In
physics,
temperature is
a
physical property of a
system that underlies the common notions of
hot and cold; something that feels hotter generally has the higher
temperature. Temperature is one of the principal parameters of
thermodynamics. If no net heat flow
occurs between two objects, the objects have the same temperature;
otherwise heat flows from the hotter object to the colder object.
This is a consequence of the
laws
of thermodynamics. On the microscopic scale, temperature can be
defined as the average energy in each
degree of freedom
of the particles in a system. Because temperature is a statistical
property, a system must contain a large number of particles for
temperature to have a useful meaning. For a solid, this energy is
found primarily in the
vibrations of its
atoms about their equilibrium positions. In an
ideal monatomic gas, energy is found in the
translational motions of the particles; with molecular gases,
vibrational and
rotational motions also provide thermodynamic
degrees of freedom.
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. The entire
scientific world (these countries included) measures temperature
using the Celsius scale and thermodynamic temperature using the
Kelvin scale, which is just the Celsius scale
shifted downwards so that 0 K= −273.15 °C, or
absolute zero. Many engineering fields in the
U.S., notably high-tech and US federal specifications (civil and
military), also use the kelvin and degrees Celsius scales. Other
engineering fields in the U.S. also rely upon the
Rankine scale (a shifted Fahrenheit scale)
when working in thermodynamic-related disciplines such as
combustion.
For a system in thermal equilibrium at a constant volume,
temperature is thermodynamically defined in terms of its
energy (
E) and
entropy (
S) as:
T \equiv \frac{\partial E}{\partial S} .
Overview
Intuitively, temperature is the measurement of how hot or cold
something is, although the most immediate way in which we can
measure this, by feeling it, is unreliable, resulting in the
phenomenon of
felt air
temperature, which can differ at varying degrees from actual
temperature. On the molecular level, temperature is the result of
the motion of particles which make up a substance. Temperature
increases as the energy of this motion increases. The motion may be
the translational motion of the particle, or the internal energy of
the particle due to molecular vibration or the excitation of an
electron energy
level. Although very specialized laboratory equipment is
required to directly detect the translational thermal motions,
thermal collisions by atoms or molecules with small particles
suspended in a
fluid produces
Brownian motion that can be seen with an
ordinary microscope. The thermal motions of atoms are
very
fast and temperatures close to
absolute
zero are required to directly observe them. For instance, when
scientists at the
NIST achieved
a record-setting low temperature of 700 nK (1 nK = 10
^{−9}
K) in 1994, they used
optical
lattice laser equipment to
adiabatically cool
caesium atoms. They then turned off the entrapment
lasers and directly measured atom velocities of 7 mm per
second in order to calculate their temperature.
Molecules, such as O
_{2}, have more
degrees of freedom than single
atoms: they can have rotational and vibrational motions as well as
translational motion. An increase in temperature will cause the
average translational energy to increase. It will also cause the
energy associated with vibrational and rotational modes to
increase. Thus a
diatomic gas, with extra
degrees of freedom rotation and vibration, will require a higher
energy input to change the temperature by a certain amount, i.e. it
will have a higher
heat capacity than
a monatomic gas.
The process of cooling involves removing energy from a system. When
there is no more energy able to be removed, the system is said to
be at
absolute zero, which is the
point on the
thermodynamic
temperature scale where all kinetic motion in the particles
comprising matter ceases and they are at complete rest in the
“classic” (non-
quantum mechanical)
sense. By definition, absolute zero is a temperature of precisely
0
kelvins (−273.15
°C or −459.68
°F).
Details
The formal properties of temperature follow from its mathematical definition (see below for the zeroth law definition and the second law definition) and are studied in thermodynamics and statistical mechanics.
Contrary to other thermodynamic quantities such as
entropy and
heat, whose
microscopic definitions are valid even far away from
thermodynamic equilibrium,
temperature being an average energy per particle can only be
defined at thermodynamic equilibrium, or at least local
thermodynamic equilibrium (see below).
As a system receives heat, its temperature rises; similarly, a loss
of heat from the system tends to decrease its temperature (at
the—uncommon—exception of negative temperature; see below).
