Physical quantity of hot and cold
Temperature
|
---|
|
Common symbols
| T
|
---|
SI unit
| K
|
---|
Other units
| °C
,
°F
,
°R
,
°Rø
,
°Re
,
°N
,
°D
,
°L
,
°W
|
---|
Intensive
?
| Yes
|
---|
Derivations from
other quantities
| ,
|
---|
Dimension
| |
---|
Temperature
is a
physical quantity
that quantitatively expresses the attribute of hotness or coldness. Temperature is
measured
with a
thermometer
. It reflects the average
kinetic energy
of the vibrating and colliding
atoms
making up a substance.
Thermometers are calibrated in various
temperature scales
that historically have relied on various reference points and thermometric substances for definition. The most common scales are the
Celsius
scale with the unit symbol °C (formerly called
centigrade
), the
Fahrenheit
scale (°F), and the
Kelvin
scale (K), the latter being used predominantly for scientific purposes. The kelvin is one of the seven base units in the
International System of Units
(SI).
Absolute zero
, i.e., zero kelvin or ?273.15 °C, is the lowest point in the
thermodynamic temperature
scale. Experimentally, it can be approached very closely but not actually reached, as recognized in the
third law of thermodynamics
. It would be impossible to extract energy as heat from a body at that temperature.
Temperature is important in all fields of
natural science
, including
physics
,
chemistry
,
Earth science
,
astronomy
,
medicine
,
biology
,
ecology
,
material science
,
metallurgy
,
mechanical engineering
and
geography
as well as most aspects of daily life.
Effects
[
edit
]
Many physical processes are related to temperature; some of them are given below:
- the physical properties of materials including the
phase
(
solid
,
liquid
,
gaseous
or
plasma
),
density
,
solubility
,
vapor pressure
,
electrical conductivity
,
hardness
,
wear resistance
,
thermal conductivity
,
corrosion resistance
, strength
- the rate and extent to which
chemical reactions
occur
[1]
- the amount and properties of
thermal radiation
emitted from the surface of an object
- air temperature
affects all living organisms
- the
speed of sound
, which in a gas is proportional to the square root of the absolute temperature
[2]
Scales
[
edit
]
Temperature scales need two values for definition: the point chosen as zero degrees and the magnitudes of the incremental unit of temperature.
The
Celsius
scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that developed historically, which led to its zero point
0 °C
being defined as the freezing point of
water
, and
100 °C
as the boiling point of water, both at
atmospheric pressure
at sea level. It was called a centigrade scale because of the 100-degree interval.
[3]
Since the standardization of the
kelvin
in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though numerically the scales differ by an exact offset of 273.15.
The
Fahrenheit
scale is in common use in the United States. Water freezes at
32 °F
and boils at
212 °F
at sea-level atmospheric pressure.
Absolute zero
[
edit
]
At the
absolute zero
of temperature, no energy can be removed from matter as heat, a fact expressed in the
third law of thermodynamics
. At this temperature, matter contains no macroscopic thermal energy, but still has quantum-mechanical
zero-point energy
as predicted by the
uncertainty principle
, although this does not enter into the definition of absolute temperature. Experimentally, absolute zero can be approached only very closely; it can never be reached (the lowest temperature attained by experiment is 38 pK).
[4]
Theoretically, in a body at a temperature of absolute zero, all classical motion of its particles has ceased and they are at complete rest in this classical sense. Absolute zero, defined as
0 K
, is exactly equal to
?273.15 °C
, or
?459.67 °F
.
Absolute scales
[
edit
]
Referring to the
Boltzmann constant
, to the
Maxwell?Boltzmann distribution
, and to the Boltzmann
statistical mechanical definition
of
entropy
, as distinct from the Gibbs definition,
[5]
for independently moving microscopic particles, disregarding interparticle potential energy, by international agreement, a temperature scale is defined and said to be absolute because it is independent of the characteristics of particular thermometric substances and
thermometer
mechanisms. Apart from absolute zero, it does not have a reference temperature. It is known as the
Kelvin scale
, widely used in science and technology. The kelvin (the unit name is spelled with a
lower-case
'k') is the unit of temperature in the
International System of Units
(SI). The temperature of a body in a state of thermodynamic equilibrium is always positive relative to absolute zero.
Besides the internationally agreed Kelvin scale, there is also a
thermodynamic temperature scale
, invented by
Lord Kelvin
, also with its numerical zero at the absolute zero of temperature, but directly relating to purely macroscopic
thermodynamic
concepts, including the macroscopic
entropy
, though microscopically referable to the Gibbs statistical mechanical definition of entropy for the
canonical ensemble
, that takes interparticle potential energy into account, as well as independent particle motion so that it can account for measurements of temperatures near absolute zero.
[5]
This scale has a reference temperature at the
triple point
of water, the numerical value of which is defined by measurements using the aforementioned internationally agreed Kelvin scale.
Kelvin scale
[
edit
]
Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the
physicist who first defined it
. It is an
absolute
scale. Its numerical zero point,
0 K
, is at the absolute zero of temperature. Since May, 2019, the kelvin has been defined
through particle kinetic theory
, and statistical mechanics. In the
International System of Units
(SI), the magnitude of the kelvin is defined in terms of the
Boltzmann constant
, the value of which is defined as fixed by international convention.
[6]
[7]
Statistical mechanical
versus
thermodynamic temperature scales
[
edit
]
Since May 2019, the magnitude of the kelvin is defined in relation to microscopic phenomena, characterized in terms of statistical mechanics. Previously, but since 1954, the International System of Units defined a scale and unit for the kelvin as a
thermodynamic temperature
, by using the reliably reproducible temperature of the
triple point
of water as a second reference point, the first reference point being
0 K
at absolute zero.
[
citation needed
]
Historically, the temperature of the triple point of water was defined as exactly 273.16 K. Today it is an empirically measured quantity. The freezing point of water at sea-level atmospheric pressure occurs at very close to
273.15 K
(
0 °C
).
Classification of scales
[
edit
]
There are various kinds of temperature scale. It may be convenient to classify them as empirically and theoretically based. Empirical temperature scales are historically older, while theoretically based scales arose in the middle of the nineteenth century.
