Thermometry

The thermometry is the field of the Physique concerning the measurement of the Température. Among the physical sizes, the temperature is one of most delicate to measure in a rigorous way for two reasons:

• it is necessary well to define the system which one measures the temperature,
  for example, leave a few moments a spoon in a very hot dish with the handle which exceeds outside, one will be able to seize to it sleeve with the hand whereas the convex part is extreme; is the spoon hot or cold?
• the definition even of the concept of temperature and its scale of measurement require to know a certain number of concepts Thermodynamique S which are far from being intuitive.

History of thermometry

The temperature being one of the fundamental concepts of thermodynamics, the history of thermometry is closely related to that of thermodynamics.

The first thermometer was the thermoscope of Santorre Santario, invented in 1612. The first alcohol thermometer was built by Réaumur in 1730.

Historically, Daniel Gabriel Fahrenheit (death in 1736), then Centigrade Anders in 1742, imposed their system on two easily identifiable reference marks:

• for Fahrenheit:
• item zero of the scale was the temperature of solidification of a mixture of water and salt in equal mass proportions;
• the high point of the scale, fixed initially at 12, was the temperature of the Sang; thereafter, it subdivided its degrees into eight, the high point thus became 8×12 = 96;
• for Celsius
• item zero was the melting ice;
• the high point was the boiling of water and is fixed at 100 degrees.

However, these points appeared inadequate. Concerning the Fahrenheit scale, the Homéothermie does not ensure a temperature interns strictly constant, it is thus illusory to make use of it like reference.

In addition, one discovered soon that the boiling point of water depended on the pressure, gold during boiling, the pressure increases in a confined enclosure.

One started to work on the normal scale with hydrogen in 1878; this device made it possible to have a great reproducibility. The device was adopted in 1887 by the International committee of the weights and measures (CIPM), then ratified in 1889 by the first General conference of the weights and measures (CGPM). The Thermomètre with platinum resistance was developed in 1888; being constant volume (its dilation is negligible), it allowed more reliable measurements at high temperature (up to 600 °C). The point high selected was then the melting point of the Soufre, estimated between 444,53 and 444,70 °C.

In 1911, the Physikalisch-Technische Bundesanstalt proposed to take the thermodynamic temperature like scale. After a long work of international collaborations, the resolution was adopted by the 7th CGPM in 1927, which founded the international scales of temperature (EIT). The degrees were called “centigrade degrees”. The methods of measurement varied according to the range of temperature:

• thermometer with resistance of platinum of -193 °C to 650 °C;
• Thermocouple platinum/rhodium platinum with 10% of 650 with 1  100 °C;
• Pyrometer beyond.

The points of reference were always points of change of state: fusion of the mercury (- 38 °C), of the Zinc (420 °C), boiling of sulfur (444,5 °C), fusion of the Antimony (631 °C), of the money (962 °C) and of the Gold (1  064 °C).

In 1937, the CIPM created an Advisory committee of thermometry and calorimetry which became then the Advisory committee of thermometry (CCT). In 1948, the name of the unit became “degree Celsius”, and the CGPM modified the scale slightly.

In 1958, the CIPM validated a scale for the very low temperatures, between 0,5 and 5,23 K (- 271 and -268 °C); it was based on the Tension of vapor of the Hélium ⁴He of 5,23 K to 2,2 K, and on thermodynamic calculations in lower part. In 1962, one used the Isotope ³He in the place.

But there remained still the problem of the dependence in pressure of the temperatures of change of state. It was thus necessary to choose like reference a system independent of the pressure. In 1954, the choice was made on a point where ice, liquid water and vapor coexist: the Point triples (however, the reference not triples was really adopted only to January 1st 1990).

But, the Centigrade scale being very largely spread, it was necessary absolutely that does not modify of anything the value measured: it was unthinkable to make change all the thermometers. As one knows well the variation of the curve of liquefaction of the ice (100 atm/°C), that gives a point triples to 1/100 °C 0°C. As the dilation coefficient is close to 0,003  661  0 (which is worth 1/273,15), the temperature of the point triple was fixed at value 273,16 for this new scale, the scale Kelvin. Thus, by a simple subtraction, one finds a boiling point of water with 100  °C; but it acts from now on more than one definition.

