Thermodynamic temperature

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This article deals with thermodynamic temperature and its underpinnings in heat energy and kinetic motions. It is intended to be suitable for high school-level students taking advanced science classes. Whereas it wouldn’t be untrue to state that “temperature is the inverse of the derivative of entropy with respect to internal energy, and absolute zero is the point where this quantity becomes zero,” every effort has been made here to explain complex thermodynamic issues while 1) using plain-speak, and 2) without sacrificing scientific rigor.

Thermodynamic temperature (previously called absolute temperature) is the absolute measure of temperature. Temperature arises from the random microscopic 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 the translational motions of its constituent particles.

The thermodynamic temperature scale’s null point, absolute zero, is the theoretical point where all molecular motion ceases and they are at complete rest (except for quantum mechanical motion).

While scientists can get ever closer to absolute zero, they can not fully achieve a state of “zero” heat energy.  Even if scientists did remove all the heat energy from matter that theoretically could be removed (another definition of absolute zero), the motion-inducing effect of quantum mechanical zero-point energy would still remain.  [Click here] for an account of zero-point energy’s 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.”  Since quantum mechanical zero-point energy is an intrinsic, all-pervasive phenomenon in all matter, absolute zero serves as the baseline atop which thermodynamics and its equations are founded. The graph at 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 energy matter retains at the “zero kelvin point.” 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.

The unit “kelvin” and its scale are, by international agreement, defined by two points: absolute zero, and the triple point of specially prepared ocean water. Absolute zero is defined as being precisely 0 K and –273.15 °C (–459.67 °F). 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).

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 The cited emission wavelengths are for true black bodies.  In this table, only the sun’s emissions are similar to a true black body.  CODATA 2002 recommended value of 2.897 7685(51) × 10–3 m K used for Wien displacement law constant b. of black-body photons
Absolute zero (precisely by definition)	0 K	–273.15 °C	∞ (No emission)
One millikelvin	0.001 K	–273.149 °C	2.89777 meters (Radio, FM band) The peak emittance wavelength of 2.897 77 m is a frequency of 103.456 MHz
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 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)]	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 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 50–100 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.	350 MK	350 million °C	8.3 × 10–3 nm (Gamma rays)
Sandia National Labs’ Z machine E 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, vol. 96, Issue 7, id. 075003. [Link to Sandia’s news release.]	2 GK	2 billion °C	1.4 × 10–3 nm (Gamma rays)F
Core of a high–mass star on its last day E 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 to a week) 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 (which decays in a two-step process to iron–56) and shortly later explodes as a 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]. Other informative links are [here,] and [here,] and a concise treatise on stars by NASA [here].	3 GK	3 billion °C	1 × 10–3 nm (Gamma rays)
Merging binary neutron star system E Based on a computer-model that predicted a peak internal temperature of 30 MeV during the merger of a binary neutron star system (which produces a gamma–ray burst).  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) here] (from Cornell University Library’s arXiv.org server). To view a browser-based summary of the research, [click here].  It’s noteworthy that at 350 GK, the average neutron is traveling at 30% the speed of light and has a relativistic mass (m) 5% greater than its rest mass (m0).	350 GK	350 billion °C	8 × 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) 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 = λmax 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.

^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 invalid 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 invalid precision in the Celsius value.
^E The 273.15 K difference between K and °C is ignored to avoid invalid 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, kinetic energy 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. At non-relativistic temperatures (less than about 30 GK), the relationship of kinetic energy, mass, and velocity is given by the formula E_k = ¹/₂m • v ². Accordingly, for simple particles like atoms and electrons, those 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 temperature of any bulk quantity of a substance arises from a specific kind of kinetic energy known as translational motion. Translational motions are ordinary, whole-body movements in 3D space whereby the particles move about and exchange energy in collisions (like rubber balls in a vigorously shaken container). These simple movements in the three X, Y, and Z–axis dimensions of space means the particles have the three spatial degrees of freedom. Translational motions are but one form of heat energy and are what give gases their pressure and, at normal earthly temperatures or greater, the vast majority of their volume.

The High Speeds of Atomic 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 fastThe average molecular translational speed (not vector-isolated velocity) of room–temperature air is approximately 1822 km/hour. This is relatively fast for 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 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 mm–Hg sea level–corrected barometric pressure. 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.Citation: Adiabatic Cooling of Cesium to 700 nK in an Optical Lattice, A. Kastberg et al., Physical Review Letters, Vol. 74 / No. 9, 27 Feb. 1995, Pg. 1542.

The Conduction and Diffusion of Heat Energy

Translational motion transfers momentum from particle to particle with each collision and this facilitates the conduction or diffusion of kinetic energy throughout the bulk of a substance. These collisions also cause atoms to emit thermal photons (known as black-body radiation). Black-body photons constitute yet another mechanism that helps diffuse heat energy as they are absorbed by neighboring particles, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment. 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 diffuses through metals extraordinarily quickly because, instead of direct molecule-to-molecule collisions, the vast majority of their heat energy is mediated, i.e. conducted, between molecules via mobile conduction electrons. This is why there is a near-perfect correlation between metals’ thermal conductivity and their electrical conductivity.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.