[8]
[9]
Empirical scales
[
edit
]
Empirically based temperature scales rely directly on measurements of simple macroscopic physical properties of materials. For example, the length of a column of mercury, confined in a glass-walled capillary tube, is dependent largely on temperature and is the basis of the very useful mercury-in-glass thermometer. Such scales are valid only within convenient ranges of temperature. For example, above the boiling point of
mercury
, a mercury-in-glass thermometer is impracticable. Most materials expand with temperature increase, but some materials, such as water, contract with temperature increase over some specific range, and then they are hardly useful as thermometric materials. A material is of no use as a thermometer near one of its phase-change temperatures, for example, its boiling-point.
In spite of these limitations, most generally used practical thermometers are of the empirically based kind. Especially, it was used for
calorimetry
, which contributed greatly to the discovery of thermodynamics. Nevertheless, empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Empirically based thermometers, beyond their base as simple direct measurements of ordinary physical properties of thermometric materials, can be re-calibrated, by use of theoretical physical reasoning, and this can extend their range of adequacy.
Theoretical scales
[
edit
]
Theoretically based temperature scales are based directly on theoretical arguments, especially those of kinetic theory and thermodynamics. They are more or less ideally realized in practically feasible physical devices and materials. Theoretically based temperature scales are used to provide calibrating standards for practical empirically based thermometers.
Microscopic statistical mechanical scale
[
edit
]
In physics, the internationally agreed conventional temperature scale is called the Kelvin scale. It is calibrated through the internationally agreed and prescribed value of the Boltzmann constant,
[6]
[7]
referring to motions of microscopic particles, such as atoms, molecules, and electrons, constituent in the body whose temperature is to be measured. In contrast with the thermodynamic temperature scale invented by Kelvin, the presently conventional Kelvin temperature is not defined through comparison with the temperature of a reference state of a standard body, nor in terms of macroscopic thermodynamics.
Apart from the absolute zero of temperature, the Kelvin temperature of a body in a state of internal thermodynamic equilibrium is defined by measurements of suitably chosen of its physical properties, such as have precisely known theoretical explanations in terms of the
Boltzmann constant
.
[
citation needed
]
That constant refers to chosen kinds of motion of microscopic particles in the constitution of the body. In those kinds of motion, the particles move individually, without mutual interaction. Such motions are typically interrupted by inter-particle collisions, but for temperature measurement, the motions are chosen so that, between collisions, the non-interactive segments of their trajectories are known to be accessible to accurate measurement. For this purpose, interparticle potential energy is disregarded.
In an
ideal gas
, and in other theoretically understood bodies, the Kelvin temperature is defined to be proportional to the average kinetic energy of non-interactively moving microscopic particles, which can be measured by suitable techniques. The proportionality constant is a simple multiple of the Boltzmann constant. If molecules, atoms, or electrons
[10]
[11]
are emitted from material and their velocities are measured, the spectrum of their velocities often nearly obeys a theoretical law called the
Maxwell?Boltzmann distribution
, which gives a well-founded measurement of temperatures for which the law holds.
[12]
There have not yet been successful experiments of this same kind that directly use the
Fermi?Dirac distribution
for thermometry, but perhaps that will be achieved in the future.
[13]
The speed of sound in a gas can be calculated theoretically from the gas's
molecular
character, temperature, pressure, and the Boltzmann constant. For a gas of known molecular character and pressure, this provides a relation between temperature and the Boltzmann constant. Those quantities can be known or measured more precisely than can the thermodynamic variables that define the state of a sample of water at its triple point. Consequently, taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.
[14]
Measurement of the spectrum of electromagnetic radiation from an ideal three-dimensional
black body
can provide an accurate temperature measurement because the frequency of maximum spectral radiance of black-body radiation is directly proportional to the temperature of the black body; this is known as
Wien's displacement law
and has a theoretical explanation in
Planck's law
and the
Bose?Einstein law
.
Measurement of the spectrum of noise-power produced by an electrical resistor can also provide accurate temperature measurement. The resistor has two terminals and is in effect a one-dimensional body. The Bose-Einstein law for this case indicates that the noise-power is directly proportional to the temperature of the resistor and to the value of its resistance and to the noise bandwidth. In a given frequency band, the noise-power has equal contributions from every frequency and is called
Johnson noise
. If the value of the resistance is known then the temperature can be found.
[15]
[16]
Macroscopic thermodynamic scale
[
edit
]
Historically, till May 2019, the definition of the Kelvin scale was that invented by Kelvin, based on a ratio of quantities of energy in processes in an ideal Carnot engine, entirely in terms of macroscopic thermodynamics.
[
citation needed
]
That Carnot engine was to work between two temperatures, that of the body whose temperature was to be measured, and a reference, that of a body at the temperature of the triple point of water. Then the reference temperature, that of the triple point, was defined to be exactly
273.16 K
. Since May 2019, that value has not been fixed by definition but is to be measured through microscopic phenomena, involving the Boltzmann constant, as described above. The microscopic statistical mechanical definition does not have a reference temperature.
Ideal gas
[
edit
]
A material on which a macroscopically defined temperature scale may be based is the
ideal gas
. The pressure exerted by a fixed volume and mass of an ideal gas is directly proportional to its temperature. Some natural gases show so nearly ideal properties over suitable temperature range that they can be used for thermometry; this was important during the development of thermodynamics and is still of practical importance today.
[17]
[18]
The ideal gas thermometer is, however, not theoretically perfect for thermodynamics. This is because the
entropy of an ideal gas
at its absolute zero of temperature is not a positive semi-definite quantity, which puts the gas in violation of the third law of thermodynamics. In contrast to real materials, the ideal gas does not liquefy or solidify, no matter how cold it is. Alternatively thinking, the ideal gas law, refers to the limit of infinitely high temperature and zero pressure; these conditions guarantee non-interactive motions of the constituent molecules.
[19]
[20]
[21]
Kinetic theory approach
[
edit
]
The magnitude of the kelvin is now defined in terms of kinetic theory, derived from the value of the
Boltzmann constant
.
Kinetic theory
provides a microscopic account of temperature for some bodies of material, especially gases, based on macroscopic systems' being composed of many microscopic particles, such as molecules and
ions
of various species, the particles of a species being all alike. It explains macroscopic phenomena through the
classical mechanics
of the microscopic particles. The
equipartition theorem
of kinetic theory asserts that each classical
degree of freedom
of a freely moving particle has an average kinetic energy of
k
B
T
/2
where
k
B
denotes the
Boltzmann constant
.
[
citation needed
]
The translational motion of the particle has three degrees of freedom, so that, except at very low temperatures where quantum effects predominate, the average translational kinetic energy of a freely moving particle in a system with temperature
T
will be
3
k
B
T
/2
.
