# Absolute temperature

Physical size
Surname Absolute temperature
(thermodynamic temperature)
Formula symbol ${\ displaystyle T}$
Size and
unit system
unit dimension
SI K θ
Planck Planck temperature ħ 1/2 · c 1/2 · G -1/2 · k -1/2

Absolute temperature , also called thermodynamic temperature , is a temperature scale that refers to the physically based absolute zero point . It is a basic term in thermodynamics and physical chemistry . In the context of the international system of units , it is measured in the unit Kelvin , in the USA the Rankine scale is also used.

Since absolute zero represents the lowest possible temperature that can only be reached theoretically (see third law of thermodynamics ), the Kelvin scale is a ratio scale . Some other temperature scales, on the other hand, refer to an arbitrarily determined zero point, such as the Celsius scale whose zero point was originally the freezing point of water, which is 273.15 K on the Kelvin scale.

## Thermodynamic definition of temperature

The thermodynamic temperature of a physical system in the state of thermal equilibrium is defined with the help of the efficiency of an ideal heat engine . The following two requirements define the thermodynamic temperature.

• First, one defines the quotient of temperatures as follows: One considers a reversibly and periodically working heat engine, which takes an ( infinitesimally small) amount of heat from a reservoir A in a period , converts part of it into mechanical work , and the rest as waste heat to a reservoir B releases. The two reservoirs A and B should each be in different thermal equilibrium states. (Both negative and positive signs for are permitted, depending on whether A is colder or warmer than B.) The relationship between the temperatures and between A and B is then defined as follows: ${\ displaystyle Q_ {A}}$ ${\ displaystyle W}$${\ displaystyle Q_ {B} = Q_ {A} -W}$${\ displaystyle W}$${\ displaystyle T_ {A}}$${\ displaystyle T_ {B}}$
${\ displaystyle {\ frac {T_ {A}} {T_ {B}}} = {\ frac {Q_ {A}} {Q_ {B}}}}$

The empirical observation behind this temperature definition is that two heat engines that work in competition for the best efficiency between two given heat baths of constant temperature each have a similar efficiency. The more both parties try to minimize energy losses from their machines, the lower the possible increases in efficiency and the lower the differences between the competitors. What is remarkable about this is that this also applies when the modes of operation of the competing machines are as different as the steam turbine , Stirling engine and Peltier element . So this definition has the advantage of universality. For any given temperature range, a physical process with a high degree of efficiency can be selected, for example magnetic effects at low temperatures, see magnetic cooling .

### Derivation from the general gas law

The absolute temperature can also be inferred from the behavior of ideal gases .

The absolute temperature can be displayed as a limit value :

${\ displaystyle T = \ lim _ {p \ to 0} {\ frac {p \ cdot v} {R}}}$

where denotes the pressure , the molar volume and the gas constant . At the limit value of pressure towards zero, the gas particles no longer interact with one another, which is also known as an ideal gas. ${\ displaystyle p}$${\ displaystyle v}$${\ displaystyle R}$

### Logical consistency of the temperature definition

The logical consistency of this temperature definition is a consequence of the second law of thermodynamics . The following applies:

• Two reversibly and periodically operating heat engines between the same reservoirs A and B have exactly the same efficiency. Otherwise, namely, one could operate the heat engine with the less efficient "reverse" as a heat pump, the engine having the higher efficiency, however, forward, in such a way that in the balance supplied to the reservoir B is much heat as is removed. Then one would have a periodically operating machine that only takes heat from reservoir A, gains mechanical work from it, but leaves reservoir B unchanged. That would be a perpetual motion machine of the second kind , which according to the second law of thermodynamics does not exist.
• Let us consider three reservoirs A, B and C, each in thermal equilibrium. The above definition then yields three temperature quotients , and . So that the temperature definition is consistent, the following consistency condition must apply:${\ displaystyle T_ {A} / T_ {B}}$${\ displaystyle T_ {B} / T_ {C}}$${\ displaystyle T_ {A} / T_ {C}}$
${\ displaystyle {\ frac {T_ {A}} {T_ {B}}} \ cdot {\ frac {T_ {B}} {T_ {C}}} = {\ frac {T_ {A}} {T_ { C}}}}$
Let us now operate a first heat engine between A and B and a second heat engine between B and C. The first machine takes a quantity of heat from reservoir A and feeds the waste heat to reservoir B. The second machine takes exactly the same amount of heat from reservoir B and supplies the waste heat to reservoir C. In the balance, the same amount of heat is supplied to reservoir B as it is withdrawn. The system of both machines can thus be seen as a heat engine between A and C. From the equation ${\ displaystyle Q_ {A}}$${\ displaystyle Q_ {B}}$${\ displaystyle Q_ {B}}$${\ displaystyle Q_ {C}}$
${\ displaystyle {\ frac {Q_ {A}} {Q_ {B}}} \ cdot {\ frac {Q_ {B}} {Q_ {C}}} = {\ frac {Q_ {A}} {Q_ { C}}}}$
the above consistency condition follows with the help of the definition of the temperature quotient.

