Temperature
Overview

Temperature is a physical property
Physical property
A physical property is any property that is measurable whose value describes a physical system's state. The changes in the physical properties of a system can be used to describe its transformations ....

of matter that quantitatively expresses the common notions of hot
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

and cold
Cold
Cold describes the condition of low temperature.Cold may also refer to:*Common cold, a contagious viral infectious disease of the upper respiratory system*Chronic obstructive pulmonary disease...

. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot. Heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

spontaneously flows from bodies of a higher temperature to bodies of lower temperature, and no net heat will be exchanged between bodies of the same temperature. Such bodies are said to be in "thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...

".

The temperature of a substance varies with the microscopic speed of the fundamental particles that it contains, raised to the second power; that is, it is proportional to the mean kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

of its particles.
Encyclopedia
Temperature is a physical property
Physical property
A physical property is any property that is measurable whose value describes a physical system's state. The changes in the physical properties of a system can be used to describe its transformations ....

of matter that quantitatively expresses the common notions of hot
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

and cold
Cold
Cold describes the condition of low temperature.Cold may also refer to:*Common cold, a contagious viral infectious disease of the upper respiratory system*Chronic obstructive pulmonary disease...

. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot. Heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

spontaneously flows from bodies of a higher temperature to bodies of lower temperature, and no net heat will be exchanged between bodies of the same temperature. Such bodies are said to be in "thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...

".

The temperature of a substance varies with the microscopic speed of the fundamental particles that it contains, raised to the second power; that is, it is proportional to the mean kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

of its particles. However any increase in temperature upon receiving external kinetic energy as heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

is also inversely proportional to heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

. Temperature can be thought of as the "concentration
Concentration
In chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...

" of kinetic energy relative to its heat capacity. A higher heat capacity implies a higher entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

in which to "spread out" the kinetic energy. A hot object must at least either have a smaller heat capacity or a larger kinetic energy than a cold object.

Quantitatively, temperature is measured with thermometers, which may be calibrated
Calibration
Calibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....

to a variety of temperature scales
Temperature conversion formulas
This is a compendium of temperature conversion formulæ and comparisons.-Celsius :The simple approximation [°F] = [°C] × 2 + 30 works well within the range of normal weather temperatures, underestimating by 6°F at −20°C and over by 4°F at 30°C.-Comparison:-Comparison of temperature scales:...

.
Temperature plays an important role in all fields of natural science, including physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, geology
Geology
Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

, chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

, atmospheric sciences
Atmospheric sciences
Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...

and biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

.

### Use in science

Many physical properties of materials including the phase (solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...

, liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

, gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

eous or plasma
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

), density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

, solubility
Solubility
Solubility is the property of a solid, liquid, or gaseous chemical substance called solute to dissolve in a solid, liquid, or gaseous solvent to form a homogeneous solution of the solute in the solvent. The solubility of a substance fundamentally depends on the used solvent as well as on...

, vapor pressure
Vapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...

, and electrical conductivity depend on the temperature. Temperature also plays an important role in determining the rate and extent to which chemical reaction
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...

s occur. This is one reason why the human body has several elaborate mechanisms for maintaining the temperature at 310 K, since temperatures only a few degrees higher can result in harmful reactions with serious consequences. Temperature also controls the thermal radiation emitted from a surface. One application of this effect is the incandescent light bulb
Incandescent light bulb
The incandescent light bulb, incandescent lamp or incandescent light globe makes light by heating a metal filament wire to a high temperature until it glows. The hot filament is protected from air by a glass bulb that is filled with inert gas or evacuated. In a halogen lamp, a chemical process...

, in which a tungsten
Tungsten
Tungsten , also known as wolfram , is a chemical element with the chemical symbol W and atomic number 74.A hard, rare metal under standard conditions when uncombined, tungsten is found naturally on Earth only in chemical compounds. It was identified as a new element in 1781, and first isolated as...

filament is electrically
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...

heated to a temperature at which significant quantities of visible light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

are emitted.

### Temperature scales

Some of the world uses the Celsius
Celsius
Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

scale (°C) for most temperature measurements. It has the same incremental scaling as the Kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

scale used by scientists, but fixes its null point, at = , the freezing point of water.Historically, the Celsius scale was a purely empirical temperature scale defined only by the freezing and boiling points of water. 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. A few countries, most notably the United States, use the Fahrenheit
Fahrenheit
Fahrenheit is the temperature scale proposed in 1724 by, and named after, the German physicist Daniel Gabriel Fahrenheit . Within this scale, the freezing of water into ice is defined at 32 degrees, while the boiling point of water is defined to be 212 degrees...

scale for common purposes, a historical scale on which water freezes at 32 °F and boils at 212 °F.