With the increase in precision of measurements, it proved that measurements gave:

under the pressure of a atmosphere (101  325 Pa), water boils with 99,98 °C.

Thus, with an unperceivable error of measurement with current thermometers (most precise being graduated with the 1/10e degree), one can always state that water boils with 100 °C. But these 0,02 °C of difference can have an importance when very precise measurements are necessary.

There is thus here the illustration of the importance of the practice and the human representation in sciences: if one considers the most current laws using the temperature (laws known as “of Arrhenius” and Statistique of Boltzmann), it would have been more “economic” to use size 1 kT , T being the absolute temperature and K the Boltzmann constant. However, on the one hand the scale is not “with human size” (the spirit handles best the orders of magnitude around 10), on the other hand, it is not intuitive (the value increases when the weather is colder), and finally a radical change involves high costs and many sources of error.

It should be noted that the scale of temperature into force currently was defined in 1990. The EIT-90 brings material changes such as the removal of the thermocouple to platinum-rhodium 10%, the thermometer with platinum resistance having been adopted for the range of the average temperatures, thus covering the fixed point of the money (approximately 962°C). But, still, fascinating of account the point triples water (0,01°C) like point of reference of the scale. However, this scale remains open and of the solutions are planned to extend the range of the fixed points for the moment reserved to the transitions from phase of bodies ideally pure (gallium, indium, tin, zinc, aluminum, money, copper, gold, etc…) in particular the eutectic materials likely to become fixed points for the thermométie of contact to higher temperatures with 1000°C, in the objective to push back still a little more the junction with the techniques of pyrometry which extrapolate the law of Planck and are thus sources of uncertainties.

Thermometers

Historically, as for the majority of the sizes, the temperature was measured by a visible consequence: the Dilation.

The discovery of the Seebeck effect in 1821 (which will give rise to the Thermocouple) and the invention of the Thermomètre to resistance of platinum in 1888 made it possible to transform the temperature into an electric value, therefore displayable on an initially analogical dial then numerical (of a Voltmètre).

Work on the black Corps gave rise to a third range of apparatuses based on the analysis of the luminous Specter emitted by an object, for example, a flame.

See the detailed article Thermometer.

The problem of sampling

The first practical problem is that of the placement of the thermometer. If one considers the thermometers of contact (thermometers with dilation or thermocouple), the posted temperature is that… thermometer itself. It thus should be made sure that the thermometer is well with thermal balance with the object in which one is interested.

The traditional example is that of weather measurement under shelters: so radiations Infrarouge S come to irradiate the thermometer (for example, radiation of the Sun, or the ground or a wall heated by the Sun), the temperature of the thermometer increases, it is not then more balances some with the atmosphere. The posted temperature is not thus any more that of the air.

In the case of a spectroscopic measurement, it should be made sure that the light collected by the thermometer comes exclusively from the object which interests us.

Lastly, the difficulty of the homogeneity of the temperature arises. If one measures the temperature in a point of the object, nothing guarantees that it will be the same one in another point.

Approaches thermodynamic

Definition of the concept temperature

One cannot measure a physical size without to have defined it (cf Epistémologie of the Mesure in physics). The size temperature is, in this respect, specimen.

Thermodynamics is the science of the systems in a thermodynamic state of balance. That wants to say

• mechanical balance: the pressure became uniform there except partition fixes preventing the transfer of volume,
  - and -
• thermal balance: the temperature became uniform there except partition insulates preventing the transfer of heat.

Thus let us take two such systems, S ₁ and S ₂, of respective temperature T ₁ and T ₂: it is already necessary to be able to say what

T ₁ means in experiments = T ₂

i.e. to define the equal sign in experiments. The equal sign in mathematics means that one established a Relation of equivalence between the states of the thermodynamic systems: there exists such a relation in thermodynamics - sometimes called “principle zero of the thermodynamics”, but which actually rises from the second principle, because it leads to this concept of state of thermodynamic balance -, it is the relation noted by a meaning Tilde “~”

“~”: “to remain in the same state of thermodynamic balance after setting in thermal contact” (i.e. via a wall diatherme).