Molecules, such as oxygen (O
2
), have more
degrees of freedom
than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase of temperature due to an increase in the average translational kinetic energy of the molecules. Heating will also cause, through
equipartitioning
, the energy associated with vibrational and rotational modes to increase. Thus a
diatomic
gas will require more energy input to increase its temperature by a certain amount, i.e. it will have a greater
heat capacity
than a monatomic gas.
As noted above, the speed of sound in a gas can be calculated from the gas's molecular character, temperature, pressure, and the Boltzmann constant. Taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.
[14]
It is possible to measure the average kinetic energy of constituent
microscopic
particles if they are allowed to escape from the bulk of the system, through a small hole in the containing wall. The spectrum of velocities has to be measured, and the average calculated from that. It is not necessarily the case that the particles that escape and are measured have the same velocity distribution as the particles that remain in the bulk of the system, but sometimes a good sample is possible.
Thermodynamic approach
[
edit
]
Temperature is one of the principal quantities in the study of
thermodynamics
. Formerly, the magnitude of the kelvin was defined in thermodynamic terms, but nowadays, as mentioned above, it is defined in terms of kinetic theory.
The thermodynamic temperature is said to be
absolute
for two reasons. One is that its formal character is independent of the properties of particular materials. The other reason is that its zero is, in a sense, absolute, in that it indicates absence of microscopic classical motion of the constituent particles of matter, so that they have a limiting specific heat of zero for zero temperature, according to the third law of thermodynamics. Nevertheless, a thermodynamic temperature does in fact have a definite numerical value that has been arbitrarily chosen by tradition and is dependent on the property of particular materials; it is simply less arbitrary than relative "degrees" scales such as
Celsius
and
Fahrenheit
. Being an absolute scale with one fixed point (zero), there is only one degree of freedom left to arbitrary choice, rather than two as in relative scales. For the Kelvin scale since May 2019, by international convention, the choice has been made to use knowledge of modes of operation of various thermometric devices, relying on microscopic kinetic theories about molecular motion. The numerical scale is settled by a conventional definition of the value of the
Boltzmann constant
, which relates macroscopic temperature to average microscopic kinetic energy of particles such as molecules. Its numerical value is arbitrary, and an alternate, less widely used absolute temperature scale exists called the
Rankine scale
, made to be aligned with the Fahrenheit scale as
Kelvin
is with Celsius.
The thermodynamic definition of temperature is due to Kelvin. It is framed in terms of an idealized device called a
Carnot engine
, imagined to run in a fictive continuous
cycle of successive processes
that traverse a cycle of states of its working body. The engine takes in a quantity of heat
Q
1
from a hot reservoir and passes out a lesser quantity of waste heat
Q
2
< 0
to a cold reservoir. The net heat energy absorbed by the working body is passed, as thermodynamic work, to a work reservoir, and is considered to be the output of the engine. The cycle is imagined to run so slowly that at each point of the cycle the working body is in a state of thermodynamic equilibrium. The successive processes of the cycle are thus imagined to run reversibly with no
entropy production
. Then the quantity of entropy taken in from the hot reservoir when the working body is heated is equal to that passed to the cold reservoir when the working body is cooled. Then the absolute or thermodynamic temperatures,
T
1
and
T
2
, of the reservoirs are defined such that
[22]
| | (1)
|
The zeroth law of thermodynamics allows this definition to be used to measure the absolute or thermodynamic temperature of an arbitrary body of interest, by making the other heat reservoir have the same temperature as the body of interest.
Kelvin's original work postulating absolute temperature was published in 1848. It was based on the work of Carnot, before the formulation of the first law of thermodynamics. Carnot had no sound understanding of heat and no specific concept of entropy. He wrote of 'caloric' and said that all the caloric that passed from the hot reservoir was passed into the cold reservoir. Kelvin wrote in his 1848 paper that his scale was absolute in the sense that it was defined "independently of the properties of any particular kind of matter". His definitive publication, which sets out the definition just stated, was printed in 1853, a paper read in 1851.
[23]
[24]
[25]
[26]
Numerical details were formerly settled by making one of the heat reservoirs a cell at the triple point of water, which was defined to have an absolute temperature of 273.16 K.
[27]
Nowadays, the numerical value is instead obtained from measurement through the microscopic statistical mechanical international definition, as above.
Intensive variability
[
edit
]
In thermodynamic terms, temperature is an
intensive variable
because it is equal to a
differential coefficient
of one
extensive variable
with respect to another, for a given body. It thus has the
dimensions
of a
ratio
of two extensive variables. In thermodynamics, two bodies are often considered as connected by contact with a common wall, which has some specific permeability properties. Such specific permeability can be referred to a specific intensive variable. An example is a diathermic wall that is permeable only to heat; the intensive variable for this case is temperature. When the two bodies have been connected through the specifically permeable wall for a very long time, and have settled to a permanent steady state, the relevant intensive variables are equal in the two bodies; for a diathermal wall, this statement is sometimes called the zeroth law of thermodynamics.
[28]
[29]
[30]
In particular, when the body is described by stating its
internal energy
U
, an extensive variable, as a function of its
entropy
S
, also an extensive variable, and other state variables
V
,
N
, with
U
=
U
(
S
,
V
,
N
), then the temperature is equal to the
partial derivative
of the internal energy with respect to the entropy:
[29]
[30]
[31]
| | (2)
|
Likewise, when the body is described by stating its entropy
S
as a function of its internal energy
U
, and other state variables
V
,
N
, with
S
=
S
(
U
,
V
,
N
)
, then the reciprocal of the temperature is equal to the partial derivative of the entropy with respect to the internal energy:
[29]
[31]
[32]
| | (3)
|
The above definition, equation (1), of the absolute temperature, is due to Kelvin. It refers to systems closed to the transfer of matter and has a special emphasis on directly experimental procedures. A presentation of thermodynamics by Gibbs starts at a more abstract level and deals with systems open to the transfer of matter; in this development of thermodynamics, the equations (2) and (3) above are actually alternative definitions of temperature.
[33]
Local thermodynamic equilibrium
[
edit
]
Real-world bodies are often not in thermodynamic equilibrium and not homogeneous. For the study by methods of classical irreversible thermodynamics, a body is usually spatially and temporally divided conceptually into 'cells' of small size. If classical thermodynamic equilibrium conditions for matter are fulfilled to good approximation in such a 'cell', then it is homogeneous and a temperature exists for it. If this is so for every 'cell' of the body, then
local thermodynamic equilibrium
is said to prevail throughout the body.