For practical purposes of scientific temperature measurement, the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI) defines a scale and unit for the thermodynamic temperature by using the easily reproducible temperature of the triple point
Triple point
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases of that substance coexist in thermodynamic equilibrium...

of water as a second reference point. For historical reasons, the triple point is fixed at 273.16 units of the measurement increment, which has been named the kelvin in honor of the Scottish physicist who first defined the scale. The unit symbol of the kelvin is K.

Absolute zero is defined as a temperature of precisely 0 kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

s, which is equal to −273.15 °C or −459.68 °F.

### Thermodynamic approach to temperature

Temperature is one of the principal quantities studied in the field of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

. Thermodynamics investigates the relation between heat and work, using a special scale of temperature called the absolute temperature, and thus relates temperature to work, as considered below. In thermodynamic terms, temperature is a macroscopic scale intensive variable
Intensive and extensive properties
In the physical sciences, an intensive property , is a physical property of a system that does not depend on the system size or the amount of material in the system: it is scale invariant.By contrast, an extensive property In the physical sciences, an intensive property (also called a bulk...

because it is independent of the bulk amount of elementary entities contained inside, be they atoms, molecules, or electrons. Real world systems are not homogeneous and for study are usually spatially and temporally divided conceptually into imagined 'cells' of small size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation (local thermodynamic equilibrium).

### Statistical mechanics approach to temperature

Statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

provides a microscopic explanation of temperature, based on macroscopic systems' being composed of many particles, such as molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s and ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

s of various species, the particles of a species being all alike. It explains macroscopic phenomena in terms of the mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

of the molecules
Molecular mechanics
Molecular mechanics uses Newtonian mechanics to model molecular systems. The potential energy of all systems in molecular mechanics is calculated using force fields...

and ions, and statistical assessments of their joint adventures
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

. In the statistical thermodynamic approach
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

On the molecular level, temperature is the result of the motion of the particles that constitute the material. Moving particles carry kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

. Temperature increases as this motion and the kinetic energy increase. The motion may be the translational motion of particles, or the energy of the particle due to molecular vibration
Molecular vibration
A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion...

or the excitation of an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

energy level
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

. 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
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

produces Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

that can be seen with an ordinary microscope. The thermal motions of atoms are very fast and temperatures close to absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

are required to directly observe them. For instance, when scientists at the NIST
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

achieved a record-setting low temperature of 700 nK (1 nK = 10−9 K) in 1994, they used laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

equipment to create an optical lattice
Optical lattice
An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate in the locations of potential minima...

In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...

cool caesium
Caesium
Caesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-gold alkali metal with a melting point of 28 °C , which makes it one of only five elemental metals that are liquid at room temperature...

atoms. They then turned off the entrapment lasers and directly measured atom velocities of per second in order to calculate their temperature.

Molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s, such as oxygen (O2), have more degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase in temperature due to an increase in the average translational energy of the molecules. Heating will also cause, through equipartitioning, the energy associated with vibrational and rotational modes to increase. Thus a diatomic
Diatomic
Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen , nitrogen , oxygen , and carbon monoxide . Seven elements exist in the diatomic state in the liquid and solid...

gas will require a higher energy input to increase its temperature by a certain amount, i.e. it will have a higher heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

than a monatomic gas.

The process of cooling involves removing thermal energy from a system. When no more energy can be removed, the system is at absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

, which cannot be achieved experimentally. Absolute zero is the null point of the thermodynamic temperature
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

scale, also called absolute temperature. If it were possible to cool a system to absolute zero, all motion of the particles comprising matter would cease and they would be at complete rest in this classical sense. Microscopically in the description of quantum mechanics, however, matter still has zero-point energy
Zero-point energy
Zero-point energy is the lowest possible energy that a quantum mechanical physical system may have; it is the energy of its ground state. All quantum mechanical systems undergo fluctuations even in their ground state and have an associated zero-point energy, a consequence of their wave-like nature...

even at absolute zero, because of the uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

.

## Basic theory

As distinct from a quantity of heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

, temperature may be viewed as a measure of a quality of a body or of heat. The quality is called hotness by some writers.

When two systems are at the same temperature, no net heat transfer occurs spontanteously, by conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

, between them. When a temperature difference does exist, and there is a thermally conductive or radiative connection between them, heat transfers spontaneously from the warmer system to the colder system, until they are at mutual thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...

. This transfer occurs by heat conduction or by thermal radiation.

Experimental physicists, for example Galileo and Newton, found that there are indefinitely many empirical temperature scales
Scale of temperature
Scale of temperature is a way to measure temperature quantitatively.-Formal description:According to the zeroth law of thermodynamics, being in thermal equilibrium is an equivalence relation. Thus all thermal systems may be divided into a quotient set by this equivalence relation, denoted below as M...

.