This relation “~” is

• reflexive: the state of S ₁ is “~” with state of S ₁-cloné;
• symmetrical: the state of S ₁ is “~” with state of S ₂ ⇔ state of S ₂ is “~” with state of S ₁;
• transitive: the state of S ₁ is “~” with state of S ₂ and state of S ₂ is “~” with state of S ₃ ⇒ state of S ₁ is “~” with state of S ₃.

Obviously, the experimental precision with which one can test this relation “~” defines the experimental capacity that there is, to distinguish 2 classes from equivalence. Each class of equivalence thus belongs to the together-quotient of the " states “~” " : in French of physicist, one says:

each class of equivalence represents systems of even temperature

what gives the possibility of writing T ₁ = T ₂ and thus to define the size temperature.

Choice of the thermometric scale

Once defined a size, it acts to specify if it is a locatable size, i.e. if one can define in experiments

T ₁ > T ₂

The answer is: yes. It is the expression of the second principle (stated of Clausius):

if T ₁ > T ₂, spontaneously, from the Chaleur passes from S ₁ towards S ₂ through the wall diatherme.

This relation is clearly antisymmetric and transitive. The whole of the classes of equivalence is thus provided with a total order.

It is now enough to find a scale, i.e. a physical size having the good taste to grow in a monotonous way with T . It is that for all the real Gaz, with constant volume, the pressure increases if T increases (Loi of Charles), therefore

a Manomètre can be used as thermometer.

As the law is not completely the same one for all gases, one agrees to choose a particular gas, in fact the Dihydrogène. One had been able to choose the Helium, but this one has the bad taste to dissolve a little in the walls of the container; however, it is necessary to operate obviously with a constant quantity of gas in time. Moreover, at the time where this choice was made, one knew well best dihydrogene and his equation of state.

From there, one can manufacture all kinds of thermometers, by graduating them by an abacus of reference with the thermometer to hydrogen.

Scale of absolute temperature T in Kelvins

However, one can go further and make temperature a measurable size: i.e. to define the ratio of 2 temperatures. As each one knows it, the weather is not three times hotter at 3  °C that with 1  °C: the scale Celsius is only one scale of locatable temperature. On the other hand, it is perfectly judicious to say that the 2 preceding temperatures are in the ratio of 276,15/274,15.

It was necessary for that to introduce the concept of absolute temperature of perfect gas , of which it is shown (cf Perfect gas, Cycle of Carnot and Loi of Joule) that it is the thermodynamic temperature defined by the equality of Clausius establishing the existence of the function of state “Entropie” (second statement of Clausius of the Second principle of thermodynamics):

$\backslash \; frac\; \{Q\_1\}\; \{T\_1\}\; +\; \backslash \; frac\; \{Q\_2\}\; \{T\_2\}\; =\; 0$ in a cycle of Carnot built on the two isotherms T ₁ and T ₂.

Under these conditions, the statement

$\backslash \; frac\; \{T\_1\}\; \{T\_2\}\; =\; -\; \backslash \; frac\; \{Q\_1\}\; \{Q\_2\}$

has a physical significance and that defines a measurable size.

Scale practices definition

Practically, the emergence of the concept of temperature and a scale for its measurement are due to an astonishing phenomenon, because independent of the nature of consdéré gas, discovered by Avogadro starting from the laws of Mariotte, Charles and Gay-Lussac:

in extreme cases of the low pressures, extrapolated with null pressure, the product of the pressure by volume statement for a mole of gas is an increasing function of the temperature

thus one can make a thermometer with this physical size statement .