[34]
[35]
[36]
[37]
[38]
It makes good sense, for example, to say of the extensive variable
U
, or of the extensive variable
S
, that it has a density per unit volume or a quantity per unit mass of the system, but it makes no sense to speak of the density of temperature per unit volume or quantity of temperature per unit mass of the system. On the other hand, it makes no sense to speak of the internal energy at a point, while when local thermodynamic equilibrium prevails, it makes good sense to speak of the temperature at a point. Consequently, the temperature can vary from point to point in a medium that is not in global thermodynamic equilibrium, but in which there is local thermodynamic equilibrium.
Thus, when local thermodynamic equilibrium prevails in a body, the temperature can be regarded as a spatially varying local property in that body, and this is because the temperature is an intensive variable.
Basic theory
[
edit
]
Temperature is a measure of a
quality
of a state of a material.
[39]
The quality may be regarded as a more abstract entity than any particular temperature scale that measures it, and is called
hotness
by some writers.
[40]
[41]
[42]
The quality of hotness refers to the state of material only in a particular locality, and in general, apart from bodies held in a steady state of thermodynamic equilibrium, hotness varies from place to place. It is not necessarily the case that a material in a particular place is in a state that is steady and nearly homogeneous enough to allow it to have a well-defined hotness or temperature. Hotness may be represented abstractly as a one-dimensional
manifold
. Every valid temperature scale has its own one-to-one map into the hotness manifold.
[43]
[44]
When two systems in thermal contact are at the same temperature no heat transfers between them. When a temperature difference does exist heat flows spontaneously from the warmer system to the colder system until they are in
thermal equilibrium
. Such heat transfer occurs by conduction or by thermal radiation.
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
Experimental physicists, for example
Galileo
and
Newton
,
[53]
found that there are indefinitely many
empirical temperature scales
. Nevertheless, the
zeroth law of thermodynamics
says that they all measure the same quality. This means that for a body in its own state of internal thermodynamic equilibrium, every correctly calibrated thermometer, of whatever kind, that measures the temperature of the body, records one and the same temperature. For a body that is not in its own state of internal thermodynamic equilibrium, different thermometers can record different temperatures, depending respectively on the mechanisms of operation of the thermometers.
Bodies in thermodynamic equilibrium
[
edit
]
For experimental physics, hotness means that, when comparing any two given bodies in their respective separate
thermodynamic equilibria
, any two suitably given empirical thermometers with numerical scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature.
[54]
This does not require the two thermometers to have a linear relation between their numerical scale readings, but it does require that the relation between their numerical readings shall be
strictly monotonic
.
[55]
[56]
A definite sense of greater hotness can be had, independently of
calorimetry
, of thermodynamics, and of properties of particular materials, from
Wien's displacement law
of
thermal radiation
: the temperature of a bath of
thermal radiation
is
proportional
, by a universal constant, to the frequency of the maximum of its
frequency spectrum
; this frequency is always positive, but can have values that
tend to zero
. Thermal radiation is initially defined for a cavity in thermodynamic equilibrium. These physical facts justify a mathematical statement that hotness exists on an ordered one-dimensional
manifold
. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.
[8]
[43]
[44]
[57]
[58]
Except for a system undergoing a
first-order
phase change
such as the melting of ice, as a closed system receives heat, without a change in its volume and without a change in external force fields acting on it, its temperature rises. For a system undergoing such a phase change so slowly that departure from thermodynamic equilibrium can be neglected, its temperature remains constant as the system is supplied with
latent heat
. Conversely, a loss of heat from a closed system, without phase change, without change of volume, and without a change in external force fields acting on it, decreases its temperature.
[59]
Bodies in a steady state but not in thermodynamic equilibrium
[
edit
]
While for bodies in their own thermodynamic equilibrium states, the notion of temperature requires that all empirical thermometers must agree as to which of two bodies is the hotter or that they are at the same temperature, this requirement is not safe for bodies that are in steady states though not in thermodynamic equilibrium. It can then well be that different empirical thermometers disagree about which is hotter, and if this is so, then at least one of the bodies does not have a well-defined absolute thermodynamic temperature. Nevertheless, any one given body and any one suitable empirical thermometer can still support notions of empirical, non-absolute, hotness, and temperature, for a suitable range of processes. This is a matter for study in
non-equilibrium thermodynamics
.
[
citation needed
]
Bodies not in a steady state
[
edit
]
When a body is not in a steady-state, then the notion of temperature becomes even less safe than for a body in a steady state not in thermodynamic equilibrium. This is also a matter for study in
non-equilibrium thermodynamics
.
Thermodynamic equilibrium axiomatics
[
edit
]
For the axiomatic treatment of thermodynamic equilibrium, since the 1930s, it has become customary to refer to a
zeroth law of thermodynamics
. The customarily stated minimalist version of such a law postulates only that all bodies, which when thermally connected would be in thermal equilibrium, should be said to have the same temperature by definition, but by itself does not establish temperature as a quantity expressed as a real number on a scale. A more physically informative version of such a law views empirical temperature as a chart on a hotness manifold.
[43]
[58]
[60]
While the zeroth law permits the definitions of many different empirical scales of temperature, the
second law of thermodynamics
selects the definition of a single preferred,
absolute temperature
, unique up to an arbitrary scale factor, whence called the
thermodynamic temperature
.
[8]
[43]
[61]
[62]
[63]
[64]
If
internal energy
is considered as a function of the volume and entropy of a homogeneous system in thermodynamic equilibrium, thermodynamic absolute temperature appears as the partial derivative of
internal energy
with respect the
entropy
at constant volume. Its natural, intrinsic origin or null point is
absolute zero
at which the entropy of any system is at a minimum. Although this is the lowest absolute temperature described by the model, the
third law of thermodynamics
postulates that absolute zero cannot be attained by any physical system.
Heat capacity
[
edit
]
When an energy transfer to or from a body is only as heat, the state of the body changes. Depending on the surroundings and the walls separating them from the body, various changes are possible in the body. They include chemical reactions, increase of pressure, increase of temperature and phase change. For each kind of change under specified conditions, the heat capacity is the ratio of the quantity of heat transferred to the magnitude of the change.