### Statistical definition and entropy

The statistical definition of temperature according to Boltzmann relates the absolute temperature to the entropy , which is a logarithmic measure for the number of microstates accessible to an isolated system (i.e. the phase space volume ) for a given macrostate: ${\ displaystyle S}$${\ displaystyle \ Omega}$

${\ displaystyle S = k _ {\ mathrm {B}} \ ln (\ Omega)}$

where the proportionality factor denotes the Boltzmann constant . The absolute temperature is then the reciprocal of the partial derivative of the entropy in terms of the internal energy : ${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle S}$${\ displaystyle U}$

${\ displaystyle {\ frac {1} {T}} = {\ frac {\ partial S} {\ partial U}}}$

For all reversible interactions in which only heat is exchanged then applies:

${\ displaystyle dS = {\ frac {\ partial S} {\ partial U}} dU = {\ frac {dU} {T}}}$

from what

${\ displaystyle dU = \ delta Q _ {\ mathrm {rev}} = TdS}$

as well as the formulation by Clausius follows:

${\ displaystyle dS = {\ frac {\ delta Q _ {\ mathrm {rev}}} {T}}}$

The symbol indicates an incomplete differential . ${\ displaystyle \ delta}$

## The temperature in statistical mechanics

Closely related to this concept of thermodynamic temperature, the temperature is in the statistical mechanics : A system of statistical mechanics in thermal equilibrium at the temperature is represented by a probability density described. Here referred to the energy function that is in the, classical physics the Hamilton function , in quantum physics the Hamiltonian . Further denotes the Boltzmann constant . The normalization constant is called the partition function. The term is called the Boltzmann factor . ${\ displaystyle T}$ ${\ displaystyle e ^ {- {\ frac {H} {k _ {\ mathrm {B}} T}}} / Z}$${\ displaystyle H}$${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle Z}$${\ displaystyle e ^ {- {\ frac {H} {k _ {\ mathrm {B}} T}}}}$

## Apparently negative values

Negative absolute temperatures are definitely used as a computational aid. For example, you can describe the state of a population inversion quite easily with this tool. However, this is only possible because it is not a state in thermodynamic equilibrium . Ideas for this were pursued as early as the 1950s by Edward Mills Purcell and Robert Pound, as well as by Norman Ramsey .

## Logarithmic scale

In the “Handbuch der Kältetechnik”, Rudolf Plank suggests a logarithmic temperature scale, where no “lowest possible” temperature occurs. The zero point corresponds to the melting point of the ice. Below that the minus degrees extend to minus infinity.

"[...] If you suddenly remove the magnetic field, the thermomagnetic cooling effect occurs . In this way, a temperature of 0.05 K was achieved with potassium chrome alum. In 1935 one had even advanced to 0.005 K. [...] In order to correctly assess the progress achieved, one would actually have to use the logarithmic temperature scale as suggested by Lord Kelvin . Accordingly, a decrease from 100 K to 10 K would have the same meaning as [...] from 1 K to 0.1 K. "

## literature

• Rudolf Plank: Handbook of Refrigeration Technology , Volume 2, Thermodynamic Basics, Springer, Berlin 1953.