### Temperature for bodies in thermodynamic equilibrium

For experimental physics, the fact of hotness means that, when comparing any two given bodies in their respective separate thermodynamic equilibria
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

, 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. 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. A definite sense of greater hotness can be had, independently of calorimetry
Calorimetry
Calorimetry is the science of measuring the heat of chemical reactions or physical changes. Calorimetry is performed with a calorimeter. The word calorimetry is derived from the Latin word calor, meaning heat...

, of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, and of properties of particular materials, from Wien's displacement law of thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

: the temperature of a bath of thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

is proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

, 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
Third law of thermodynamics
The third law of thermodynamics is a statistical law of nature regarding entropy:For other materials, the residual entropy is not necessarily zero, although it is always zero for a perfect crystal in which there is only one possible ground state.-History:...

. 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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium.

Except for a system undergoing a first-order phase change such as the melting of ice, as a closed system receives heat, without change in its volume and without 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
Latent heat
Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...

. Conversely, a loss of heat from a closed system, without phase change, without change of volume, and without change in external force fields acting on it, decreases its temperature.

### Temperature for bodies in a steady state but not in thermodynamic equilibrium

While for bodies in their own thermodynamic equilibrium states, the notion of temperature safely 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 the 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
Non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...

.

### Temperature for bodies not in a steady state

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
Non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...

.

### Thermodynamic equilibrium axiomatics

For axiomatic treatment of thermodynamic equilibrium, since the 1930's, it has become customary to refer to a zeroth law of thermodynamics
Zeroth law of thermodynamics
The zeroth law of thermodynamics is a generalization principle of thermal equilibrium among bodies, or thermodynamic systems, in contact.The zeroth law states that if two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.Systems are said to...

. 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. While the zeroth law permits the definitions of many different empirical scales of temperature, the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

selects the definition of a single preferred, absolute temperature, unique up to an arbitrary scale factor, whence called the thermodynamic temperature
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

. If internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

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
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

with respect the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

at constant volume. Its natural, intrinsic origin or null point is absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

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
Third law of thermodynamics
The third law of thermodynamics is a statistical law of nature regarding entropy:For other materials, the residual entropy is not necessarily zero, although it is always zero for a perfect crystal in which there is only one possible ground state.-History:...

postulates that absolute zero cannot be attained by any physical system.

## Heat capacity

When a sample is heated, meaning it receives thermal energy from an external source, some of the introduced heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

is converted into kinetic energy, the rest to other forms of internal energy, specific to the material. The amount converted into kinetic energy causes the temperature of the material to rise. The introduced heat () divided by the observed temperature change is the heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

(C) of the material.

If heat capacity is measured for a well defined amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

, 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, to raise the temperature of water by one kelvin (equal to one degree Celsius) requires 4186 joules per kilogram
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

(J/kg)..

## Temperature measurement

Temperature measurement
Temperature measurement
Attempts of standardized temperature measurement have been reported as early as 170 AD by Claudius Galenus. The modern scientific field has its origins in the works by Florentine scientists in the 17th century. Early devices to measure temperature were called thermoscopes. The first sealed...

using modern scientific thermometer
Thermometer
Developed during the 16th and 17th centuries, a thermometer is a device that measures temperature or temperature gradient using a variety of different principles. A thermometer has two important elements: the temperature sensor Developed during the 16th and 17th centuries, a thermometer (from the...

s and temperature scales goes back at least as far as the early 18th century, when Gabriel Fahrenheit
Gabriel Fahrenheit
Daniel Gabriel Fahrenheit was a German physicist, engineer, and glass blower who is best known for inventing the alcohol thermometer and the mercury thermometer , and for developing a temperature scale now named after him.- Biography :Fahrenheit was born in 1686 in Danzig , the Polish-Lithuanian...

adapted a thermometer (switching to mercury
Mercury (element)
Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...

) 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
Calibration
Calibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....

to a variety of temperature scales
Temperature conversion formulas
This is a compendium of temperature conversion formulæ and comparisons.-Celsius :The simple approximation [°F] = [°C] × 2 + 30 works well within the range of normal weather temperatures, underestimating by 6°F at −20°C and over by 4°F at 30°C.-Comparison:-Comparison of temperature scales:...

. In most of the world (except for Belize
Belize
Belize is a constitutional monarchy and the northernmost country in Central America. Belize has a diverse society, comprising many cultures and languages. Even though Kriol and Spanish are spoken among the population, Belize is the only country in Central America where English is the official...

, Myanmar
Myanmar
Burma , officially the Republic of the Union of Myanmar , is a country in Southeast Asia. Burma is bordered by China on the northeast, Laos on the east, Thailand on the southeast, Bangladesh on the west, India on the northwest, the Bay of Bengal to the southwest, and the Andaman Sea on the south....

, Liberia
Liberia
Liberia , officially the Republic of Liberia, is a country in West Africa. It is bordered by Sierra Leone on the west, Guinea on the north and Côte d'Ivoire on the east. Liberia's coastline is composed of mostly mangrove forests while the more sparsely populated inland consists of forests that open...

and the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...

