Physical principles

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Temperature is one of the fundamental thermodynamic properties and probably the most difficult to measure. The current official temperature scale is the International Temperature scale of 1990 (ITS-90). It is the scale which accredited laboratories refer to in calibration certificates and which is to be used for practical measurements. The ITS-90 is a close approximation to the fundamental thermodynamic temperature scale.

Temperature T is the basic thermodynamic property. It governs heat transfer between two systems in thermal contact. It occurs in most relations between other thermodynamic variables such as, for example, the equation of state. Temperature governs the efficiency of heat engines and the emission of heat radiation, and in statistical thermodynamics it is a parameter of probability distributions.

For many industrial production processes, temperature itself is a main parameter – just think of steel or glass production, chemical processes or food preparation. In addition, practically all other macroscopic physical quantities of interest to industry and research are more or less strongly influenced by temperature. Therefore, they can easily be compared only if they are determined at the same temperature. For this reason temperature is one of the physical quantities most frequently measured, and several million temperature sensors and temperature measuring devices are every year being newly installed worldwide.

Thermodynamic temperature and primary thermometers

The temperature T occurring in fundamental physical laws is thermodynamic temperature. The thermodynamic temperature scale can be defined in various ways, all of which can be shown to be equivalent. Some of the definitions are rather abstract and not directly useful for temperature measurement. An example is the definition via the efficiency of ideal Carnot heat machines. Perhaps the most comprehensible definition is that the thermodynamic temperature scale is identical to the ideal-gas scale based on the equation of state of the ideal gas

pV = NkT,

where p and V are the gas pressure and volume, respectively, N is the number of gas particles (very large!), and k is Boltzmann's constant.

It is exactly the same temperature that goes into other fundamental laws such as Planck‘s radiation law for blackbodies (or the Stefan-Boltzmann law which can be derived from it), the Nyquist formula for thermal noise or the expression for the Doppler broadening width of a spectral line emitted or absorbed by a gas that has the Maxwell distribution of gas particle velocities.

These laws are also the relations which can serve as a basis for primary thermometers capable of measuring thermodynamic temperature. Such a primary thermometer does not have to refer to other temperature measurements (i.e. it requires no calibration) but derives the temperature from measurements of other quantities such as pressure, radiant power or noise voltage. Therefore, it can be used to set up the thermodynamic temperature scale.

Primary thermometer

Underlying thermodynamic relation

Meaning of the symbols

Gas thermometer
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p V = N kT

p pressure, V volume, N number of gas particles, k Boltzmann constant

Dielectric-constant
gas thermometer
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ε = ε0 + α0 N V
p
= kT (εε0) ∕α0

ε permittivity (“dielectric constant”) of gas, ε0 electric constant, α0 static dipole polarisability of single gas atom

Acoustic thermometer
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ca2 = (cp cV) kT m

m mass of gas particle, cp/cV ratio of specific heat capacities at constant pressure or volume (= 5/3 for monatomic ideal gas)

Noise thermometer
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< (ΔU)2 > = 4 kT R Δν
(valid in low-frequency / high-temperature limit)

ΔU thermal noise of electrical voltage drop U across resistor with resistance R occurring in narrow frequency band Δν

Spectral-band radiation thermometer
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Lν,BB = (2 3 c2 ) ∕
{exp[∕(kT)] − 1}

Lν,BB spectral radiance (with respect to frequency ν ) of blackbody radiation, h Planck constant (“quantum of action”),
c speed of light

Total radiation thermometer
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LBB = σT4∕π
= 2π4 (kT)4∕(15c2h3 )

LBB (total) radiance of blackbody radiation, σ Stefan-Boltzmann constant

Doppler broadening thermometer
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ΔνD= [2 kT∕(mc2)]1/2 ν0

ΔνD Doppler frequency width of spectral line with central frequency ν0 , absorbed (or emitted) by atoms or molecules of an ideal gas at temperature T

Thermodynamic reasoning (and all of the thermodynamic relations in the table above) shows that there is an absolute zero of temperature at which, for instance, the ideal gas pressure at constant volume, the emission of heat radiation, and the thermal noise voltage would all vanish. The unit of thermodynamic temperature can be defined with the help of one additional temperature 'fixed point', as proposed by William Thomson (later Lord Kelvin) as early as 1854.