The Internal Motions of Molecules and Specific Heat

There are other forms of heat energy besides translational motions.  Molecules also have various internal vibrational and rotational degrees of freedom. This is because molecules are complex objects; they are a population of atoms that can move about within a molecule in different ways. Heat energy is stored in these internal motions which gives molecules an internal temperature. Even though these motions are called “internal,” the external portions of molecules still move—rather like the jiggling of a 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 internal and translational kinetic energies (temperatures) simultaneously diminish in equal proportions. The heat energy stored internally in molecules does not contribute to the temperature of a bulk quantity of a substance. This is because any kinetic energy that is, at a given instant, bound up in internal motions is not at that same instant contributing to the molecules’ translational motions. Individual molecules with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. Since the internal temperature of the molecules in any bulk quantity of a substance are, on average, equal to the temperature of their translational motions, the distinction is usually of interest only in the detailed study of certain thermodynamic phenomenon 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 (specific number of particles) with only a modest temperature change. This property is known as a substance’s specific heat. High specific heat capacity arises because a substance’s molecules possess a greater number of degrees of freedom. Water has six active degrees of freedom, the maximum available. Not surprisingly, water gas molecules (steam molecules) have twiceUnder constant pressure (C_p) steam has 1.803 times: 37.47 J mol^{–1 K–1 (100 °C, 101.325 kPa) v.s. 20.7862 J mol–1 K–1 for the monatomic gases. Under constant volume (C_v) steam has 2.247 times: 28.03 J mol–1 K–1 (100 °C, 101.325 kPa) v.s. 12.4717 J mol–1 K–1 for the monatomic gases. This is a 2.025 average for C_p and C_v. Citation: ''Water Structure and Behavior, Specific heat capacity (by London South Bank University). [Link to Web site.]} the specific heat capacity per mole as do the monatomic gases such as helium and argon that consist of individual atoms and which move only within the three degrees of freedom comprising translational motion.

Heat Energy and Absolute Zero

As a substance cools, all 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; 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, electrons still move between atoms at the Fermi velocity of about 1.6 × 10⁶ m/s. and black-body radiation’s wavelength increases (the photons’ energy decreases). When no more heat energy remains in a substance the molecules are at complete rest (except for quantum mechanical motion), the substance is at absolute zero.

The Heat of Phase Changes

The kinetic energy of 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). These processes are known as phase transitions. Anyone who has compared the 100 °C air from a hair dryer to 100 °C steam knows that steam condensing on skin can 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 and compounds exhibit no temperature change whatsoever while they undergo them (see graph 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. In the graph, the melting of water 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. So during melting, every joule of heat energy that is added to a substance only causes the bonds of a specific quantity of atoms or molecules to jump to the next quantum state; no kinetic energy is added to translational motion (which gives a bulk quantity of any substance its 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.

The heat energy required for a phase transition is called latent heat. In the specific case of melting, 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.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.} 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.Xenon value citation: [This link] to WebElements’ xenon data (available values range from 2.3 to 3.1 kJ mole^–1).  It is also noteworthy that helium’s heat of fusion of only 0.021 kJ mole^–1 is so weak of a bonding force that zero-point energy prevents helium from freezing unless it is at a pressure of at least 25 atmospheres. Relatively speaking, phase transitions are 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.Citation: Data from CRC Handbook of Chemistry and Physics, 1st Student Edition and [this link] to Web Elements’ home page. 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.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.] This is why steam can burn flesh so easily.

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 deepest ocean depths are never colder than 277 K (4 °C). Even the world-record cold surface temperature established on July 21, 1983 at Vostok 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. The Sun constantly replenishes heat energy lost to space. Because matter is absolutely 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 287–288 K (14–15 °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.

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:

[\textrm = \frac } = \frac = 1 - \frac] (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 = 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 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:

[f(T_1,T_3) = \frac = \frac = f(T_1,T_2)f(T_2,T_3)]

Consequently, we have:

[f(T_2,T_3) = \frac = \frac]

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 = f(T_H,T_C) = \frac] (3)

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

[\textrm = 1 - \frac = 1 - \frac] (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.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, Charles M. Schroeder et al., Science, Vol. 301, 12 Sept. 2003, Pg. 1515. Note: This record-setting temperature was too impractical for inclusion in the table of temperatures due to its unwieldily equivalent Celsius temperature of –273.149 999 999 55 °C. The record’s peak emittance black-body wavelength is 6,400 kilometers (which is roughly the radius of Earth). Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives:

[\frac - \frac = 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 }] (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}]

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

[\frac = \frac]

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

See also

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• Absolute zero
• Celsius
• Fermi velocity
• Heat
• ITS-90

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• Ideal gas law
• Kelvin
• Kinetic energy
• Quantum mechanics
• Rankine

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• Temperature
• Thermodynamic equations
• Triple point
• Universal gas constant
• Zero-point energy

|}

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

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.

 

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