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 temperature at which the particle constituents of matter have minimal motion and can become no colder.^{[1]} ^{[2]}
At its simplest, temperature arises from the kinetic energy of the vibrational motions of matter's particle constituents (molecules, atoms, and subatomic particles). The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, contribute the total thermal energy (loosely, the heat energy) within a substance. Thus, thermal energy may be stored in a number of ways within a substance, but only the kinetic energy of particles contributes to the substance's temperature.
Temperature arises from the random submicroscopic vibrations of the particle constituents of matter. These motions comprise the kinetic energy in a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy of a certain kind of vibrational motion of its constituent particles called translational motions. Translational motions are ordinary, wholebody movements in threedimensional space whereby particles move about and exchange energy in collisions. Figure 1 below shows translational motion in gases; Figure 4 below shows translational motion in solids. Thermodynamic temperature's null point, absolute zero, is the temperature at which the particle constituents of matter are as close as possible to complete rest; that is, they have minimal motion, retaining only quantum mechanical motion.^{[3]} Zero kinetic energy remains in a substance at absolute zero (see Heat energy at absolute zero, below).
Throughout the scientific world where measurements are made in SI units, thermodynamic temperature is measured in kelvins (symbol: K). Many engineering fields in the U.S. however, measure thermodynamic temperature using the Rankine scale.
By international agreement, the unit kelvin and its scale are defined by two points: absolute zero, and the triple point of Vienna Standard Mean Ocean Water (water with a specified blend of hydrogen and oxygen isotopes). Absolute zero, the lowest possible temperature, is defined as being precisely 0 K and −273.15 °C. The triple point of water is defined as being precisely 273.16 K and 0.01 °C. This definition does three things:
Temperatures expressed in kelvins are converted to degrees Rankine simply by multiplying by 1.8 as follows: T_{°R} = 1.8T_{K}, where T_{K} and T_{°R} are temperatures in kelvin and degrees Rankine respectively. Temperatures expressed in degrees Rankine are converted to kelvins by dividing by 1.8 as follows: T_{K} = .
See main article: ITS90.
Although the Kelvin and Celsius scales are defined using absolute zero (0 K) and the triple point of water (273.16 K and 0.01 °C), it is impractical to use this definition at temperatures that are very different from the triple point of water. ITS90 is then designed to represent the thermodynamic temperature as closely as possible throughout its range. Many different thermometer designs are required to cover the entire range. These include helium vapor pressure thermometers, helium gas thermometers, standard platinum resistance thermometers (known as SPRTs, PRTs or Platium RTDs) and monochromatic radiation thermometers.
At its simplest, temperature arises from the kinetic energy of the vibrational motions of matter's particle constituents (molecules, atoms, and subatomic particles). The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, contribute the total thermal energy (loosely, the heat energy) within a substance. Thus, thermal energy may be stored in a number of ways within a substance, but only the kinetic energy of particles contributes to the substance's temperature. The heat capacity, which relates heat input and temperature change, is discussed below.
The relationship of kinetic energy, mass, and velocity is given by the formula E_{k} = mv^{2}.^{[4]} Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy, and precisely the same temperature, as those with four times the mass but half the velocity.
The thermodynamic temperature of any bulk quantity of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as translational motion. These simple movements in the three x, y, and z–axis dimensions of space means the particles move in the three spatial degrees of freedom. This particular form of kinetic energy is sometimes referred to as kinetic temperature. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the ideal gas law's formula pV = nRT and is embodied in the gas laws.
The extent to which the kinetic energy of translational motion of an individual atom or molecule (particle) in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the Boltzmann constant (symbol: k_{B}). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an individual particle's translational motion as follows:
\bar{E}=  3 
2 
k_{B}T
where:
\scriptstyle\bar{E}
While the Boltzmann constant is useful for finding the mean kinetic energy of a particle, it's important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in Figure 1 above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in Fig. 2 shows the speed distribution of 5500 K helium atoms. They have a most probable speed of 4.780 km/s. However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the x–axis to the right). This graph uses inverse speed for its x–axis so the shape of the curve can easily be compared to the curves in Figure 5 below. In both graphs, zero on the x–axis represents infinite temperature. Additionally, the x and y–axis on both graphs are scaled proportionally.
Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are very fast^{[5]} and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a recordsetting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool caesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second in order to calculate their temperature.^{[6]} Formulas for calculating the velocity and speed of translational motion are given in the following footnote.^{[7]}
There are other forms of heat energy besides the kinetic energy of translational motion. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain their internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements. These are all types of internal degrees of freedom. This makes molecules distinct from monatomic substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom. Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an internal temperature. Even though these motions are called internal, the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the twoway exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as heat is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of nonlocal thermodynamic equilibrium (LTE) phenomena such as combustion, the sublimation of solids, and the diffusion of hot gases in a partial vacuum.
The kinetic energy stored internally in molecules causes substances to contain more heat energy at any given temperature and to absorb additional heat energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions is not at that same instant contributing to the molecules' translational motions.^{[8]} This extra kinetic energy simply increases the amount of heat energy a substance absorbs for a given temperature rise. This property is known as a substance's specific heat capacity.
Different molecules absorb different amounts of heat energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances' molecules possess more internal degrees of freedom than others do. For instance, roomtemperature nitrogen, which is a diatomic molecule, has five active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has fivethirds the specific heat capacity per mole (a specific number of molecules) as do the monatomic gases.^{[9]} Another example is gasoline (see table showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.
Heat conduction is the diffusion of heat energy from hot parts of a system to cold. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term bulk in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever heat energy diffuses within an isolated system, temperature differences within the system decrease (and entropy increases).
One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in Fig. 1. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can move forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; convection speeds this process even more.^{[10]}
Translational motion in solids however, takes the form of phonons (see Fig. 4 at right). Phonons are constrained, quantized wave packets traveling at the speed of sound for a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phononbased heat conduction is usually inefficient^{[11]} and such solids are considered thermal insulators (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.
Metals however, are not restricted to only phononbased heat conduction. Heat energy conducts through metals extraordinarily quickly because instead of direct moleculetomolecule collisions, the vast majority of heat energy is mediated via very light, mobile conduction electrons. This is why there is a nearperfect correlation between metals' thermal conductivity and their electrical conductivity.^{[12]} Conduction electrons imbue metals with their extraordinary conductivity because they are delocalized (i.e., not tied to a specific atom) and behave rather like a sort of quantum gas due to the effects of zeropoint energy (for more on ZPE, see Note 1 below). Furthermore, electrons are relatively light with a rest mass only th that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Isaac Newton wrote with his third law of motion,
However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, heat energy is readily borne by mobile conduction electrons. Additionally, because they're delocalized and very fast, kinetic heat energy conducts extremely quickly through metals with abundant conduction electrons.
Thermal radiation is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal photons (known as blackbody radiation). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even individual molecules with internal temperatures greater than absolute zero also emit blackbody radiation from their atoms. In any bulk quantity of a substance at equilibrium, blackbody photons are emitted across a range of wavelengths in a spectrum that has a bell curvelike shape called a Planck curve (see graph in Fig. 5 at right). The top of a Planck curve (the peak emittance wavelength) is located in a particular part of the electromagnetic spectrum depending on the temperature of the blackbody. Substances at extreme cryogenic temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short gamma rays (see Table of common temperatures).
Blackbody radiation diffuses heat energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Blackbody photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.
As established by the Stefan–Boltzmann law, the intensity of blackbody radiation increases as the fourth power of absolute temperature. Thus, a blackbody at 824 K (just short of glowing dull red) emits 60 times the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, blackbody radiation can be the principal mechanism by which heat energy escapes a system.
The full range of the thermodynamic temperature scale, from absolute zero to absolute hot, and some notable points between them are shown in the table below.
kelvin  Peak emittance wavelength^{[13]} of blackbody photons  
Absolute zero (precisely by definition)  0 K  ∞> 
Coldest measured temperature^{ }^{[14]}  450 pK  6,400 kilometers 
One millikelvin (precisely by definition)  0.001 K  2.897 77 meters (Radio, FM band)^{[15]} 
Water's triple point (precisely by definition)  273.16 K  10,608.3 nm (Long wavelength I.R.) 