On the one hand it is the same function for all real gases, which is not intuitive, and on the other hand, considering the theorem of Carnot,

$\backslash \; frac\; \{P\_1\; V\_1\}\; \{P\_2\; V\_2\}\; =\; -\; \backslash \; frac\; \{Q\_1\}\; \{Q\_2\}$

thus

statement = R · T (for a mole)

where R is a constant chosen arbitrarily. The choice for R was made on a zero-variable reference mark of temperature, i.e. not varying with the pressure, which was not the case of the old reference mark the Centigrade one (the temperatures of change of state of water vary according to the pressure): the choice is the Point triples Eau, i.e. the exact temperature where ice, liquid water and the steam are simultaneously balance some (this triple point is single, and is with a pressure of approximately 615 Pascal S).

Then, for a mole of gas whatever it is, the limit of the product statement when P tends towards 0 is worth

$\backslash \; lim\_\; \{P\; \backslash \; rightarrow\; 0\}\; statement\; =\; 101\; \backslash \; 325\; \backslash \; times\; \backslash \; left\; (22,414\; \backslash \; 816\; (40)\; \backslash \; times\; 0,001\; \backslash \; right)$ J/mol

and one declares that this temperature is worth 273,16 Kelvins, by definition of the scale, known as Kelvin, of absolute temperature .

Comparison enters the Celsius scale and the Kelvin scale

Whereas one with the practice to have simple values of reference and whole, the values of reference of the temperature appear terribly complex. This is not a whim of the organizations of standardization (in particular of the International office of the weights and measures), but a consequence of the history (cf history of thermometry).

These values were given so that with the pressure of a atmosphere and the temperature of balance water/ice, one finds the already tabulées values well, for example that of the molar Volume 22,414 L. Simply, thanks to the new definition of the temperature, one is able to have a value much more precise of this size: 22,413  996 (39) L.

From where the value of the Constant of perfect gases

R = 8,314 472 (15) J/K·mol

by dividing this value by the Number of Avogadro $N\_A\; =\; 6,022\; \backslash \; 141\; \backslash \; 99\; (47)\; 10^\; \{23\}\; \{\backslash \; rm\; mol\}\; ^\; \{-\; 1\}$, one obtains the conversion factor of the joules into Kelvins (i.e. the relationship between the kinetic energy average in joules of the Molécule S of a gas and its temperature):

the Boltzmann constant K = 1,380 650 5 (24)·10^-23 J/K

or in electronvolt S (1 eV is worth 1/6  241  509  629  152  650  000 C·V):

a gas whose molecules have a average kinetic energy of 1 eV has a temperature of 11  604,505 (24)   K ~ 11  600  K

Standard

A standard is a system on which physical size considered is measured by an apparatus of comparison, called “primary education standard”. The physical size is:

$\backslash \; lim\_\; \{P\; \backslash \; rightarrow\; 0\}\; PV$

for a thermometer with gas dihydrogene with volume constant V° (thus container without dilation). Having beforehand traced all the isotherms of this gas, statement = ƒ ( P ), for each temperature, one can extrapolate these isotherms when P tends towards zero and thus number very precisely the temperature T of each isotherm.

For the measurement of a Temperature T ' of an enclosure: one plunges the thermometer in the enclosure, and one records the pressure P ' for V° volume. The point P ' V ° falls on an isotherm determined, which one reads the temperature.

As the gas is very dilatable, the apparatus is sensitive. The process is stable if one can maintain the same quantity of hydrogen. On the other hand, it is not fine : measurement disturbs the value to be measured, but one arranges oneself to have to measure only thermostated systems, the smoothness then does not have more importance).

But especially, the system is absolutely not practical.

One will thus use secondary standards, or “practical realization of the absolute scale of temperature”.

Realization practices scale of temperature

Gràce with the thermometer with perfect gas, one calibrates the temperature of points as fixed as possible: for example of the triple points, if one can carry them out. Or of the melting points which do not depend much on the pressure, and that one as far as possible carries out under 101325 pascals. Once his determined temperatures, one produces thermometers as sensitive as possible, faithful and PRACTICES which one calibrates gràce at these points reference mark: the computerized abacus makes it possible to go back immediately to the absolute temperature. In the ordinary temperatures (of 20K with 2000K), one arrives at 5 significant figures. In differential temperature one can reach 10 microkelvin.

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