Thermodynamic temperature

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

The Z machine at Sandia National Laboratory in Albuquerque, New Mexico, U.S.A., set a record man–made temperature for a bulk quantity of matter of greater than two billion kelvin. Courtesy, Sandia National Laboratories.

Thermodynamic temperature (previously called absolute temperature) is the absolute measure of temperature. Temperature arises from the random microscopic vibrations of the particle constituents of matter. These motions comprise the kinetic energy in a substance. More specifically, the thermodynamic temperature of any bulk quantity of matter is the measure of the average kinetic energy of the translational motions of its constituent particles.

The thermodynamic temperature scale’s null point, absolute zero, is the theoretical point where all molecular motion ceases and they are at complete rest (except for quantum mechanical motion). Throughout the scientific world where measurements are made in SI units, thermodynamic temperature is measured in kelvins (symbol: K). Many engineering fields in the U.S. measure thermodynamic temperature using the Rankine scale.

The unit “kelvin” and its scale are, by international agreement, defined by two points: absolute zero, and the triple point of specially prepared water. Absolute zero is defined as being precisely 0 K and –273.15 °C (–459.67 °F). The triple point of water is defined as being precisely 273.16 K and 0.01 °C. This definition does three things: 1) it fixes the magnitude of the kelvin unit as being precisely 1 part in 273.16 parts the difference between absolute zero and the triple point of water; 2) it establishes that one kelvin has precisely the same magnitude as a one degree increment on the Celsius scale; and 3) it establishes the difference between the two scales’ null points as being precisely 273.15 kelvins (0 K = –273.15 °C and 273.16 K = 0.01 °C).

The full range of the thermodynamic temperature scale and some notable points along it are shown in the table below.

	kelvin	Celsius	Peak emittance wavelength of black-body photons
Absolute zero (precisely by definition)	0 K	–273.15 °C	∞ (No emission)
One millikelvin	0.001 K	–273.149 °C	2.897 77 meters (Radio, FM band)
Water’s triple point (precisely by definition)	273.16 K	0.01 °C	10,608.3 nm (Long wavelength I.R.)
Water’s boiling point A	373.1339 K	99.9839 °C	7766.03 nm (Mid wavelength I.R.)
Incandescent lampB	2500 K	~2200 °C	1160 nm (Near infrared)C
Sun’s visible surfaceD	5778 K	5505 °C	501.5 nm (Green light)
Lightning bolt’s channel E	28,000 K	28,000 °C	100 nm (Far Ultraviolet light)
Sun’s core E	16 MK	16 million °C	0.18 nm (X-rays)
Thermonuclear weapon (peak temperature)E	350 MK	350 million °C	8.3 × 10–3 nm (Gamma rays)
Sandia National Labs’ Z machine E	2 GK	2 billion °C	1.4 × 10–3 nm (Gamma rays)F
Core of a high–mass star on its last day E	3 GK	3 billion °C	1 × 10–3 nm (Gamma rays)
Merging binary neutron star system E	350 GK	350 billion °C	8 × 10–6 nm (Gamma rays)
Universe 5.391 × 10–44 s after the Big Bang E	1.417 × 1032 K	1.417 × 1032 °C	1.616 × 10–26 nm (Planck frequency)

^A For Vienna Standard Mean Ocean Water at one standard atmosphere (101.325 kPa) when calibrated strictly per the two-point definition of thermodynamic temperature.
^B The 2500 K value is approximate. The 273.15 K difference between K and °C is rounded to 300 K to avoid invalid precision in the Celsius value.
^C For a true blackbody (which tungsten filaments are not). Tungsten filaments’ emissivity is greater at shorter wavelengths which makes them appear whiter.
^D Effective photosphere temperature. The 273.15 K difference between K and °C is rounded to 273 K to avoid invalid precision in the Celsius value.
^E The 273.15 K difference between K and °C is ignored to avoid invalid precision in the Celsius value.
^F For a true blackbody (which the plasma was not). The Z machine’s dominant emission originated from 40 MK electrons (soft x-ray emissions) within the plasma.