[65]
For example, if the change is an increase in temperature at constant volume, with no phase change and no chemical change, then the temperature of the body rises and its pressure increases. The quantity of heat transferred,
Δ
Q
, divided by the observed temperature change,
Δ
T
, is the body's
heat capacity
at constant volume:
If heat capacity is measured for a well-defined
amount of substance
, the
specific heat
is the measure of the heat required to increase the temperature of such a unit quantity by one unit of temperature. For example, raising the temperature of water by one kelvin (equal to one degree Celsius) requires 4186
joules
per
kilogram
(J/kg).
Measurement
[
edit
]
Temperature measurement
using modern scientific
thermometers
and temperature scales goes back at least as far as the early 18th century, when
Daniel Gabriel Fahrenheit
adapted a thermometer (switching to
mercury
) and a scale both developed by
Ole Christensen Rømer
. Fahrenheit's scale is still in use in the United States for non-scientific applications.
Temperature is measured with
thermometers
that may be
calibrated
to a variety of
temperature scales
. In most of the world (except for
Belize
,
Myanmar
,
Liberia
and the
United States
), the Celsius scale is used for most temperature measuring purposes. Most scientists measure temperature using the Celsius scale and thermodynamic temperature using the
Kelvin
scale, which is the Celsius scale offset so that its null point is
0 K
=
?273.15 °C
, or
absolute zero
. Many engineering fields in the US, notably high-tech and US federal specifications (civil and military), also use the Kelvin and Celsius scales. Other engineering fields in the US also rely upon the
Rankine scale
(a shifted Fahrenheit scale) when working in thermodynamic-related disciplines such as
combustion
.
Units
[
edit
]
The basic unit of temperature in the
International System of Units
(SI) is the
kelvin
. It has the symbol K.
For everyday applications, it is often convenient to use the Celsius scale, in which
0 °C
corresponds very closely to the
freezing point
of water and
100 °C
is its
boiling point
at sea level. Because liquid droplets commonly exist in clouds at sub-zero temperatures,
0 °C
is better defined as the melting point of ice. In this scale, a temperature difference of 1 degree Celsius is the same as a
1
kelvin
increment, but the scale is offset by the temperature at which ice melts (
273.15 K
).
By international agreement,
[66]
until May 2019, the Kelvin and Celsius scales were defined by two fixing points:
absolute zero
and the
triple point
of
Vienna Standard Mean Ocean Water
, which is water specially prepared with a specified blend of hydrogen and oxygen isotopes. Absolute zero was defined as precisely
0 K
and
?273.15 °C
. It is the temperature at which all classical translational motion of the particles comprising matter ceases and they are at complete rest in the classical model. Quantum-mechanically, however, zero-point motion remains and has an associated energy, the
zero-point energy
. Matter is in its
ground state
,
[67]
and contains no
thermal energy
. The temperatures
273.16 K
and
0.01 °C
were defined as those of the triple point of water. This definition served the following purposes: it fixed the magnitude of the kelvin as being precisely 1 part in 273.16 parts of the difference between absolute zero and the triple point of water; it established that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it established the difference between the null points of these scales as being
273.15 K
(
0 K
=
?273.15 °C
and
273.16 K
=
0.01 °C
). Since 2019, there has been a new definition based on the Boltzmann constant,
[68]
but the scales are scarcely changed.
In the United States, the Fahrenheit scale is the most widely used. On this scale the freezing point of water corresponds to
32 °F
and the boiling point to
212 °F
. The Rankine scale, still used in fields of chemical engineering in the US, is an absolute scale based on the Fahrenheit increment.
Historical scales
[
edit
]
The following temperature scales are in use or have historically been used for measuring temperature:
Plasma physics
[
edit
]
The field of
plasma physics
deals with phenomena of
electromagnetic
nature that involve very high temperatures. It is customary to express temperature as energy in a unit related to the
electronvolt
or kiloelectronvolt (
eV/
k
B
or keV/
k
B
). The corresponding energy, which is
dimensionally distinct
from temperature, is then calculated as the product of the
Boltzmann constant
and temperature,
. Then, 1
eV/
k
B
is
11
605
K
. In the study of
QCD matter
one routinely encounters temperatures of the order of a few hundred MeV/
k
B
, equivalent to about
10
12
K
.
Continuous or discrete
[
edit
]
When one measures the variation of temperature across a region of space or time, do the temperature measurements turn out to be continuous or discrete? There is a widely held misconception that such temperature measurements must always be continuous.
[69]
This misconception partly originates from the historical view associated with the continuity of classical
physical quantities
, which states that physical quantities must assume every intermediate value between a starting value and a final value.
[69]
[70]
However, the classical picture is only true in the cases where temperature is measured in a system that is in
equilibrium
, that is, temperature may not be continuous outside these conditions.
[69]
For systems outside equilibrium, such as at interfaces between materials (e.g., a metal/non-metal interface or a liquid-vapour interface) temperature measurements may show steep discontinuities in time and space.
[69]
For instance, Fang and Ward were some of the first authors to successfully report temperature discontinuities of as much as 7.8 K at the surface of evaporating water droplets.
[71]
This was reported at inter-molecular scales, or at the scale of the
mean free path
of molecules which is typically of the order of a few micrometers in gases
[72]
at room temperature. Generally speaking, temperature discontinuities are considered to be norms rather than exceptions in cases of interfacial heat transfer.
[73]
This is due to the abrupt change in the vibrational or
thermal properties of the materials
across such interfaces which prevent instantaneous transfer of heat and the establishment of thermal equilibrium (a prerequisite for having a uniform equilibrium temperature across the interface).
[74]
[75]
Further, temperature measurements at the macro-scale (typical observational scale) may be too coarse-grained as they average out the microscopic thermal information based on the scale of the representative sample volume of the control system, and thus it is likely that temperature discontinuities at the micro-scale may be overlooked in such averages.
[69]
Such an averaging may even produce incorrect or misleading results in many cases of temperature measurements, even at macro-scales, and thus it is prudent that one examines the micro-physical information carefully before averaging out or smoothing out any potential temperature discontinuities in a system as such discontinuities cannot always be averaged or smoothed out.
[69]
[76]
Temperature discontiuities, rather than merely being anomalies, have actually substantially improved our understanding and predictive abilities pertaining to heat transfer at small scales.
