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Thermodynamic temperature

2007 Schools Wikipedia Selection. Related subjects: General Physics

   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.

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.
   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.

   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—the coldest possible
   temperature—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].

Table of thermodynamic temperatures

   The full range of the thermodynamic temperature scale and some notable
   points along it are shown in the table below.
   kelvin Celsius Peak emittance
   wavelength of
   black-body photons
   Absolute zero

   (precisely by definition)
   0 K −273.15 °C    ∞
   Coldest measured
   temperature^  450 pK –273.149 999 999 55 °C 6,400 kilometers
   One millikelvin

   (precisely by definition)
   0.001 K −273.149 °C 2.897 77 meters
   (Radio, FM band)
   Water’s triple point

   (precisely by definition)
   273.16 K 0.01 °C 10,608.3 nm
   (Long wavelength I.R.)
   Water’s boiling point ^A 373.1339 K 99.9839 °C 7766.03 nm
   (Mid wavelength I.R.)
   Incandescent lamp^B 2500 K ≈2200 °C 1160 nm
   (Near infrared)^C
   Sun’s visible surface^D 5778 K 5505 °C 501.5 nm
   ( Green light)
   Lightning bolt’s
   channel ^E 28,000 K 28,000 °C 100 nm
   (Far Ultraviolet light)
   Sun’s core ^E 16 MK 16 million °C 0.18 nm ( X-rays)
   Thermonuclear weapon
   (peak temperature)^E 350 MK 350 million °C 8.3 × 10^−3 nm
   ( Gamma rays)
   Sandia National Labs’
   Z machine ^E 2 GK 2 billion °C 1.4 × 10^−3 nm
   (Gamma rays)^F
   Core of a high–mass
   star on its last day ^E 3 GK 3 billion °C 1 × 10^−3 nm
   (Gamma rays)
   Merging binary neutron
   star system ^E 350 GK 350 billion °C 8 × 10^−6 nm
   (Gamma rays)
   Relativistic Heavy
   Ion Collider ^E 1 TK 1 trillion °C 3 × 10^−6 nm
   (Gamma rays)
   CERN’s proton vs.
   nucleus collisions ^E 10 TK 10 trillion °C 3 × 10^−7 nm
   (Gamma rays)
   Universe 5.391 × 10^−44 s
   after the Big Bang ^E 1.417 × 10^32 K 1.417 × 10^32 °C 1.616 × 10^−26
   nm
   (Planck frequency)

   ^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.

The relationship of temperature, motions, conduction, and heat energy

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/[2]m • v^ 2. 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.
   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. 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[B]T

                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.

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 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.  Formulas for
   calculating the velocity and speed of translational motion are given in
   the following footnote.

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.
   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. 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. Larger, more complex molecules can have
   dozens of internal degrees of freedom.

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.
   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
   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.
   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/[1836]th 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.

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

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

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.
   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.

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; 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. 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. 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). Lastly, it is always true to say that all T=0
   substances have zero kinetic heat energy.^

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.
   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.

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
       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
       Carolus Linnaeus
       Coincident with the death of Anders Celsius, the famous botanist
       Carolus Linnaeus (1707 – 1778) effectively reversed^  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.

     * 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 V[1]/T[1] = V[2]/T[2].

     * 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 behaviour 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
       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
       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.”

     * 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 renamed the unit increment
       of thermodynamic temperature “kelvin”, symbol K, replacing “degree
       absolute”, symbol °K. Further, feeling it useful to more explicitly
       define the magnitude of the unit increment, the 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.

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, T[2]/T[1], 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:

          \textrm{efficiency} = \frac {w_{cy}}{q_H} = \frac{q_H-q_C}{q_H}
          = 1 - \frac{q_C}{q_H} (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:

          \frac{q_C}{q_H} = f(T_H,T_C) (2)

   Carnot's theorem states that all reversible engines operating between
   the same heat reservoirs are equally efficient. Thus, a heat engine
   operating between T[1] and T[3] must have the same efficiency as one
   consisting of two cycles, one between T[1] and T[2], and the second
   between T[2] and T[3]. This can only be the case if:

          f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} =
          f(T_1,T_2)f(T_2,T_3)

   Consequently, we have:

          f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot
          f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}

   where T[1] is the temperature of the triple point of water. So we can
   define the thermodynamic temperature as:

          T = 273.16 \cdot f(T_1,T) \!

   This temperature scale has the property that:

          \frac{q_C}{q_H} = f(T_H,T_C) = \frac{T_C}{T_H} (3)

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

          \textrm{efficiency} = 1 - \frac{q_C}{q_H} = 1 - \frac{T_C}{T_H}
          (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. Subtracting the right hand side of Equation 4 from the middle
   portion and rearranging gives:

          \frac {q_H}{T_H} - \frac{q_C}{T_C} = 0

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

          dS = \frac {dq_\mathrm{rev}}{T} (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:

          T = \frac{dq_\mathrm{rev}}{dS}

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

          \frac{1}{T} = \frac{dS}{dE}

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

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