When two systems are at the same temperature, no heat transfer
occurs between them. When a temperature difference does exist, heat
will tend to move from the
higher-temperature system to
the
lower-temperature system, until they are at thermal
equilibrium. This heat transfer may occur via
conduction,
convection or
radiation or combinations of them (see
heat for additional discussion of the various
mechanisms of heat transfer) and some ions may vary.
Temperature is also related to the amount of
internal energy and
enthalpy of a system: the higher the temperature of
a system, the higher its internal energy and enthalpy.
Temperature is an
intensive
property of a system, meaning that it does not depend on the
system size, the amount or type of material in the system, the same
as for the
pressure and
density. By contrast,
mass,
volume, and
entropy
are
extensive
properties, and depend on the amount of material in the
system.
The role of temperature in nature
A map of monthly mean
temperatures
Water freezes at 0 °C.
The frost shown here is at -17 °C.
Temperature plays an important role in almost all fields of
science, including physics, geology, chemistry, and biology.
Many physical properties of materials including the
phase (
solid,
liquid,
gaseous or
plasma),
density,
solubility,
vapor pressure, and
electrical conductivity depend on
the temperature. Temperature also plays an important role in
determining the rate and extent to which
chemical reactions occur. This is one
reason why the human body has several elaborate mechanisms for
maintaining the temperature at 37 °C, since temperatures only a few
degrees higher can result in harmful reactions with serious
consequences. Temperature also controls the type and quantity of
thermal radiation emitted from a surface. One application of this
effect is the
incandescent light
bulb, in which a
tungsten filament is
electrically heated to a temperature at
which significant quantities of visible
light
are emitted.
Temperature-dependence of the
speed of sound in air
c, density of
air
ρ and
acoustic
impedance Z vs. temperature °C
Effect of temperature on
speed of sound, air density and acoustic impedance at sea
level |
T in °C |
c in m/s |
ρ in kg/m³ |
Z in N·s/m³ |
−10 |
325.4 |
1.341 |
436.5 |
−5 |
328.5 |
1.316 |
432.4 |
0 |
331.5 |
1.293 |
428.3 |
5 |
334.5 |
1.269 |
424.5 |
10 |
337.5 |
1.247 |
420.7 |
15 |
340.5 |
1.225 |
417.0 |
20 |
343.4 |
1.204 |
413.5 |
25 |
346.3 |
1.184 |
410.0 |
30 |
349.2 |
1.164 |
406.6 |
Temperature measurement
Temperature measurement using modern scientific
thermometers and temperature scales goes back at
least as far as the early 18th century, when
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 USA, with the
Celsius scale in
use in the rest of the world and the
kelvin
scale.
Units of temperature
The basic unit of temperature (symbol:
T) in the
International System of Units is the
kelvin (Symbol: K). The kelvin and Celsius scales
are, by
international agreement, defined by two points:
absolute zero, and the
triple point of
Vienna Standard Mean Ocean
Water (water specially prepared with a specified blend of
hydrogen and oxygen isotopes). Absolute zero is defined as being
precisely 0 K
and −273.15 °C. Absolute zero is
where all
kinetic motion in the particles
comprising matter ceases and they are at complete rest in the
“classic” (non-
quantum mechanical)
sense. At absolute zero, matter contains no
thermal energy. Also, 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
kelvin (0 K = −273.15 °C and 273.16 K =
0.01 °C). Formulas for converting from these defining units of
temperature to other scales can be found at
Temperature conversion
formulas.
In the field of
plasma physics,
because of the high temperatures encountered and the
electromagnetic nature of the
phenomena involved, it is customary to express temperature in
electronvolts (eV) or kiloelectronvolts
(keV), where 1 eV = 11,604 K. In the study of
QCD matter one routinely meets temperatures of
the order of a few hundred
MeV, equivalent to
about 10
^{12} K.
For everyday applications, it's very often convenient to use the
Celsius scale, in which 0 °C
corresponds to the temperature at which water
freezes and 100 °C corresponds to the
boiling point of water at sea level.