The relationship of temperature, kinetic energy and heat energy

The Nature of Kinetic Energy, Translational Motion, and Temperature

At its simplest, “temperature” is the measure of the kinetic energy resulting from the motions of matter’s particle constituents (molecules, atoms, and subatomic particles). The full variety of these kinetic motions contribute to the total heat energy in a substance. At non-relativistic temperatures (less than about 30 GK), the relationship of kinetic energy, mass, and velocity is given by the formula E_k = ¹/₂m • v ². Accordingly, for simple particles like atoms and electrons, those with one unit of mass moving at one unit of velocity have the same kinetic energy—and the same temperature—as those with twice the mass but only 70.7% of the velocity.

The thermodynamic temperature of any bulk quantity of a substance is directly proportional to the kinetic energy of a specific kind of particle motion known as translational motion. Translational motions are ordinary, whole-body movements in 3D space whereby the particles move about and exchange energy in collisions (like rubber balls in a vigorously shaken container). These simple movements in the three X, Y, and Z–axis dimensions of space means the particles have the three spatial degrees of freedom. Translational motions are but one form of heat energy and are what give gases their pressure and, at normal earthly temperatures or greater, the vast majority of their volume.

The High Speeds of Atomic Motion

Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are very fast and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool cesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature.

The Conduction and Diffusion of Heat Energy

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

The Internal Motions of Molecules and Specific Heat

There are other forms of heat energy besides translational motions. Molecules also have various internal vibrational and rotational degrees of freedom. This is because molecules are complex objects; they are a population of atoms that can move about within a molecule in different ways. Heat energy is stored in these internal motions which gives molecules an internal temperature. Even though these motions are called “internal,” the external portions of molecules still move—rather like the jiggling of a water balloon. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as heat is removed from molecules, both their internal and translational kinetic energies (temperatures) simultaneously diminish in equal proportions.

The heat energy stored internally in molecules does not contribute to the temperature of a bulk quantity of a substance. This is because any kinetic energy that is, at a given instant, bound up in internal motions is not at that same instant contributing to the molecules’ translational motions. Individual molecules with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. Since the internal temperature of the molecules in any bulk quantity of a substance are, on average, equal to the temperature of their translational motions, the distinction is usually of interest only in the detailed study of certain thermodynamic phenomenon such as the sublimation of solids and the diffusion of hot gases in a partial vacuum.

Different molecules absorb different amounts of heat energy for each incremental increase in temperature. Water for instance, can absorb a large amount of heat energy per mole (specific number of particles) with only a modest temperature change. This property is known as a substance’s specific heat. High specific heat capacity arises because a substance’s molecules possess a greater number of degrees of freedom. Water has six active degrees of freedom, the maximum available. Not surprisingly, water gas molecules (steam molecules) have twice the specific heat capacity per mole as do the monatomic gases such as helium and argon that consist of individual atoms and which move only within the three degrees of freedom comprising translational motion.

Heat Energy and Absolute Zero

As a substance cools, all forms of heat energy and their related effects simultaneously decrease in magnitude: the translational motions of atoms diminish, both the internal and translational motions of molecules diminish, conduction electrons (if the substance is an electrical conductor) travel somewhat slower, and black-body radiation’s wavelength increases (the photons’ energy decreases). When no more heat energy remains in a substance the molecules are at complete rest (except for quantum mechanical motion), the substance is at absolute zero.

The Heat of Phase Changes

Water’s temperature does not change during phase transitions as heat flows into our out of it. The total heat capacity of a mole of water in its liquid phase (the green line) is 7.5507 kJ.

The kinetic energy of motion is just one contributor to the total heat energy in a substance. The other is the potential energy of molecular bonds that can yet form in a substance as it cools (such as during condensing and freezing). These processes are known as phase transitions. Anyone who has compared the 100 °C air from a hair dryer to 100 °C steam knows that the steam can cause severe burns whereas the air can not. The burn occurs because a large amount of heat energy is liberated as steam condenses into liquid water on the skin. Even though heat energy is liberated or absorbed during phase transitions, pure chemical elements and compounds exhibit no temperature change whatsoever while they undergo them (see graph at right). This phenomenon can be readily understood by examining one particular type of phase transition: the melting of a solid.

When a solid melts, crystal lattice chemical bonds break apart; the substance has gone from what is known as a more ordered state to a less ordered state. In the graph, the melting of 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 the lattice bonds can withstand without breaking and jumping to a higher quantum energy state. Quantum transitions are a complete jump from one energy level to another; no intermediate values are possible. So during melting, every joule of heat energy that is added to a substance only causes the bonds of a specific quantity of atoms or molecules to jump to the next quantum state; no kinetic energy is added to translational motion (which gives a bulk quantity of any substance its temperature). The effect is rather like popcorn: at a certain temperature, additional heat energy can’t make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), heat energy must be removed from a substance.