), the Celsius scale is used for most temperature measuring purposes. Most scientist measures temperature using the Celsius scale and the thermodynamic temperature using the Kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

scale, which is the Celsius scale offset so that its null point is = , or absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

. Many engineering fields in the U.S., notably high-tech and US federal specifications (civil and military), also use the Kelvin and 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
Combustion
Combustion or burning is the sequence of exothermic chemical reactions between a fuel and an oxidant accompanied by the production of heat and conversion of chemical species. The release of heat can result in the production of light in the form of either glowing or a flame...

.

### Units

The basic unit of temperature in the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI) is the kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

. It has the symbol K.

For everyday applications, it is often convenient to use the Celsius scale, in which corresponds very closely to the freezing point
Freezing Point
Freezing Point is a news journal in the People's Republic of China which has been the subject of controversy over its criticism of Communist Party officials and the sympathetic ear it lent to a Chinese historian who had criticized official history textbooks...

of water and is its boiling point
Boiling point
The boiling point of an element or a substance is the temperature at which the vapor pressure of the liquid equals the environmental pressure surrounding the liquid....

at sea level. Because liquid droplets commonly exist in clouds at sub-zero temperatures, is better defined as the melting point of ice. In this scale a temperature difference of 1 degree Celsius is the same as a increment, but the scale is offset by the temperature at which ice melts (273.15 K).

By international agreement the Kelvin and Celsius scales are defined by two fixing points: absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

and the triple point
Triple point
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases of that substance coexist in thermodynamic equilibrium...

of Vienna Standard Mean Ocean Water, which is water specially prepared with a specified blend of hydrogen and oxygen isotopes. Absolute zero is defined as precisely and . 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
Zero-point energy
Zero-point energy is the lowest possible energy that a quantum mechanical physical system may have; it is the energy of its ground state. All quantum mechanical systems undergo fluctuations even in their ground state and have an associated zero-point energy, a consequence of their wave-like nature...

. Matter is in its ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

, and contains no thermal energy
Thermal energy
Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

. The triple point of water is defined as and . This definition serves the following purposes: it fixes 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 establishes that one kelvin has precisely the same magnitude as one degree on the Celsius scale; and it establishes the difference between the null points of these scales as being ( = and = ).

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 Rankine scale, still used in fields of chemical engineering in the U.S., is an absolute scale based on the Fahrenheit increment.

#### Conversion

The following table shows the temperature conversion formulas
Temperature conversion formulas
This is a compendium of temperature conversion formulæ and comparisons.-Celsius :The simple approximation [°F] = [°C] × 2 + 30 works well within the range of normal weather temperatures, underestimating by 6°F at −20°C and over by 4°F at 30°C.-Comparison:-Comparison of temperature scales:...

for conversions to and from the Celsius scale.

#### Plasma physics

The field of plasma physics deals with phenomena of electromagnetic
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

nature that involve very high temperatures. It is customary to express temperature in electronvolt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...

s (eV) or kiloelectronvolts (keV), where 1 eV = . In the study of QCD matter
QCD matter
Quark matter or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high temperatures and densities, billions of times higher than can be produced in equilibrium in laboratories...

one routinely encounters temperatures of the order of a few hundred MeV
MEV
MeV and meV are multiples and submultiples of the electron volt unit referring to 1,000,000 eV and 0.001 eV, respectively.Mev or MEV may refer to:In entertainment:* Musica Elettronica Viva, an Italian musical group...

## Theoretical foundation

Historically, there are several scientific approaches to the explanation of temperature: the classical thermodynamic description based on 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
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...

and quantum theory
Quantum theory
Quantum theory may mean:In science:*Quantum mechanics: a subset of quantum physics explaining the physical behaviours at atomic and sub-atomic levels*Old quantum theory under the Bohr model...

. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics and temperature. 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, using natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

, 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
Metric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...

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
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

is based on a model that decomposes a system into its fundamental particles of matter or into a set of classical or quantum-mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

oscillators and considers the system as a statistical ensemble
Statistical ensemble (mathematical physics)
In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1878, an ensemble is an idealization consisting of a large number of mental copies of a system, considered all at once, each of which represents a possible state that the real...

of microstates. As a collection of classical material particles, temperature is a measure of the mean energy of motion, called kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

, of the particles, whether in solids, liquids, gases, or plasmas. Kinetic energy, a concept of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, is one half the product of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

and the square of a particle's velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. 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 monoatomic perfect gas
Perfect gas
In physics, a perfect gas is a theoretical gas that differs from real gases in a way that makes certain calculations easier to handle. Its behavior is more simplified compared to an ideal gas...

es and, approximately, in most gases, temperature is a measure of the mean particle kinetic energy. It also determines the probability distribution function of the 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
Thermodynamic limit
In thermodynamics, particularly statistical mechanics, the thermodynamic limit is reached as the number of particles in a system, N, approaches infinity...

, meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model.

In the context of thermodynamics, the kinetic energy is also referred to as thermal energy
Thermal energy
Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

. The thermal energy may be partitioned into independent components attributed to the degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

of the particles or to the modes of oscillators in a thermodynamic system
Thermodynamic system
A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....

. In general, the number of these degrees of freedom that are available for the equipartitioning
Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...

of energy depend on the temperature, i.e. the energy region of the interactions under consideration. For solids, the thermal energy is associated primarily with the vibrations
Atom vibrations
The atoms and ions, which are bonded with each other with considerable interatomic forces, are not motionless. Due to the consistent vibrating movements, they are permanently deviating from their equilibrium position. Elastic waves of different lengths, frequencies, and amplitudes run through...

of its atoms or molecules about their equilibrium position. In an ideal monatomic gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems, vibration
Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...

al and rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

al motions also contribute degrees of freedom.

### Kinetic theory of gases

The kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

of gases uses the model of the ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

to relate temperature to the average kinetic energy of the atoms in a container of gas. Classical mechanics defines the kinetic energy as follows:,
where m is the particle mass and v its velocity. 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:,
for each degree of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

, where (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 , where is the thermal de Broglie wavelength. A monoatomic gas has only the three translational degrees of freedom.

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 masses, the heaviest particles will move slower than lighter particles, but have the same average kinetic energy. A neon
Neon
Neon is the chemical element that has the symbol Ne and an atomic number of 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in either low-voltage neon glow lamps or...

atom moves slower relative to a hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

molecule of the same kinetic energy; a pollen particle suspended in water moves in a slow Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

among fast moving water molecules.

### Zeroth law of thermodynamics

It has long been recognized that if two bodies of different temperatures are brought into thermal connection, conductive or radiative, they exchange heat accompanied by changes of other state variables. Left isolated from other bodies, the two connected bodies eventually reach a state of thermal equilibrium
Thermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...

in which no further changes occur. This basic knowledge is relevant to thermodynamics. Some approaches to thermodynamics take this basic knowledge as axiomatic, other approaches select only one narrow aspect of this basic knowledge as axiomatic, and use other axioms to justify and express deductively the remaining aspects of it. The one aspect chosen by the latter approaches is often stated in textbooks as the zeroth law of thermodynamics, but other statements of this basic knowledge are made by various writers.

The usual textbook statement of the zeroth law of thermodynamics
Zeroth law of thermodynamics
The zeroth law of thermodynamics is a generalization principle of thermal equilibrium among bodies, or thermodynamic systems, in contact.The zeroth law states that if two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.Systems are said to...

is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other. This statement is taken to justify a statement that all three systems have the same temperature, but, by itself, it does not justify the idea of temperature as a numerical scale for a concept of hotness which exists on a one-dimensional manifold with a sense of greater hotness. Sometimes the zeroth law is stated to provide the latter justification. For suitable systems, an empirical temperature scale may be defined by the variation of one of the other state variables, such as pressure, when all other coordinates are fixed. The second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

is used to define an absolute thermodynamic temperature scale for systems in thermal equilibrium.

A temperature scale is based on the properties of some reference system to which other thermometers may be calibrated. One such reference system is a fixed quantity of gas. The ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...

indicates that the product of the pressure (p) and volume (V) of a gas is directly proportional to the thermodynamic temperature:
where T is temperature, n is the number of moles of gas and R = is the gas constant
Gas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...

.
Reformulating the pressure-volume term as the sum of classical mechanical particle energies in terms of particle mass, m, and root-mean-square particle speed v, the ideal gas law directly provides the relationship between kinetic energy and temperature:

Thus, one can define a scale for temperature based on the corresponding pressure and volume of the gas: the temperature in kelvins is the pressure in pascals of one mole of gas in a container of one cubic metre, divided by the gas constant. In practice, such a gas thermometer is not very convenient, but other thermometers 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, however, to extrapolate to absolute zero by using the ideal gas law.

### Second law of thermodynamics

In the previous section certain properties of temperature were expressed by the zeroth law of thermodynamics. It is also possible to define temperature in terms of the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

which deals with entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

. Entropy is often thought of as 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.

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 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 come 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 it was just shown that the universe tends to progress so as to maximize entropy, which 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
Heat engine
In thermodynamics, a heat engine is a system that performs the conversion of heat or thermal energy to mechanical work. It does this by bringing a working substance from a high temperature state to a lower temperature state. A heat "source" generates thermal energy that brings the working substance...

is a device for converting thermal energy into mechanical energy, resulting in the performance of work, and analysis of the Carnot heat engine
Carnot heat engine
A Carnot heat engine is a hypothetical engine that operates on the reversible Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in 1824...

