Thermodynamic temperature

From The UCSC Wikipedia Trust Project

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 and no heat energy remains in a substance.

Contents 1 Overview 2 Table of thermodynamic temperatures 3 The relationship of temperature, motions, conduction, and heat energy 3.1 The nature of kinetic energy, translational motion, and temperature 3.2 The high speeds of translational motion 3.3 The internal motions of molecules and specific heat 3.4 The diffusion of heat energy: Entropy, phonons, and mobile conduction electrons 3.5 The diffusion of heat energy: Black-body radiation 3.6 Heat energy and absolute zero 3.7 The heat of phase changes 3.8 Internal energy 3.9 The origin of heat energy 4 History of thermodynamic temperature 5 Derivations of thermodynamic temperature 6 See also 7 Notes 8 External links

Overview

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) and zero heat energy remains in a substance.^[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.

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 [2] of black-body photons
Absolute zero (precisely by definition)	0 K	−273.15 °C	∞ [1]
One millikelvin (precisely by definition)	0.001 K	−273.149 °C	2.897 77 meters (Radio, FM band) [3]
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 lampB	2500 K	≈2200 °C	1160 nm (Near infrared)C
Sun’s visible surfaceD [4]	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 [5]	350 MK	350 million °C	8.3 × 10−3 nm (Gamma rays)
Sandia National Labs’ Z machine E [6]	2 GK	2 billion °C	1.4 × 10−3 nm (Gamma rays)F
Core of a high–mass star on its last day E [7]	3 GK	3 billion °C	1 × 10−3 nm (Gamma rays)
Merging binary neutron star system E [8]	350 GK	350 billion °C	8 × 10−6 nm (Gamma rays)
Relativistic Heavy Ion Collider E [9]	1 TK	1 trillion °C	3 × 10−6 nm (Gamma rays)
Universe 5.391 × 10−44 s after the Big Bang E	1.417 × 1032 K	1.417 × 1032 °C	1.616 × 10−26 nm (Planck frequency) [10]

^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” is the measure of the kinetic energy resulting from the 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.^[11] 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.

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.^[12] 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 relationship between thermodynamic temperature and the kinetic energy of the translational motion of a given particle is given by the Boltzmann constant (symbol: K_b). The Boltzmann constant also relates the thermodynamic temperature of a bulk quantity of a substance to the mean energy of the translational motions of its constituent particles 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 particles, 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^[13] 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.^[14]  Formulas for calculating the velocity and speed of translational motion are given in the following footnote.^[15]

The internal motions of molecules and specific heat

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.^[12] Heat 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 heat 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 the temperature of their translational motions, 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, 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.^[16] Larger, more complex molecules can have dozens of internal degrees of freedom.

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

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^[17] 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.^[18] 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.

The diffusion of heat energy: Black-body radiation

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.

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 heat energy 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.

Heat energy and absolute zero

As a substance cools, many forms of heat energy and their related effects simultaneously decrease in magnitude: the translational motions of atoms diminish, both the internal and translational motions of molecules diminish, conduction electrons (if the substance is an electrical conductor) travel somewhat slower,^[19] and black-body radiation’s wavelength increases (the photons’ energy decreases). When no more heat energy remains in a substance and the molecules are as close as possible to complete rest (retaining only quantum mechanical motion), the substance is at absolute zero.^[1]

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. 6, at 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. 6, 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. 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 causes the bonds of a specific quantity of its atoms or molecules to release from the