[69]
[73]
[74]
[75]
[76]
Theoretical foundation
[
edit
]
Historically, there are several scientific approaches to the explanation of temperature: the classical thermodynamic description based on macroscopic empirical variables that can be measured in a laboratory; the
kinetic theory of gases
which relates the macroscopic description to the probability distribution of the energy of motion of gas particles; and a microscopic explanation based on
statistical physics
and
quantum mechanics
. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics and temperature.
[77]
Statistical physics provides a deeper understanding by describing the atomic behavior of matter and derives macroscopic properties from statistical averages of microscopic states, including both classical and quantum states. In the fundamental physical description, the temperature may be measured directly in units of energy. However, in the practical systems of measurement for science, technology, and commerce, such as the modern
metric system
of units, the macroscopic and the microscopic descriptions are interrelated by the
Boltzmann constant
, a proportionality factor that scales temperature to the microscopic mean kinetic energy.
The microscopic description in
statistical mechanics
is based on a model that analyzes a system into its fundamental particles of matter or into a set of classical or
quantum-mechanical
oscillators and considers the system as a
statistical ensemble
of
microstates
. As a collection of classical material particles, the temperature is a measure of the mean energy of motion, called translational
kinetic energy
, of the particles, whether in solids, liquids, gases, or plasmas. The kinetic energy, a concept of
classical mechanics
, is half the
mass
of a particle times its
speed
squared. In this mechanical interpretation of thermal motion, the kinetic energies of material particles may reside in the velocity of the particles of their translational or vibrational motion or in the inertia of their rotational modes. In monatomic
perfect gases
and, approximately, in most gas and in simple metals, the temperature is a measure of the mean particle translational kinetic energy, 3/2
k
B
T
. It also determines the probability distribution function of energy. In condensed matter, and particularly in solids, this purely mechanical description is often less useful and the oscillator model provides a better description to account for quantum mechanical phenomena. Temperature determines the statistical occupation of the microstates of the ensemble. The microscopic definition of temperature is only meaningful in the
thermodynamic limit
, meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model.
Kinetic energy is also considered as a component of
thermal energy
. The thermal energy may be partitioned into independent components attributed to the
degrees of freedom
of the particles or to the modes of oscillators in a
thermodynamic system
. In general, the number of these degrees of freedom that are available for the
equipartitioning
of energy depends on the temperature, i.e. the energy region of the interactions under consideration. For solids, the thermal energy is associated primarily with the
vibrations
of its atoms or molecules about their equilibrium position. In an
ideal monatomic gas
, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems,
vibrational
and
rotational
motions also contribute degrees of freedom.
Kinetic theory of gases
[
edit
]
Maxwell
and
Boltzmann
developed a
kinetic theory
that yields a fundamental understanding of temperature in gases.
[78]
This theory also explains the
ideal gas
law and the observed heat capacity of
monatomic
(or
'noble'
) gases.
[79]
[80]
[81]
The
ideal gas law
is based on observed
empirical relationships
between pressure (
p
), volume (
V
), and temperature (
T
), and was recognized long before the kinetic theory of gases was developed (see
Boyle's
and
Charles's
laws). The ideal gas law states:
[82]
where
n
is the number of
moles
of gas and
R
=
8.314
462
618
... J?mol
?1
?K
?1
[83]
is the
gas constant
.
This relationship gives us our first hint that there is an
absolute zero
on the temperature scale, because it only holds if the temperature is measured on an absolute scale such as Kelvin's. The
ideal gas law
allows one to measure temperature on this absolute scale using the
gas thermometer
. The temperature in kelvins can be defined as the pressure in pascals of one mole of gas in a container of one cubic meter, divided by the gas constant.
Although it is not a particularly convenient device, the
gas thermometer
provides an essential theoretical basis by which all thermometers can be calibrated. As a practical matter, it is not possible to use a gas thermometer to measure absolute zero temperature since the gases condense into a liquid long before the temperature reaches zero. It is possible, however, to extrapolate to absolute zero by using the ideal gas law, as shown in the figure.
The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational
kinetic energy
. Using a sophisticated symmetry argument,
[84]
Boltzmann
deduced what is now called the
Maxwell?Boltzmann probability distribution
function for the velocity of particles in an ideal gas. From that
probability distribution
function, the average
kinetic energy
(per particle) of a
monatomic
ideal gas
is
[80]
[85]
where the
Boltzmann constant
k
B
is the
ideal gas constant
divided by the
Avogadro number
, and
is the
root-mean-square speed
.
[86]
This direct proportionality between temperature and mean molecular kinetic energy is a special case of the
equipartition theorem
, and holds only in the
classical
limit of a
perfect gas
. It does not hold exactly for most substances.
Zeroth law of thermodynamics
[
edit
]
When two otherwise isolated bodies are connected together by a rigid physical path impermeable to matter, there is the spontaneous transfer of energy as heat from the hotter to the colder of them. Eventually, they reach a state of mutual
thermal equilibrium
, in which heat transfer has ceased, and the bodies' respective state variables have settled to become unchanging.
[87]
[88]
[89]
One statement of the
zeroth law of thermodynamics
is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other.
[90]
[91]
[92]
This statement helps to define temperature but it does not, by itself, complete the definition. An empirical temperature is a numerical scale for the hotness of a thermodynamic system. Such hotness may be defined as existing on a
one-dimensional manifold
, stretching between hot and cold. Sometimes the zeroth law is stated to include the existence of a unique universal hotness manifold, and of numerical scales on it, so as to provide a complete definition of empirical temperature.
[60]
To be suitable for empirical thermometry, a material must have a monotonic relation between hotness and some easily measured state variable, such as pressure or volume, when all other relevant coordinates are fixed. An exceptionally suitable system is the
ideal gas
, which can provide a temperature scale that matches the absolute Kelvin scale. The Kelvin scale is defined on the basis of the second law of thermodynamics.
Second law of thermodynamics
[
edit
]
As an alternative to considering or defining the zeroth law of thermodynamics, it was the historical development in thermodynamics to define temperature in terms of the
second law of thermodynamics
which deals with
entropy
.
[
citation needed
]
The second law states that any process will result in either no change or a net increase in the entropy of the universe. This can be understood in terms of probability.
For example, in a series of coin tosses, a perfectly ordered system would be one in which either every toss comes up heads or every toss comes up tails. This means the outcome is always 100% the same result. In contrast, many mixed (
disordered
) outcomes are possible, and their number increases with each toss. Eventually, the combinations of ~50% heads and ~50% tails dominate, and obtaining an outcome significantly different from 50/50 becomes increasingly unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.