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, qH and the heat ejected at the low temperature, qC. The efficiency is the work divided by the heat put into the system or: (2)

where wcy is the work done per cycle. The efficiency depends only on qC/qH. Because qC and qH correspond to heat transfer at the temperatures TC and TH, respectively, qC/qH should be some function of these temperatures: (3)

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:

which implies:

Since the first function is independent of T2, this temperature must cancel on the right side, meaning f(T1,T3) is of the form g(T1)/g(T3) (i.e. f(T1,T3) = f(T1,T2)f(T2,T3) = g(T1)/g(T2g(T2)/g(T3) = g(T1)/g(T3)), where g is a function of a single temperature. A temperature scale can now be chosen with the property that:
(4)

Substituting Equation 4 back into Equation 2 gives a relationship for the efficiency in terms of temperature: (5)

Notice that for TC = 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:

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by: (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: (7)

For a system, where entropy S(E) is a function of its energy E, the temperature T is given by: (8),

i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy.

### Definition from statistical mechanics

The previous section elaborated the historical derivation relating entropy and heat. A modern definition of temperature is given by statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

. It is defined in terms of the fundamental degrees of freedom of a system. Eq.(8) of the previous section is taken to be the defining relation of the temperature. Eq. (7) can be derived from first principles.

### Generalized temperature from single particle statistics

It is possible to extend the definition of temperature even to systems of few particles, like in a quantum dot
Quantum dot
A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...

. 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 partition ensemble, obtained under the hypothesis of ergodicity
Ergodicity
In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.-Etymology:The word ergodic is...

and orthodicity, allows to express the generalized temperature from the ratio of the average time of occupation 1 and 2 of the single/double occupancy system :
where EF is the Fermi energy which tends to the ordinary temperature when N goes to infinity.

### Negative temperature

On the empirical temperature scales, which are not referenced to absolute zero, a negative temperature is one below the zero-point of the scale used. For example, dry ice
Dry ice
Dry ice, sometimes referred to as "Cardice" or as "card ice" , is the solid form of carbon dioxide. It is used primarily as a cooling agent. Its advantages include lower temperature than that of water ice and not leaving any residue...

has a sublimation temperature of which is equivalent to . On the absolute Kelvin scale, however, this temperature is 194.6 K. On the absolute scale of thermodynamic temperature no material can exhibit a temperature smaller than or equal to 0K, both of which are forbidden by the third law of thermodynamics
Third law of thermodynamics
The third law of thermodynamics is a statistical law of nature regarding entropy:For other materials, the residual entropy is not necessarily zero, although it is always zero for a perfect crystal in which there is only one possible ground state.-History:...

.

In the quantum mechanical description of electron and nuclear spin systems that have a limited number of possible states, and therefore a discrete upper limit of energy they can attain, it is possible to obtain a negative temperature
Negative temperature
In physics, certain systems can achieve negative temperatures; that is, their thermodynamic temperature can be a negative quantity. Negative temperatures can be expressed as negative numbers on the kelvin scale....

, which is numerically indeed less than absolute zero. However, this is not the macroscopic temperature of the material, but instead the temperature of only very specific degrees of freedom, that are isolated from others and do not exchange energy by virtue of the equipartition theorem
Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...

.

A negative temperature is experimentally achieved with suitable radio frequency techniques that cause a population inversion
Population inversion
In physics, specifically statistical mechanics, a population inversion occurs when a system exists in state with more members in an excited state than in lower energy states...

of spin states from the ground state. As the energy in the system increases upon population of the upper states, the entropy increases as well, as the system becomes less ordered, but attains a maximum value when the spins are evenly distributed among ground and excited states, after which it begins to decrease, once again achieving a state of higher order as the upper states begin to fill exclusively. At the point of maximum entropy, the temperature function shows the behavior of a singularity
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

, because the slope of the entropy function decreases to zero at first and then turns negative. Since temperature is the inverse of the derivative of the entropy, the temperature formally goes to infinity at this point, and switches to negative infinity as the slope turns negative. At energies higher than this point, the spin degree of freedom therefore exhibits formally a negative thermodynamic temperature. As the energy increases further by continued population of the excited state, the negative temperature approaches zero asymptotically. As the energy of the system increases in the population inversion, a system with a negative temperature is not colder than absolute zero, but rather it has a higher energy than at positive temperature, and may be said to be in fact hotter at negative temperatures. When brought into contact with a system at a positive temperature, energy will be transferred from the negative temperature regime to the positive temperature region.

## Examples of temperature

Temperature Peak emittance wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

Wien's displacement law
Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph...