Incandescent lamp^{B}  2500 K  1160 nm (Near infrared)^{C} 
Sun’s visible surface^{C}^{[16]}  5778 K  501.5 nm (Green light) 
Lightning bolt’s channel  28,000 K  100 nm (Far Ultraviolet light) 
Sun’s core  16 MK  0.18 nm (Xrays) 
Thermonuclear weapon (peak temperature)^{[17]}  350 MK  8.3 × 10^{−3} nm (Gamma rays) 
Sandia National Labs’ Z machine^{ D}^{[18]}  2 GK  1.4 × 10^{−3} nm (Gamma rays) 
Core of a high–mass star on its last day^{[19]}  3 GK  1 × 10^{−3} nm (Gamma rays) 
Merging binary neutron star system ^{[20]}  350 GK  8 × 10^{−6} nm (Gamma rays) 
Gammaray burst progenitors^{[21]}  1 TK  3 × 10^{−6} nm (Gamma rays) 
Relativistic Heavy Ion Collider^{[22]}  1 TK  3 × 10^{−6} nm (Gamma rays) 
CERN’s proton vs. nucleus collisions^{[23]}  10 TK  3 × 10^{−7} nm (Gamma rays) 
Universe 5.391 × 10^{−44} s after the Big Bang  1.417 × 10^{32} K  1.616 × 10^{−26} nm (Planck frequency)^{[24]} 
The kinetic energy of particle motion is just one contributor to the total heat energy in a substance; another is phase transitions, which are the potential energy of molecular bonds that can form in a substance as it cools (such as during condensing and freezing). The heat energy required for a phase transition is called latent heat. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to break chemical bonds (such as during evaporation and melting). Almost everyone is familiar with the effects of phase transitions; for instance, steam at 100 °C can cause severe burns much faster than the 100 °C air from a hair dryer. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.
Even though heat energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic alloys exhibit no temperature change whatsoever while they undergo them (see Fig. 7, below right). Consider one particular type of phase transition: melting. When a solid is melting, crystal lattice chemical bonds are being broken apart; the substance is transitioning from what is known as a more ordered state to a less ordered state. In Fig. 7, the melting of ice is shown within the lower left box heading from blue to green.
At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are, on average, at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are allornothing forces: they either hold fast, or break; there is no inbetween state. Consequently, when a substance is at its melting point, every joule of added heat energy only breaks the bonds of a specific quantity of its atoms or molecules,^{[25]} converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional heat energy can't make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), heat energy must be removed from a substance.
As stated above, the heat energy required for a phase transition is called latent heat. In the specific cases of melting and freezing, it's called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements.^{[26]} If the substance is one of the monatomic gases, (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole.^{[27]} Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the heat energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals' ratios are even greater, typically in the range of 400 to 1200 times.^{[28]} And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a onedegree increase.^{[29]}
Water's sizable enthalpy of vaporization is why one's skin can be burned so quickly as steam condenses on it (heading from red to green in Fig. 7 above). In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a subambient wetbulb temperature that is dependent on relative humidity). Water's highly energetic enthalpy of vaporization is also an important factor underlying why solar pool covers (floating, insulated blankets that cover swimming pools when not in use) are so effective at reducing heating costs: they prevent evaporation. For instance, the evaporation of just 20 mm of water from a 1.29meterdeep pool chills its water 8.4 degrees Celsius (15.1 °F).
The total kinetic energy of all particle motion, including that of conduction electrons, plus the potential energy of phase changes, plus zeropoint energy^{[3]} comprise the internal energy of a substance, which is its total heat energy. The term internal energy mustn't be confused with internal degrees of freedom. Whereas the internal degrees of freedom of molecules refers to one particular place where kinetic energy is bound, the internal energy of a substance comprises all forms of heat energy.