The heat energy required for a phase transition is called latent heat. In the specific case of melting, it’s called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements. 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. 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. And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a one–degree increase. Water’s sizable enthalpy of vaporization is why one’s skin gets burned so quickly as steam condenses on it (heading from red to green in the graph above). And in the opposite direction, this is why one’s skin feels cool as liquid water on it evaporates (a process that occurs at a sub–ambient wet–bulb temperature that is dependent on relative humidity).

The Origin of Heat Energy

Earth‘s proximity to the Sun is why most-everything near Earth’s surface is warm with a temperature substantially above absolute zero. The Sun constantly replenishes heat energy lost to space. Because matter is absolutely everywhere in the natural world, and because of the wide variety of heat diffusion mechanisms (one of which is black-body radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287–288 K (14–15 °C). The more an object’s or system’s temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.

Derivations of thermodynamic temperature

Strictly speaking, the temperature of a system is well-defined only if its particles (atoms, molecules, electrons, photons) are at equilibrium, and so obey a Boltzmann distribution (or its quantum mechanical counterpart). There are many possible scales of temperature, derived from a variety of observations of physical phenomena. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratios of temperatures, T₂/T₁, are the same in all absolute scales.

Loosely stated, temperature controls the flow of heat between two systems and the Universe, as we would expect any natural system, tends to progress so as to maximize entropy. Thus, we would expect there to be some relationship between temperature and entropy. In order to find this relationship let's first consider the relationship between heat, work and temperature. A heat engine is a device for converting heat into mechanical work and analysis of the Carnot heat engine provides the necessary relationships we seek. The work from a heat engine corresponds to the difference between the heat put into the system at the high temperature, q_H and the heat ejected at the low temperature, q_C. The efficiency is the work divided by the heat put into the system or:

${\textrm {efficiency}}={\frac {w_{cy}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}$ (1)

where w_cy is the work done per cycle. We see that the efficiency depends only on q_C/q_H. Because q_C and q_H correspond to heat transfer at the temperatures T_C and T_H, respectively, q_C/q_H should be some function of these temperatures:

${\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})$ (2)

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T₁ and T₃ must have the same efficiency as one consisting of two cycles, one between T₁ and T₂, and the second between T₂ and T₃. This can only be the case if:

$f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3})$

Consequently, we have:

$f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}$

where $T_{1}$ is the temperature of the triple point of water. So we can define the thermodynamic temperature as:

$T=273.16\cdot f(T_{1},T)\!$

This temperature scale has the property that:

${\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}}{T_{H}}}$ (3)

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

${\textrm {efficiency}}=1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}}{T_{H}}}$ (4)

Notice that for T_C=0 K the efficiency is 100% and that efficiency becomes greater than 100% below 0 K. Since an efficiency greater than 100% violates the first law of thermodynamics, this requires that 0 K must be the minimum possible temperature. This makes intuitive sense; since temperature is the motion of particles, no system can, on average, have less motion than the minimum permitted by quantum physics. In fact, as of June 2006, the coldest man-made temperature was 450 pK. Subtracting the right hand side of Equation 4 from the middle portion and rearranging gives:

${\frac {q_{H}}{T_{H}}}-{\frac {q_{C}}{T_{C}}}=0$

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

$dS={\frac {dq_{\mathrm {rev} }}{T}}$ (5)

where the subscript indicates a reversible process. The change of this state function around any cycle is zero, as is necessary for any state function. This function corresponds to the entropy of the system, which we described previously. We can rearranging Equation 5 to get a new definition for temperature in terms of entropy and heat:

$T={\frac {dq_{\mathrm {rev} }}{dS}}$

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

${\frac {1}{T}}={\frac {dS}{dE}}$

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

See also

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

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

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

External links

• Kinetic Molecular Theory of Gases. An excellent explanation (with interactive animations) of the kinetic motion of molecules and how it affects matter. By David N. Blauch, Department of Chemistry, Davidson College.

Notes

In the following notes, wherever numeric equalities are shown in concise form — such as 1.854 87(14) × 10⁴³ — the two digits between the parentheses denotes the uncertainty (the standard deviation at 68.27% confidence level) in the two least significant digits of the mantissa.