In this scale a temperature difference of 1 degree is the same as a
1 K temperature difference, so the scale is essentially the
same as the kelvin scale, but offset by the temperature at which
water freezes (273.15 K). Thus the following equation can be used
to convert from degrees Celsius to kelvin.\mathrm{K = [^\circ C]
\left(\frac{1 \, K}{1\, ^\circ C}\right) + 273.15\, K}
In the
United
States, the Fahrenheit scale is
widely used. On this scale the freezing point of water
corresponds to 32 °F and the boiling point to 212 °F. The following
conversion formulas may be used to convert between Fahrenheit
(
F) and Celsius (
C) temperature values:
- C = \frac{5}{9} \left({F - 32}\right) and F = \frac{9}{5}{C +
32}.
See
temperature
conversion formulas for conversions between most temperature
scales.
Negative temperatures
In the macroscopic sense relevant to most people, a negative
temperature is one below the zero-point of the measurement system
used. For example, a temperature of 100 K is equivalent to −173.15
°C. Temperatures of
macroscopic systems
may have negative values in the Celsius and Fahrenheit, but not in
the Kelvin or Rankine scales.
However, for some systems and specific definitions of temperature,
it is possible to obtain a
negative
temperature, which is numerically less than absolute zero.
However, a system with a negative temperature is not colder than
absolute zero, but rather it is, in a
sense,
hotter than infinite
temperature.
Theoretical foundation
Definition Based on Zeroth Law of Thermodynamics
If two systems with fixed volumes are brought together in thermal
contact, changes will most likely take place in the properties of
both systems. These changes are caused by the transfer of heat
between the systems. A state must be reached in which no further
changes occur, to put the objects into
thermal equilibrium.
A basis for the definition of temperature can therefore be obtained
from the
Zeroth Law of
Thermodynamics which states that if two systems, A and B,
are in thermal equilibrium and a third system C is in thermal
equilibrium with system A then systems B and C will also be in
thermal equilibrium (being in thermal equilibrium is a
transitive relation; moreover, it is an
equivalence relation). This is
an empirical fact, based on observation rather than theory. Since
A, B, and C are all in thermal equilibrium, it is reasonable to say
each of these systems shares a common value of some property. This
property is called temperature.
Generally, it is not convenient to place any two arbitrary systems
in thermal contact to see if they are in thermal equilibrium and
thus have the same temperature. Also, it would only provide an
ordinal
scale.
Therefore, it is useful to establish a temperature scale based on
the properties of some reference system. Then, a measuring device
can be calibrated based on the properties of the reference system
and used to measure the temperature of other systems. One such
reference system is a fixed quantity of gas. The
ideal gas law indicates that the product of
the pressure and volume (P · V) of a gas is
directly proportional to the
temperature:P \cdot V = n \cdot R \cdot T (1)
where 'T is temperature, n is the number of
mole of gas and R is the
gas constant. Thus, one can define a scale for
temperature based on the corresponding pressure and volume of the
gas: the temperature in kelvin is the pressure in pascals of one
mole of gas in a container of one cubic metre, divided by 8.31...
In practice, such a
gas thermometer is not very
convenient, but other measuring instruments can be calibrated to
this scale.
The pressure, volume, and the number of moles of a substance are
all inherently greater than or equal to zero, suggesting that
temperature must also be greater than or equal to zero. As a
practical matter it is not possible to use a gas thermometer to
measure absolute zero temperature since the gasses tend to condense
into a liquid long before the temperature reaches zero. It is
possible to extrapolate how many degrees below the present
temperature the absolute zero is from the temperature range where
Equation 1 works.
Temperature in gases
For an
ideal gas the
kinetic theory of gases uses
statistical mechanics to relate the
temperature to the average kinetic energy of the atoms in the
system. This average energy is independent of particle mass, which
seems counter-intuitive. Temperature is related only to the
average kinetic energy of the particles in a gas - each
particle has its own energy which may or may not correspond to the
average; the distribution of energies (and thus speeds) of the
particles in any gas are given by the
Maxwell-Boltzmann
distribution.The temperature of a classical ideal gas is
related to its average kinetic energy via the equation:
- \overline{E}_\text{k} = \begin{matrix} \frac{3}{2} \end{matrix}
kT , where k = R/n (n= Avogadro
number, R= ideal gas
constant). This relation is valid in the classical regime, i.e.
when the particle density is much less than 1/\Lambda^{3}, where
\Lambda is the thermal de
Broglie wavelength.