Kelvin Degrees Celsius
Absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

(precisely by definition)
0 K −273.15 °C Infinite
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

Coldest measured
temperature
450 pK  °C 6,400 km
One millikelvin
(precisely by definition)
0.001 K −273.149 °C  m
FM broadcasting is a broadcasting technology pioneered by Edwin Howard Armstrong which uses frequency modulation to provide high-fidelity sound over broadcast radio. The term "FM band" describes the "frequency band in which FM is used for broadcasting"...

)
Water's triple point
Triple point
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases of that substance coexist in thermodynamic equilibrium...

(precisely by definition)
273.16 K 0.01 °C 10,608.3 nm
(long wavelength I.R.
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

)
Water's boiling point
Boiling point
The boiling point of an element or a substance is the temperature at which the vapor pressure of the liquid equals the environmental pressure surrounding the liquid....

373.1339 K 99.9839 °C 7,766.03 nm
(mid wavelength I.R.)
Incandescent lamp
Incandescent light bulb
The incandescent light bulb, incandescent lamp or incandescent light globe makes light by heating a metal filament wire to a high temperature until it glows. The hot filament is protected from air by a glass bulb that is filled with inert gas or evacuated. In a halogen lamp, a chemical process...

2500 K ≈2,200 °C 1,160 nm
(near infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...

)
Sun's
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

visible surface
5,778 K 5,505 °C 501.5 nm
(green-blue light)
Lightning bolt's
Lightning
Lightning is an atmospheric electrostatic discharge accompanied by thunder, which typically occurs during thunderstorms, and sometimes during volcanic eruptions or dust storms...

channel
28 kK 28,000 °C 100 nm
(far ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...

light)
Sun's core 16 MK 16 million °C 0.18 nm (X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

s)
Thermonuclear weapon
Nuclear weapon
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission or a combination of fission and fusion. Both reactions release vast quantities of energy from relatively small amounts of matter. The first fission bomb test released the same amount...

(peak temperature)
350 MK 350 million °C 8.3×10−3 nm
(gamma ray
Gamma ray
Gamma radiation, also known as gamma rays or hyphenated as gamma-rays and denoted as γ, is electromagnetic radiation of high frequency . Gamma rays are usually naturally produced on Earth by decay of high energy states in atomic nuclei...

s)
Sandia National Labs'
Z machine
Z machine
The Z machine is the largest X-ray generator in the world and is designed to test materials in conditions of extreme temperature and pressure. Operated by Sandia National Laboratories, it gathers data to aid in computer modeling of nuclear weapons...

2 GK 2 billion °C 1.4×10−3 nm
(gamma rays)
Core of a high-mass
star on its last day
Silicon burning process
In astrophysics, silicon burning is a very brief sequence of nuclear fusion reactions that occur in massive stars with a minimum of about 8–11 solar masses. Silicon burning is the final stage of fusion for massive stars that have run out of the fuels that power them for their long lives in the main...

3 GK 3 billion °C 1×10−3 nm
(gamma rays)
Merging binary neutron
star
Neutron star
A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...

system
350 GK 350 billion °C 8×10−6 nm
(gamma rays)
Relativistic Heavy
Ion Collider
Relativistic Heavy Ion Collider
The Relativistic Heavy Ion Collider is one of two existing heavy-ion colliders, and the only spin-polarized proton collider in the world. It is located at Brookhaven National Laboratory in Upton, New York and operated by an international team of researchers...

1 TK 1 trillion °C 3×10−6 nm
(gamma rays)
CERN's
CERN
The European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...

proton vs
nucleus collisions
10 TK 10 trillion °C 3×10−7 nm
(gamma rays)
Universe 5.391×10−44 s
Planck time
In physics, the Planck time, , is the unit of time in the system of natural units known as Planck units. It is the time required for light to travel, in a vacuum, a distance of 1 Planck length...

after the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

1.417×1032 K
Planck temperature
Planck temperature is the greatest physically-possible temperature, according the set of theories proposed by the German physicist Max Planck. It's part of a system of five natural units known as Planck units, based on universal physical constants....

1.417×1032 °C 1.616×10−26 nm
(Planck Length)

• For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated strictly per the two-point definition of thermodynamic temperature.

• The 2500 K value is approximate. The 273.15 K difference between K and °C is rounded to 300 K to avoid false precision
False precision
False precision occurs when numerical data are presented in a manner that implies better precision than is actually the case; since precision is a limit to accuracy, this often leads to overconfidence in the accuracy as well.In science and engineering, convention dictates that...

in the Celsius value.

• For a true black-body (which tungsten filaments are not). Tungsten filaments' emissivity is greater at shorter wavelengths, which makes them appear whiter.

• 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.

• The 273.15 K difference between K and °C is without the precision of these values.

• 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.

• Scale of temperature
Scale of temperature
Scale of temperature is a way to measure temperature quantitatively.-Formal description:According to the zeroth law of thermodynamics, being in thermal equilibrium is an equivalence relation. Thus all thermal systems may be divided into a quotient set by this equivalence relation, denoted below as M...