A hundred years later, the 10th General Conference on Weights and Measures of the Metre Convention followed this suggestion. As the fixed point, the triple point of water (TPW) was chosen, the unique point in the phase diagram where vapour, fluid and ice coexist. The triple-point temperature TTPW was assigned the temperature 273.16 K, so that the unit of temperature, the kelvin, becomes

1 K = TTPW / 273.16.

        

The numerical value 273.16 was chosen for the kelvin to be in as close agreement as possible with the degree Celsius used before, which was one hundredth of the temperature difference between the boiling point and the melting point of water at normal atmospheric pressure, p atm  = 101.325 kPa. In everyday life (and in the temperature range above 0 °C, also in thermometry) the degree Celsius with the unit symbol °C is still in use though only as a specific name for the kelvin for the statement of Celsius temperatures t . As the melting temperature of water at atmospheric pressure is slightly lower than the triple point temperature (the triple point pressure is only 0.6 kPa), the relation

t /°C = T /K - 273.15

is valid if t and T designate the same temperature. The Celsius temperature scale is thus a kelvin temperature scale with the zero point shifted to 273.15 K. Accordingly, a given temperature difference has the same numerical value in both the Kelvin and the Celsius scales:

Δt /°C = ΔT/K.

Note that the unit name kelvin (K) is no longer accompanied by 'degree' or the degree symbol '°'. It is true that the temperature unit derived from the TPW temperature originally was named 'degree Kelvin' in 1954. However, it was renamed into kelvin by the General Conference on Weights and Measures ( CGPM, resolution 3 ) in 1967. Still using 'degree Kelvin' today thus indicates that your knowledge of thermometry is somewhat antiquated. Note also, that kelvin as an SI unit is correctly written with lowercase k (except at the beginning of a sentence).

It should also be noted that the magnitude of the unit of thermodynamic temperature cannot be determined by thermodynamic considerations. This is because T occurs always in the combination kT in all fundamental physical laws (for examples, see the table above). This combination is often referred to loosely as thermal energy , since the mean kinetic energy of a gas particle is 3kT/2 at temperature T . Strictly speaking, a primary thermometer therefore does not measure T, but rather measures the thermal energy kT . Hence, one can rescale T to aT , if this is accompanied and compensated by rescaling the Boltzmann constant k to k / a in order to retain the value of kT .

Essentially, there are two ways of extracting temperature T from a measured value of thermal energy kT .

The choice of defining TTPW to be exactly 273.16 K corresponds to the choice of a specific value of the 'scale factor' a and, thus, implicitly determines also the numerical value of the Boltzmann constant k which has to be determined by measurement (preferably at the TPW). This is the way chosen by the current SI definition of the kelvin, and the Boltzmann constant in current SI units is

k = 1.380 6505 x 10-23 J/K = 8.617 343 x 10-5 eV/K,

with a relative standard uncertainty of 1.8 x 10-6 .

This definition takes advantage of the fact that different realisations of the TPW temperature have been shown to agree nearly perfectly with one another, the typical relative variation of careful state-of-the-art realisations being as small as 3 x 10-7 , i. e. nearly an order of magnitude smaller than the uncertainty of the measured value of the Boltzmann constant. As a disadvantage, increasing the measurement uncertainty particularly at very low or very high temperatures, temperature measurements have to be traceable in some way to a measurement done at the TPW.

Another possibility of extracting T from kT is presently under serious discussion: as an alternative to the current SI definition of the kelvin via the TPW temperature, the Boltzmann constant might be assigned a value by definition - and so have no uncertainty associated with it. This redefinition of the kelvin would have the advantage of not favouring a temperature value or measurement method. Linking the temperature unit to the appropriate fundamental constant would also be more satisfying intellectually than linking it to a poorly understood material property, which is certainly not fundamental. Fixing the value of the Boltzmann constant as, for example, k = 1.380 6505 x 10−23 J/K would link the kelvin with the unit of energy, the joule, in the same way as the unit of length, the metre, is currently linked with the unit of time, the second, by assigning the exact value of c = 299 792 458 m/s to the speed of light in vacuum.

The discussion about a possible redefinition of the kelvin has recently become rather lively, see for example the presentations given at a recent Workshop on Methods for New Determinations of the Boltzmann Constant .



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