In the case of a monoatomic gas, the
kinetic energy is:
- E_\text{k} = \begin{matrix} \frac{1}{2} \end{matrix} mv^2
(Note that a calculation of the kinetic energy of a more
complicated object, such as a molecule, is slightly more involved.
Additional
degrees of
freedom are available, so molecular rotation or vibration must
be included.)
The second law of thermodynamics states that any two given systems
when interacting with each other will later reach the same average
energy per particle (and hence the same temperature).In a mixture
of particles of various mass, the heaviest particles will move more
slowly than lighter counterparts, but will still have the same
average energy. A
neon atom moves slower
relative to a
hydrogen molecule of the same
kinetic energy; a pollen particle moves in a slow
Brownian motion among fast moving water
molecules, etc. A visual illustration of this
from Oklahoma State University makes the point more
clear. Particles with different mass have different velocity
distributions, but the average kinetic energy is the same because
of the
ideal gas law.
Temperature of the vacuum
It is possible to use the zeroth law definition of temperature to
assign a temperature to something not normally associated with
temperatures, like a perfect vacuum. Because all objects emit
black body radiation, a thermometer in a
vacuum away from thermally radiating sources will radiate away its
own thermal energy; decreasing in temperature indefinitely until it
reaches the
zero-point energy
limit. At that point it can be said to be in equilibrium with the
vacuum and by definition at the same temperature. A gas that
behaved ideally all the way down to absolute zero, obeying the
kinetic theory of gases, would achieve zero kinetic energy per
particle, and thereby achieve absolute zero temperature. Thus, by
the zeroth law a perfect, isolated vacuum is at absolute zero
temperature. Note that in order to behave ideally in this context
it is necessary for the atoms of the gas to have no zero point
energy. It will turn out not to matter that this is not possible
because the second law definition of temperature will yield the
same result for any unique vacuum state.
More realistically, no such ideal vacuum exists. For instance a
thermometer in a vacuum chamber which is maintained at some finite
temperature (say, chamber is in the lab at room temperature) will
equilibrate with the thermal radiation it receives from the chamber
and with time reaches the temperature of the chamber. If a
thermometer orbiting the Earth is exposed to
sunlight, then it equilibrates at the temperature
at which power received by the thermometer from the Sun is exactly
equal to the power radiated away by thermal radiation of the
thermometer. For a black body this equilibrium temperature is about
281 K (+8 °C). Since Earth has an
albedo of
30%, average temperature as seen from space is lower than for a
black body, 254 K, while the surface temperature is considerably
higher due to the
greenhouse
effect.
A thermometer isolated from solar radiation (in the shade of the
Earth, for example) is still exposed to thermal radiation of Earth
- thus will show some equilibrium temperature at which it receives
and radiates equal amount of energy. If this thermometer is close
to Earth then its equilibrium temperature is about 236 K (-37 °C)
provided that Earth surface is at 281 K.
A thermometer far away from the Solar system still receives
Cosmic microwave
background radiation. Equilibrium temperature of such
thermometer is about 2.725 K, which is the temperature of a photon
gas constituting black body microwave background radiation at
present state of expansion of Universe. This temperature is
sometimes referred to as the temperature of space. This temperature
is thus like a
test charge in that it
facilitates a measure of the system even though temperature is not
strictly defined there.
Phenomenological definition based on second law of
thermodynamics
In the previous section temperature was defined in terms of the
Zeroth Law of thermodynamics. It is also possible to define
temperature in terms of the
second law of thermodynamics,
which deals with
entropy. Entropy is a
measure of the disorder in a system. 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. Consider 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 that for a perfectly ordered
set of coin tosses, there is only one set of toss outcomes
possible: the set in which 100% of tosses came up the same.