• Atmospheric temperature
Atmospheric temperature
Atmospheric temperature is a measure of temperature at different levels of the Earth's atmosphere. It is governed by many factors, including incoming solar radiation, humidity and altitude...

• Color temperature
Color temperature
Color temperature is a characteristic of visible light that has important applications in lighting, photography, videography, publishing, manufacturing, astrophysics, and other fields. The color temperature of a light source is the temperature of an ideal black-body radiator that radiates light of...

• Dry-bulb temperature
Dry-bulb temperature
The dry-bulb temperature is the temperature of air measured by a thermometer freely exposed to the air but shielded from radiation and moisture. Dry bulb temperature is the temperature that is usually thought of as air temperature, and it is the true thermodynamic temperature. It is the...

• Heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

• Heat convection
• ISO 1
ISO 1
ISO 1 is an international standard that specifies the standard reference temperature for geometrical product specification and verification. The temperature is fixed at 20 degrees centigrade, which is equal to 293.15 Kelvins and 68 degrees Fahrenheit....

• ITS-90
International Temperature Scale of 1990
The International Temperature Scale of 1990 is an equipment calibration standard for making measurements on the Kelvin and Celsius temperature scales. ITS–90 is an approximation of the thermodynamic temperature scale that facilitates the comparability and compatibility of temperature measurements...

• Maxwell's demon
Maxwell's demon
In the philosophy of thermal and statistical physics, Maxwell's demon is a thought experiment created by the Scottish physicist James Clerk Maxwell to "show that the Second Law of Thermodynamics has only a statistical certainty." It demonstrates Maxwell's point by hypothetically describing how to...

• Orders of magnitude (temperature)
Orders of magnitude (temperature)
-Detailed list for 100 K to 1000 K:Most ordinary human activity takes place at temperatures of this order of magnitude. Circumstances where water naturally occurs in liquid form are shown in light grey.-External links:*...

• Outside air temperature
Outside air temperature
In aviation terminology, the outside air temperature or static air temperature refers to the temperature of the air around an aircraft, but unaffected by the passage of the aircraft through it.-Aviation usage:...

• Planck temperature
Planck temperature
Planck temperature is the greatest physically-possible temperature, according the set of theories proposed by the German physicist Max Planck. It's part of a system of five natural units known as Planck units, based on universal physical constants....

• Rankine scale
• Relativistic heat conduction
Relativistic heat conduction
The theory of relativistic heat conduction claims to be the only model for heat conduction that is compatible with the theory of special relativity, the second law of thermodynamics, electrodynamics, and quantum mechanics, simultaneously...

• Stagnation temperature
Stagnation temperature
In thermodynamics and fluid mechanics, stagnation temperature is the temperature at a stagnation point in a fluid flow. At a stagnation point the speed of the fluid is zero and all of the kinetic energy has been converted to internal energy and is added to the local static enthalpy...

Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

• Thermoception
Thermoception
Thermoception or thermoreception is the sense by which an organism perceives temperature. The details of how temperature receptors work are still being investigated. Ciliopathy is associated with decreased ability to sense heat, thus cilia may aid in the process...

• Thermodynamic (absolute) temperature
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

• Thermography
Thermography
Infrared thermography, thermal imaging, and thermal video are examples of infrared imaging science. Thermal imaging cameras detect radiation in the infrared range of the electromagnetic spectrum and produce images of that radiation, called thermograms...

• Thermometer
Thermometer
Developed during the 16th and 17th centuries, a thermometer is a device that measures temperature or temperature gradient using a variety of different principles. A thermometer has two important elements: the temperature sensor Developed during the 16th and 17th centuries, a thermometer (from the...

• Body temperature
Thermoregulation
Thermoregulation is the ability of an organism to keep its body temperature within certain boundaries, even when the surrounding temperature is very different...

(Thermoregulation)
• Virtual temperature
Virtual temperature
In atmospheric thermodynamics, the virtual temperature T_v of a moist air parcel is the temperature at which a theoretical dry air parcel would have a total pressure and density equal to the moist parcel of air.-Description:...

• Wet Bulb Globe Temperature
Wet Bulb Globe Temperature
The Wet Bulb Globe Temperature is a composite temperature used to estimate the effect of temperature, humidity, wind speed and solar radiation on humans. It is used by industrial hygienists, athletes, and the military to determine appropriate exposure levels to high temperatures...

• Wet-bulb temperature
Wet-bulb temperature
The wet-bulb temperature is a type of temperature measurement that reflects the physical properties of a system with a mixture of a gas and a vapor, usually air and water vapor. Wet bulb temperature is the lowest temperature that can be reached by the evaporation of water only. It is the...

• 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. Princeton, N.J.: Van Nostrand.
• T. J. Quinn (1983), Temperature, Academic Press, London.