As a substance cools, different forms of heat energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions are liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic temperature decreases); the internal motions of molecules diminish (their internal temperature decreases); conduction electrons (if the substance is an electrical conductor) travel somewhat slower;^{[30]} and blackbody radiation's peak emittance wavelength increases (the photons' energy decreases). When the particles of a substance are as close as possible to complete rest and retain only ZPEinduced quantum mechanical motion, the substance is at the temperature of absolute zero (T=0).
Note that whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero heat energy; one must be very precise with what one means by heat energy. Often, all the phase changes that can occur in a substance, will have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, T=0 helium remains liquid at room pressure and must be under a pressure of at least 25abbr=onNaNabbr=on to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21 joules per mole) that the motioninducing effect of zeropoint energy is sufficient to prevent it from freezing at lower pressures. Only if under at least 25bar of pressure will this latent heat energy be liberated as helium freezes while approaching absolute zero. A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars, or hundreds of gigapascals). These are known as solidsolid phase transitions wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one.
The above complexities make for rather cumbersome blanket statements regarding the internal energy in T=0 substances. Regardless of pressure though, what can be said is that at absolute zero, all solids with a lowestenergy crystal lattice such those with a closestpacked arrangement (see Fig. 8, above left) contain minimal internal energy, retaining only that due to the everpresent background of zeropoint energy.^{[3]} ^{ }^{[31]} One can also say that for a given substance at constant pressure, absolute zero is the point of lowest enthalpy (a measure of work potential that takes internal energy, pressure, and volume into consideration).^{[32]} Lastly, it is always true to say that all T=0 substances contain zero kinetic heat energy.^{[3]} ^{ }^{[7]}
Thermodynamic temperature is useful not only for scientists, it can also be useful for laypeople in many disciplines involving gases. By expressing variables in absolute terms and applying Gay–Lussac's law of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a relatively cold pressure of 200 kPagage, then in absolute terms (relative to a vacuum), its pressure is 300 kPaabsolute.^{[33]} ^{}
>^{[34]} ^{ }>^{[35]} Room temperature ("cold" in tire terms) is 296 K. If the tire pressure is 20 °C hotter (20 kelvins), the solution is calculated as = 6.8% greater thermodynamic temperature and absolute pressure; that is, a pressure of 320 kPaabsolute, which is 220 kPagage.Earth's proximity to the Sun is the reason why almost everything near Earth's surface is warm with a temperature substantially above absolute zero.^{[36]} Solar radiation constantly replenishes heat energy that Earth loses into space and a relatively stable state of near equilibrium is achieved. Because of the wide variety of heat diffusion mechanisms (one of which is blackbody radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287 to 288 K (14 to 15 °C). The more an object's or system's temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.
Strictly speaking, the temperature of a system is welldefined only if its particles (atoms, molecules, electrons, photons) are at equilibrium, so that their energies obey a Boltzmann distribution (or its quantum mechanical counterpart). There are many possible scales of temperature, derived from a variety of observations of physical phenomena. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratio T_{2}/T_{1} of two temperaturesT_{1} andT_{2} is the same in all absolute scales.
Loosely stated, temperature controls the flow of heat between two systems, and the universe as a whole, as with any natural system, tends to progress so as to maximize entropy. This suggests that there should be a relationship between temperature and entropy. To elucidate this, consider first the relationship between heat, work and temperature. One way to study this is to analyze a heat engine, which is a device for converting heat into mechanical work, such as the Carnot heat engine. Such a heat engine functions by using a temperature gradient between a high temperatureT_{H} and a low temperature T_{C} to generate work, and the work done (per cycle, say) by the heat engine is equal to the difference between the heat energy q_{H} put into the system at the high temperature and the heat q_{C} ejected at the low temperature (in that cycle). The efficiency of the engine is the work divided by the heat put into the system or
rm{efficiency}=
w_{cy}  
q_{H} 
=
q_{Hq}_{C}  
q_{H} 
=1
q_{C}  
q_{H} 
(1)
where w_{cy} is the work done per cycle. Thus the efficiency depends only on q_{C}/q_{H}.
Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient.Thus, any reversible heat engine operating between temperatures T_{1} and T_{2} must have the same efficiency, that is to say, the effiency is the function of only temperatures
q_{C}  
q_{H} 
=f(T_{H,T}_{C) }(2).
In addition, a reversible heat engine operating between temperatures T_{1} and T_{3} must have the same efficiency as one consisting of two cycles, one between T_{1} and another (intermediate) temperature T_{2}, and the second between T_{2} andT_{3}. A quick way to see this is that should this not be the case, then energy (in the form of Q) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles.
With this understanding of Q_{1}, Q_{2} and Q_{3}, we note also that mathematically,
f(T_{1,T}_{3)}=
q_{3}  
q_{1} 
=
q_{2}q_{3}  
q_{1}q_{2} 
=f(T_{1,T}_{2)f(T}_{2,T}_{3). }
But the first function is NOT a function of T_{2}, therefore the product of the final two functions MUST result in the removal of T_{2} as a variable. The only way is therefore to define the function f as follows:
f(T_{1,T}_{2)}=
g(T_{1)}  
g(T_{2)} 
.
and
f(T_{2,T}_{3)}=
g(T_{2)}  
g(T_{3)} 
.
so that
f(T_{1,T}_{3)}=
g(T_{1)}  
g(T_{3)} 
=
q_{1}  
q_{3} 
.
i.e. The ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our
f(T)
f(T)=T
It is to be noted that such a definition coincides with that of the ideal gas derivation; also it is this definition of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of T_{H} and T_{L}, and hence derive that the (complete) Carnot cycle is isentropic:
q_{C}  
q_{H} 
=f(T_{H,T}_{C)}=
T_{C}  
T_{H} 
. (3).
Substituting this back into our first formula for efficiency yields a relationship in terms of temperature:
rm{efficiency}=1
q_{C}  
q_{H} 
=1
T_{C}  
T_{H} 
(4).
Notice that for T_{C}=0 the efficiency is 100% and that efficiency becomes greater than 100% for T_{C}. Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives
q_{H}  
T_{H} 

q_{C}  
T_{C} 
=0,
where the negative sign indicates heat ejected from the system. The generalization of this equation is Clausius theorem, which suggests the existence of a state function S (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by
dS=
dq_{rev}  
T 
(5),
where the subscript indicates heat transfer in a reversible process. The function S corresponds to the entropy of the system, mentioned previously, and the change of S around any cycle is zero (as is necessary for any state function). Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid logic loop, we should first define entropy through statistical mechanics):
T=
dq_{rev}  
dS 
.
For a system in which the entropy S is a function S(E) of its energy E, the thermodynamic temperature T is therefore given by
1  
T 
=
dS  
dE 
,
so that the reciprocal of the thermodynamic temperature is the rate of increase of entropy with energy.
In the following notes, wherever numeric equalities are shown in concise form, such as, the two digits between the parentheses denotes the uncertainty at 1σ (1 standard deviation, 68% confidence level) in the two least significant digits of the significand.
\bar{v}=\sqrt{
k_{B}T  
m 
where:
\bar{v}
In the above formula, molecular mass, m, in kilograms per particle is the quotient of a substance's molar mass (also known as atomic weight, atomic mass, relative atomic mass, and unified atomic mass units) in g/mol or daltons divided by (which is the Avogadro constant times one thousand). For diatomic molecules such as H_{2}, N_{2}, and O_{2}, multiply atomic weight by two before plugging it into the above formula.
The mean speed (not vectorisolated velocity) of an atom or molecule along any arbitrary path is calculated as follows:
\bar{s}=\bar{v}\sqrt{3}
where:
\bar{s}
Note that the mean energy of the translational motions of a substance's constituent particles correlates to their mean speed, not velocity. Thus, substituting
\bar{s}
Note too that the Boltzmann constant and its related formulas establish that absolute zero is the point of both zero kinetic energy of particle motion and zero kinetic velocity (see also Note 1 above).