On the other hand, there are multiple combinations that can result
in disordered or mixed systems, where some fraction are heads and
the rest tails. A disordered system can be 90% heads and 10% tails,
or it could be 98% heads and 2% tails, et cetera. As the number of
coin tosses increases, the number of possible combinations
corresponding to imperfectly ordered systems increases. For a very
large number of coin tosses, the combinations to ~50% heads and
~50% tails dominates and obtaining an outcome significantly
different from 50/50 becomes extremely unlikely. Thus the system
naturally progresses to a state of maximum disorder or
entropy.
It has been previously stated that temperature controls the flow of
heat between two systems and was just shown that the universe tends
to progress so as to maximize entropy (this is expected of any
natural system). Thus, it is expected that there is some
relationship between temperature and entropy. To find this
relationship, the relationship between heat, work and temperature
is first considered. A
heat engine is a
device for converting heat into mechanical work and analysis of the
Carnot heat engine provides the
necessary relationships. The work from a heat engine corresponds to
the difference between the heat put into the system at the high
temperature,
q_{H} and the heat ejected at the low
temperature,
q_{C}. The efficiency is the work
divided by the heat put into the system or:\textrm{efficiency} =
\frac {w_{cy}}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H}
(2)
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:\frac{q_C}{q_H} = f(T_H,T_C)
(3)
Carnot's theorem
states that all reversible engines operating between the same heat
reservoirs are equally efficient. Thus, a heat engine operating
between
T_{1} and
T_{3} must have
the same efficiency as one consisting of two cycles, one between
T_{1} and
T_{2}, and the second
between
T_{2} and
T_{3}. This can
only be the case if:q_{13} = \frac{q_1}{o_7}
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:
\frac{q_C}{q_H} = \frac{T_C}{T_H} (4)
Substituting Equation 4 back into Equation 2 gives a relationship
for the efficiency in terms of temperature:\textrm{efficiency} = 1
- \frac{q_C}{q_H} = 1 - \frac{T_C}{T_H} (5)
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 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 Equation 5 from the middle
portion and rearranging gives:\frac {q_H}{T_H} - \frac{q_C}{T_C} =
0
where the negative sign indicates heat ejected from the system.
This relationship suggests the existence of a state function,
S, defined by:dS = \frac {dq_\mathrm{rev}}{T} (6)
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 was described previously. Rearranging Equation 6
gives a new definition for temperature in terms of entropy and
heat:T = \frac{dq_\mathrm{rev}}{dS} (7)
For a system, where entropy
S may be a function
S(
E) of its energy
E, the temperature
T is given by:\frac{1}{T} = \frac{dS}{dE} (8)
ie. the reciprocal of the temperature is the rate of increase of
entropy with respect to energy.
Definition of temperature in Statistical mechanics
The argument in the previous section is how the relation between
entropy and heat was arrived at historically. Modern definition of
temperature is given in
Statistical mechanics and it is
defined in terms of the fundamental degrees of freedom of a system
(see the article
entropy for details).
Eq.(8) of the previous section is then taken to be the
defining relation of the temperature. Eq. (7) can
be derived from the definition of entropy,
see e.g. here.
Importance of temperature
The following table demonstrates that the properties of air change
significantly with temperature.
Table — speed of sound in air
c, density of air ρ,acoustic impedance Z vs.
temperature
\vartheta
Temperature and specific heat
Specific heat is the measure of the
energy required to increase the temperature of a unit quantity of a
substance by a unit of temperature. For example, the energy
required to raise water’s temperature by one kelvin (equal to one
degree Celsius) is 4186 J/kg.
See also
Notes
- This means "zero kelvin"; the unit kelvin is not used with a
degree symbol because it is an absolute scale, i.e. the notation 0
°K is not correct according to international standards
- Kittel and Kroemer, pp. 462
- Vu-Quoc, L., Configuration integral (statistical mechanics),
2008
References
- Chang, Hasok (2004). Inventing Temperature: Measurement and
Scientific Progress. Oxford: Oxford University Press. ISBN
9780195171273.
- Zemansky, Mark Waldo (1964). Temperatures Very Low and Very
High. Princeton, N.J.: Van Nostrand.
External links