## 7 The Definition of Temperature

The Themenheft 2007-3 of the Physikalisch-Technische Bundesanstalt PTB in Braunschweig, Germany is completely dedicated to the notion of temperature. In the first article of this publication, Peter Strehlow and Joachim Seidel show, how the *existence* of an absolute thermodynamic temperature and the state variable entropy can be derived from a small set of axioms. In the fourth section of this paper they further show, that this thermodynamic temperature is (up to a constant scale factor) identical to the temperature measured with a gas thermometer, if the gas follows the Ideal Gas Law. So, the following more practical approach to the definition of temperature is theoretically well-founded.

Our definition of temperature is based on the Ideal Gas Law. Use a certain amount of an ideal gas (actually Helium is used under low pressure) and do measure temperatures using the relation

*T* n= *P · V* · *T _{o}* / (

*P*)

_{o}· V_{o}where the reference values *T _{o}* ,

*V*jand

_{o}*P*jcan be freely chosen. A helium gas thermometer as a primary thermometer further eliminates the need to measure volumes: The volume is kept constant through pressure adjustment, and so we get

_{o}*T* n= *P ·* *T _{o}* /

*P*mmmm if

_{o}*V*jjis constant

The triple point of water is taken as the reference temperature *T _{o}*j. It is assigned the value of j273.160 K

**by definition**. This defines the above mentioned scale factor between thermodynamic temperature and gas temperature. The measurement of temperature is thus reduced to a pressure measurement.

The Physikalisch-Technische Bundesanstalt in Braunschweig, Germany is one of very few metrological organizations worldwide that operate such a helium-gas thermometer as a primary-thermometer. Reference temperatures for triple points and melting points of all sorts of pure materials are measured. These reference temperatures are easily reproducible in every laboratory worldwide, and thus calibration curves for simpler thermometers (for example, resistance thermometers or mercury thermometers) can be determined.

The temperature scale is thus *defined* by the gas law, and this definition can be *realized* with a gas thermometer in the range of approximately j-j250° C to +j1500° C which is, metrologically, of critical importance. At higher temperatures, temperature measurement relies on the laws of radiation, for low temperatures reference values are negotiated at conferences, which are periodically refined. At low temperatures, for example, thermal noise is used to implement a primary thermometer. Another current primary method is based on measuring the speed of sound in gas.

Much information on this subject is provided by the website of the Physikalisch-Technische Bundesanstalt and in the above mentioned Themenheft.

Metrologically, it is *impossible* to define temperature via the average energy of particles, via a change in entropy or via heat quantities; since no one knows how to measure these quantities with reasonable accuracy. Temperature is defined as a macroscopic quantity before it is associated with microscopic terms and long before one can estimate changes in total entropy. Temperature is - like time and mass - *a fundamental unit of physics*. This does not mean, however, that it has to be relativistically invariant.

Furthermore, the concept of temperature gives time its direction in physics! Thermal equilibrium is reached when two bodies are at the same temperature, and two bodies in thermal contact always strive to reach the same temperature. The concepts of entropy and heat are not needed to introduce the famous "arrow of time" - temperature is sufficent.