (Boltzmann constant) = 1.380 6505(24) × 10⁻²³ J/K

    T = thermodynamic temperature in kelvins

    m = molecular mass of substance in kg/particle

    Note that molecular mass, m, in kg/particle is the quotient of a substance’s molar mass (also known as atomic weight, atomic mass, relative atomic mass, and unified atomic mass units) in g/mol or daltons divided by 6.022 1415(10) × 10²⁶ (which is the Avogadro constant times one thousand). For diatomic molecules such as H₂, N₂, and O₂, multiply atomic weight by two before plugging it into the above formula.
    
    The mean speed (not vector-isolated velocity) of an atom or molecule along any arbitrary path is calculated as follows:
    
    

    

    where…

    = mean speed of translational particle motion in m/s
    

    Note that the mean energy of the translational motions of a substance’s constituent particles correlates to their mean speed, not velocity. Thus, substituting for v in the classic formula for kinetic energy, E_k = ¹/₂m • v ² produces precisely the same value as does E_mean = 3/2k_BT (as shown in the section titled The nature of kinetic energy, translational motion, and temperature).
    
    Note too that the Boltzmann constant and its related formulas establish that absolute zero is the point of both zero kinetic energy of particle motion and zero kinetic velocity (see also Note 1 above).
18. ^ Nitrogen’s C_vH (100 kPa, 20 °C) = 20.8 J mol^–1 K^–1 vs. the monatomic gases, which equal 12.4717 J mol^–1 K^–1. Citations: W.H. Freeman’s Physical Chemistry, Part 3: Change (422 kB PDF, here), Exercise 21.20b, Pg. 787. Also Georgia State University’s Molar Specific Heats of Gases.
19. ^ Diamond is a notable exception. Due to the highly quantized modes of phonon vibration occurring in its rigid crystal lattice, not only does diamond have exceptionally poor specific heat capacity, it also has exceptionally high thermal conductivity.
20. ^ Correlation is 752 (W m⁻¹ K⁻¹) / (MS•cm), σ = 81, through a 7:1 range in conductivity. Value and standard deviation based on data for Ag, Cu, Au, Al, Ca, Be, Mg, Rh, Ir, Zn, Co, Ni, Os, Fe, Pa, Pt, and Sn. Citation: Data from CRC Handbook of Chemistry and Physics, 1st Student Edition and this link to Web Elements’ home page.
21. ^ Water’s enthalpy of fusion is 6.0095 kJ mol⁻¹ K⁻¹ (0 °C, 101.325 kPa). Citation: Water Structure and Behavior, Enthalpy of fusion, (0 °C, 101.325 kPa) (by London South Bank University). Link to Web site. The only metals with enthalpies of fusion not in the range of 6–30 J mol⁻¹ K⁻¹ are (on the high side): Ta, W, and Re; and (on the low side) most of the group 1 (alkaline) metals plus Ga, In, Hg, Tl, Pb, and Np. Citation: This link to Web Elements’ home page.
22. ^ Xenon value citation: This link to WebElements’ xenon data (available values range from 2.3 to 3.1 kJ mol⁻¹). It is also noteworthy that helium’s heat of fusion of only 0.021 kJ mol⁻¹ is so weak of a bonding force that zero-point energy prevents helium from freezing unless it is under a pressure of at least 25 atmospheres.
23. ^ Citation: Data from CRC Handbook of Chemistry and Physics, 1st Student Edition and this link to Web Elements’ home page.
24. ^ H₂O specific heat capacity, C_p = 0.075327 kJ mol⁻¹ K⁻¹ (25 °C); Enthalpy of fusion = 6.0095 kJ mol⁻¹ (0 °C, 101.325 kPa); Enthalpy of vaporization (liquid) = 40.657 kJ mol⁻¹ (100 °C). Citation: Water Structure and Behavior (by London South Bank University). Link to Web site.
25. ^  Mobile conduction electrons are delocalized, i.e. not tied to a specific atom, and behave rather like a sort of “quantum gas” due to the effects of zero-point energy. Consequently, even at absolute zero, conduction electrons still move between atoms at the Fermi velocity of about 1.6 × 10⁶ m/s. Kinetic heat energy adds to this speed and also causes delocalized electrons to travel farther away from the nuclei.