As temperature governs the transfer of heat between two systems and the universe tends to progress toward a maximum of entropy, it is expected that there is some relationship between temperature and entropy. A
heat engine
is a device for converting thermal energy into mechanical energy, resulting in the performance of work. An analysis of the
Carnot heat engine
provides the necessary relationships. According to energy conservation and energy being a
state function
that does not change over a full cycle, the work from a heat engine over a full cycle is equal to the net heat, i.e. the sum of the heat put into the system at high temperature,
q
H
> 0, and the waste heat given off at the low temperature,
q
C
< 0.
[93]
The efficiency is the work divided by the heat input:
| | (4)
|
where
w
cy
is the work done per cycle. 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:
| | (5)
|
Carnot's theorem
states that all reversible engines operating between the same heat reservoirs are equally efficient.
[
citation needed
]
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
which implies
Since the first function is independent of
T
2
, this temperature must cancel on the right side, meaning
f
(
T
1
,
T
3
) is of the form
g
(
T
1
)/
g
(
T
3
) (i.e.
f
(
T
1
,
T
3
)
=
f
(
T
1
,
T
2
)
f
(
T
2
,
T
3
)
=
g
(
T
1
)/
g
(
T
2
) ·
g
(
T
2
)/
g
(
T
3
)
=
g
(
T
1
)/
g
(
T
3
))
, where
g
is a function of a single temperature. A temperature scale can now be chosen with the property that
| | (6)
|
Substituting (6) back into (4) gives a relationship for the efficiency in terms of temperature:
| | (7)
|
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 implies that 0
K is the minimum possible temperature. In fact, the lowest temperature ever obtained in a macroscopic system was 20
nK, which was achieved in 1995 at NIST. Subtracting the right hand side of (5) from the middle portion and rearranging gives
[22]
[93]
where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function,
S
, whose change characteristically vanishes for a complete cycle if it is defined by
| | (8)
|
where the subscript indicates a reversible process. This function corresponds to the entropy of the system, which was described previously. Rearranging (8) gives a formula for temperature in terms of fictive infinitesimal quasi-reversible elements of entropy and heat:
| | (9)
|
For a constant-volume system where entropy
S
(
E
) is a function of its energy
E
, d
E
= d
q
rev
and (9) gives
| | (10)
|
i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy at constant volume.
Definition from statistical mechanics
[
edit
]
Statistical mechanics
defines temperature based on a system's fundamental degrees of freedom. Eq.(10) is the defining relation of temperature, where the entropy
is defined (up to a constant) by the logarithm of the number of
microstates
of the system in the given macrostate (as specified in the
microcanonical ensemble
):
where
is the Boltzmann constant and
W
is the number of microstates with the energy
E
of the system (degeneracy).
When two systems with different temperatures are put into purely thermal connection, heat will flow from the higher temperature system to the lower temperature one; thermodynamically this is understood by the second law of thermodynamics: The total change in entropy following a transfer of energy
from system 1 to system 2 is:
and is thus positive if
From the point of view of statistical mechanics, the total number of microstates in the combined system 1 + system 2 is
, the logarithm of which (times the Boltzmann constant) is the sum of their entropies; thus a flow of heat from high to low temperature, which brings an increase in total entropy, is more likely than any other scenario (normally it is much more likely), as there are more microstates in the resulting macrostate.
Generalized temperature from single-particle statistics
[
edit
]
It is possible to extend the definition of temperature even to systems of few particles, like in a
quantum dot
. The generalized temperature is obtained by considering time ensembles instead of configuration-space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of
fermions
(
N
even less than 10) with a single/double-occupancy system. The finite quantum
grand canonical ensemble
,
[94]
obtained under the hypothesis of
ergodicity
and orthodicity,
[95]
allows expressing the generalized temperature from the ratio of the average time of occupation
and
of the single/double-occupancy system:
[96]
where
E
F
is the
Fermi energy
. This generalized temperature tends to the ordinary temperature when
N
goes to infinity.
Negative temperature
[
edit
]
On the empirical temperature scales that are not referenced to absolute zero, a negative temperature is one below the zero-point of the scale used. For example,
dry ice
has a sublimation temperature of
?78.5 °C
which is equivalent to
?109.3 °F
.
[97]
On the absolute Kelvin scale this temperature is
194.6 K
. No body can be brought to exactly
0 K
(the temperature of the ideally coldest possible body) by any finite practicable process; this is a consequence of the
third law of thermodynamics
.
[98]
[99]
[100]
The internal kinetic theory temperature of a body cannot take negative values. The thermodynamic temperature scale, however, is not so constrained.
For a body of matter, there can sometimes be conceptually defined, in terms of microscopic degrees of freedom, namely particle spins, a subsystem, with a temperature other than that of the whole body. When the body is in its own state of internal thermodynamic equilibrium, the temperatures of the whole body and of the subsystem must be the same. The two temperatures can differ when, by work through externally imposed force fields, energy can be transferred to and from the subsystem, separately from the rest of the body; then the whole body is not in its own state of internal thermodynamic equilibrium. There is an upper limit of energy such a spin subsystem can attain.
Considering the subsystem to be in a temporary state of virtual thermodynamic equilibrium, it is possible to obtain a
negative temperature
on the thermodynamic scale. Thermodynamic temperature is the inverse of the derivative of the subsystem's entropy with respect to its internal energy. As the subsystem's internal energy increases, the entropy increases for some range, but eventually attains a maximum value and then begins to decrease as the highest energy states begin to fill. At the point of maximum entropy, the temperature function shows the behavior of a
singularity
, because the slope of the entropy as a function of energy decreases to zero and then turns negative. As the subsystem's entropy reaches its maximum, its thermodynamic temperature goes to positive infinity, switching to negative infinity as the slope turns negative. Such negative temperatures are hotter than any positive temperature. Over time, when the subsystem is exposed to the rest of the body, which has a positive temperature, energy is transferred as heat from the negative temperature subsystem to the positive temperature system.
[101]
The kinetic theory temperature is not defined for such subsystems.
Examples
[
edit
]
- 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 black-body (which tungsten filaments are not). Tungsten filament 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 within the precision of these values.
- F
For a true black-body (which the plasma was not). The Z machine's dominant emission originated from
40 MK
electrons (soft x-ray emissions) within the plasma.