26. ^ No other crystal structure can exceed the 74.048% packing density of a closest-packed arrangement.  The two regular crystal lattices found in nature that have this density are hexagonal close packed (HCP) and face-centered cubic (FCC). These regular lattices are at the lowest possible energy state. Diamond is a closest-packed structure with an FCC crystal lattice. Note too that crystalline chemical compounds, which usually are comprised of atoms of different sizes, are not in the strictest sense considered as “closest-packed structures” since the term applies only to spheres of equal size. However, any crystalline compound with one of the regular lattices listed here is in its lowest possible energy state. One such compound is the common mineral known as magnesium aluminum spinel (MgAl₂O₄). It has a face-centered cubic crystal lattice and no change in pressure can produce a lattice with a lower energy state.
27. ^ Nearly half of the 92 naturally occurring chemical elements featuring a closest-packed crystal lattice can also freeze under a vacuum (technically, under their T=0 vapor pressures). This set includes beryllium, osmium, neon, and iridium (but excludes helium). At T=0, these crystals not only have minimal internal energy (retaining only that due to ZPE), but also have zero enthalpy under a perfect vacuum.
28. ^ The deepest ocean depths (3 to 10 km) are no colder than about 274.7 – 275.7 K (1.5 – 2.5 °C). Even the world-record cold surface temperature established on July 21, 1983 at Vostok Station, Antarctica is 184 K (a reported value of −89.2 °C). The residual heat of gravitational contraction left over from earth’s formation, tidal friction, and the decay of radioisotopes in earth’s core provide insufficient heat to maintain earth’s surface, oceans, and atmosphere “substantially above” absolute zero in this context. Also, the qualification of “most-everything” provides for the exclusion of lava flows, which derive their temperature from these deep-earth sources of heat.
29. ^ Citations: Thermodynamics-information.net, A Brief History of Temperature Measurement and; Uppsala University (Sweden), Linnaeus’ thermometer
30. ^ ^{a b} According to The Oxford English Dictionary (OED), the term “Celsius’s thermometer” had been used at least as early as 1797. Further, the term “The Celsius or Centigrade thermometer” was again used in reference to a particular type of thermometer at least as early as 1850. The OED also cites this 1928 reporting of a temperature: “My altitude was about 5,800 metres, the temperature was 28° Celsius.” However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards the terminology used throughout the history of science. According to several writings of Dr. Terry Quinn CBE FRS, Director of the BIPM (1988 – 2004), including Temperature Scales from the early days of thermometry to the 21st century (148 kB PDF, here) as well as Temperature (2nd Edition / 1990 / Academic Press / 0125696817), the term Celsius in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM wasn’t even aware that “degree Celsius” was in sporadic, non-scientific use before that time. It’s also noteworthy that the twelve-volume, 1933 edition of OED didn’t even have a listing for the word Celsius (but did have listings for both centigrade and centesimal in the context of temperature measurement). The 1948 adoption of Celsius accomplished three objectives:
    

    1) All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.
    2) Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius’s name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name centigrade could continue to be used and would simultaneously inherit an intuitive association with the new name.
    3) The new name eliminated the ambiguity of the term “centigrade,” freeing it to refer exclusively to the French-language name for the unit of angular measurement.

[edit] External links

• Kinetic Molecular Theory of Gases. An excellent explanation (with interactive animations) of the kinetic motion of molecules and how it affects matter. By David N. Blauch, Department of Chemistry, Davidson College.

• Zero Point Energy and Zero Point Field. A Web site with in-depth explanations of a variety of quantum effects. By Bernard Haisch, of Calphysics Institute.