See also
[
edit
]
Notes and references
[
edit
]
- Notes
- ^
The cited emission wavelengths are for black bodies in equilibrium. CODATA 2006 recommended value of
2.897
7685
(51)
×
10
?3
m K
used for Wien displacement law constant
b
.
- ^
A temperature of 450 ±80 pK in a Bose?Einstein condensate (BEC) of sodium atoms was achieved in 2003 by researchers at
MIT
. Citation:
Cooling Bose?Einstein Condensates Below 500 Picokelvin
, A.E. Leanhardt
et al
., Science
301
, 12 Sept. 2003, p. 1515. This record's peak emittance black-body wavelength of 6,400 kilometers is roughly the radius of Earth.
- ^
The peak emittance wavelength of
2.897
77
m
is a frequency of
103.456 MHz
- ^
Since 2019, Kelvin is now defined on the
Boltzmann constant
, so that the triple point is
273.16
±
0.0001 K
- ^
Measurement was made in 2002 and has an uncertainty of ±3 kelvins. A
1989 measurement
Archived
2010-02-11 at the
Wayback Machine
produced a value of 5,777.0±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).
- ^
The 350 MK value is the maximum peak fusion fuel temperature in a thermonuclear weapon of the Teller?Ulam configuration (commonly known as a
hydrogen bomb
). Peak temperatures in Gadget-style fission bomb cores (commonly known as an
atomic bomb
) are in the range of 50 to 100 MK. Citation:
Nuclear Weapons Frequently Asked Questions, 3.2.5 Matter At High Temperatures.
Link to relevant Web page.
Archived
2007-05-03 at the
Wayback Machine
All referenced data was compiled from publicly available sources.
- ^
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).
Ion Viscous Heating in a Magnetohydrodynamically Unstable Z Pinch at Over
2
×
10
9
Kelvin
, M.G. Haines
et al.
, Physical Review Letters
96
(2006) 075003.
Link to Sandia's news release.
Archived
2010-05-30 at the
Wayback Machine
- ^
Core temperature of a high?mass (>8?11 solar masses) star after it leaves the
main sequence
on the
Hertzsprung?Russell diagram
and begins the
alpha process
(which lasts one day) of
fusing silicon?28
into heavier elements in the following steps: sulfur?32 → argon?36 → calcium?40 → titanium?44 → chromium?48 → iron?52 → nickel?56. Within minutes of finishing the sequence, the star explodes as a Type II
supernova
. Citation:
Holland, Arthur; Williams, Mark.
"Stellar Evolution: The Life and Death of Our Luminous Neighbors"
.
GS265
. University of Michigan.
Archived
from the original on 2009-01-16.
More informative links can be found here
"Chapter 21 Stellar Explosions"
. Archived from
the original
on 2013-04-11
. Retrieved
2016-02-08
.
, and here
"Trans"
. Archived from
the original
on 2011-08-14
. Retrieved
2016-02-08
.
, and a concise treatise on stars by NASA is here
"NASA - Star"
. Archived from
the original
on 2010-10-24
. Retrieved
2010-10-12
.
.
- ^
Based on a computer model that predicted a peak internal temperature of 30 MeV (350 GK) during the merger of a binary neutron star system (which produces a gamma?ray burst). The neutron stars in the model were 1.2 and 1.6 solar masses respectively, were roughly
20 km
in diameter, and were orbiting around their barycenter (common center of mass) at about
390 Hz
during the last several milliseconds before they completely merged. The 350 GK portion was a small volume located at the pair's developing common core and varied from roughly
1 to 7 km
across over a time span of around 5 ms. Imagine two city-sized objects of unimaginable density orbiting each other at the same frequency as the G4 musical note (the 28th white key on a piano). It's also noteworthy that at 350
GK, the average neutron has a vibrational speed of 30% the speed of light and a relativistic mass (
m
) 5% greater than its rest mass (
m
0
).
Torus Formation in Neutron Star Mergers and Well-Localized Short Gamma-Ray Bursts
Archived
2017-11-22 at the
Wayback Machine
, R. Oechslin
et al
. of
Max Planck Institute for Astrophysics.
Archived
2005-04-03 at the
Wayback Machine
, arXiv:astro-ph/0507099 v2, 22 Feb. 2006.
An html summary
Archived
2010-11-09 at the
Wayback Machine
.
- Citations
- ^
Agency, International Atomic Energy (1974).
Thermal discharges at nuclear power stations: their management and environmental impacts: a report prepared by a group of experts as the result of a panel meeting held in Vienna, 23?27 October 1972
. International Atomic Energy Agency.
- ^
Watkinson, John (2001).
The Art of Digital Audio
. Taylor & Francis.
ISBN
978-0-240-51587-8
.
- ^
Middleton, W.E.K. (1966), pp. 89?105.
- ^
Joanna Thompson (2021-10-14).
"Scientists just broke the record for the coldest temperature ever recorded in a lab"
.
Live Science
. Retrieved
2023-04-28
.
- ^
a
b
Jaynes, E.T. (1965), pp. 391?398.
- ^
a
b
Cryogenic Society
Archived
2020-11-07 at the
Wayback Machine
(2019).
- ^
a
b
Draft Resolution A "On the revision of the International System of Units (SI)" to be submitted to the CGPM at its 26th meeting (2018)
(PDF)
, archived from
the original
(PDF)
on 2018-04-29
, retrieved
2019-10-20
- ^
a
b
c
Truesdell, C.A. (1980), Sections 11 B, 11H, pp. 306?310, 320?332.
- ^
Quinn, T. J. (1983).
- ^
Germer, L.H. (1925). 'The distribution of initial velocities among thermionic electrons',
Phys. Rev.
,
25
: 795?807.
here
- ^
Turvey, K. (1990). 'Test of validity of Maxwellian statistics for electrons thermionically emitted from an oxide cathode',
European Journal of Physics
,
11
(1): 51?59.
here
- ^
Zeppenfeld, M., Englert, B.G.U., Glockner, R., Prehn, A., Mielenz, M., Sommer, C., van Buuren, L.D., Motsch, M., Rempe, G. (2012).
- ^
Miller, J. (2013).
- ^
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Further reading
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- Chang, Hasok (2004).
Inventing Temperature: Measurement and Scientific Progress
. Oxford: Oxford University Press.
ISBN
978-0-19-517127-3
.
- Zemansky, Mark Waldo (1964).
Temperatures Very Low and Very High
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External links
[
edit
]