3.5 The thermodynamic temperature scale

 

Temperature is measured with the aid of instruments based on the determination of one or another property of substance varying with temperature. These instruments are calibrated in accordance with the temperature scale generally accepted. When devising a particular temperature scale, however, difficulties arise, traced to the fact that the properties of every substance change in the same temperature interval in a different way. The design of many thermometers, for instance, is based on the phenomenon of the expansion of liquid with a rise in temperature; thermometers of this kind include the commonly known liquid-filled (mercury or alcohol) thermometers, in which the length of the column of liquid increases with a rise in temperature. The coefficient of thermal expansion of one and the same liquid is different at different temperatures, which hampers the establishment of a temperature scale. In 1742 the Swedish astronomer A. Celsius suggested to assign the temperature of 0°C to the melting point of ice and 100 °C to the boiling point of water and to divide the distance between the two points into one hundred equal intervals.[1] However, if the column of mercury, filling the distance between the marks of the ice melting point and of the boiling point of water, is divided into 100 equal intervals, then, taking into account the dependence of the mercury coefficient of expansion on temperature, we shall find that one and the same increase in the length of the column of mercury will correspond to different temperature increments. The value of one division of a uniform temperature scale based on various thermometric liquids will be different. If, for instance, a thermometer is filled with water, then when this thermometer is heated from the melting point of ice, a strange phenomenon can be observed: instead of rising with an increase in temperature, the column of water falls below the level (or mark) corresponding to the melting point of ice. The fact is that at atmospheric pressure the density of water is maximum at a temperature of 3.98°C. Hence, when heated from 0 to 3.98 °C, the volume of the water filling the thermometer decreases (and so the top of the column of water moves down in the thermometer).

In the past temperature scales were established based on different thermometric substances, but then the ideal gas was found to be one of the most suitable thermometric substances. In fact, Clapeyron's equation permits temperature to be determined from the relationship

 

                                                                 

 

One of the temperature scales (the so-called ideal-gas scale) can be established by measuring the pressure of a gas[2], whose properties make it close, to an ideal one, kept in a vessel of constant volume (v = const). The advantage of this scale is that at v = const the volume of an ideal gas is related to the temperature by a linear dependence.[3] Since, as was already noted in Sec. 1.3, even at low pressures a real gas differs somewhat from an ideal gas in respect to properties, the realization of the ideal-gas temperature scale also involves a number of difficulties.

The second law of thermodynamics permits the establishment of a temperature scale that does not depend on the properties of the thermometric substance. Indeed, according to the Carnot theorem, which states that the thermal efficiency of a reversible Carnot cycle does not depend on the properties of the working medium, it can be argued that the thermal efficiency of the cycle depends only on the temperatures of the high- and low-temperature sources:

 

                                                                                                (3.49)

 

where θ1 and θ2 are the temperatures of the high- and low-temperature sources.

Formula (3.49) yields

 

                                                                                                                     (3.50)

 

Consider three heat sources which are at temperatures θ1, θ2, θ3 provided θ1 > θ2 > θ3. Assume that the following reversible Carnot cycles are realized between these sources: cycle I between the heat sources at temperatures θ1 and θ2, cycle II between the heat sources at temperatures θ1 and θ3, and cycle III between the heat sources at temperatures θ1 and θ3 (Fig. 3.10).

 

 

Fig. 3.10

 

In cycle I heat Q1 is removed from the source with temperature θ1 (the high-temperature source for this cycle) and heat Q2 is transferred to the source with temperature θ2 (the low-temperature source for this cycle). In cycle II the same amount of heat Q2 is removed from the heat source with temperature θ2 (for cycle II this source serves at the high-temperature source), previously received by this source when cycle I was realized; the heat added to the source with temperature θ3 (the low-temperature source for this cycle) we denote by Q3. Finally, in cycle III an amount of heat Q1, equal to that involved in cycle I, is removed from the high-temperature source with temperature θ1 the heat transferred during this cycle to the low-temperature source with temperature θ3 we denote by Q4. By analogy with Eq. (3.50), for cycle I

 

                                                                                                                    (3.51)

 

for cycle II

 

                                                                                                                     (3.52)

 

and for cycle III

 

                                                                                                                     (3.53)

 

The jointly accomplished cycles I and II can be regarded as a combined reversible cycle realized between the same heat sources of cycle III, with temperatures θ1 and θ3. The thermal efficiency of this cycle is obviously

 

                                                                                                                     (3.54)

 

The thermal efficiency of cycle III is in turn equal to

 

                                                                                                                     (3.55)

 

The thermal efficiency of any reversible cycle realized between two heat sources will be shown in Sec. 3.6 to be equal to the thermal efficiency of a reversible Carnot cycle with the same heat sources.

It follows then that the thermal efficiency of the combined reversible cycle considered will be equal to the thermal efficiency of the reversed cycle III. But according to Eqs. (3.54) and (3.55), this is possible provided

 

                                                                                                                               (3.56)

 

Dividing Eq. (3.53) by Eq. (3.52) and taking condition (3.56) into account, we obtain:

 

                                                                                                                   (3.57)

 

On the other hand, according to Eq. (3.50),

 

                                                            

 

Comparing the above expression with Eq. (3.57), we find that

 

                                                      

 

But this is only possible if

 

                                                         

 

where the functions ψ (θ1) and ψ (θ2) do not depend on temperature θ3.

Consequently,

 

                                                              

 

and the expression (3.41) for the thermal efficiency of a reversible Carnot cycle takes the following form:

 

                                                                                                                          (3.58)

 

For a reversible Carnot cycle realized within an infinitely small temperature interval, i.e. θ1 = θ2 +d θ, Eq. (3.58) takes the form

 

                                                                                            (3.59)

 

We expand the function ψ (θ1 - dθ) in a Taylor series:

 

                                                             (3.60)

 

Confining ourselves to the first two terms in the expansion, from Eq. (3.59) we find that

 

                                                                                                          (3.61)

 

Since the temperature θ1 is chosen arbitrarily, it will be denoted by θ in the following text.

Also introduce the following notation:

 

                                                                                                                       (3.62)

 

 

With account taken of this notation, for a reversible Carnot cycle realized within an infinitely small temperature interval, Eq. (3.61) takes the following appearance:

 

                                                                                                             (3.63)

 

This differential equation relates unambiguously the thermal efficiency of a reversible Carnot cycle and temperature θ, referred to as the thermodynamic temperature. Since the thermal efficiency of a reversible Carnot cycle does not depend on the properties of the working medium (or substance), the thermodynamic temperature θ, determined via η, does not depend on these properties, too.

Integrating Eq. (3.63), we obtain:

 

                                                                                                               (3.64)

 

where θ0 is some constant temperature, and Q0 is the amount of heat Q corresponding to this temperature.

From equation (3.64) it follows that using the thermal efficiency of a reversible engine operating in the Carnot cycle[4], it is possible to establish a number of thermodynamic temperature scales by specifying the function F (θ) and the temperature θ0 assigned to the thermal state selected.

Kelvin suggested selecting the function F (θ) so that the temperature intervals on the scale would be proportional to the thermal efficiency increments. As Eq. (3.63) indicates, the function F (θ) must then be assumed equal to some constant

 

                                                                                                                               (3.65)

 

Taking (3.65) into account, Eq. (3.64) gives:

 

                                                                                                                 (3.66)

 

Let us now select the thermodynamic temperatures θ1 and θ1 whose corresponding quantities in Eq. (3.66) are Q1 and Q2, i.e.

 

                                                                                                                 (3.67)

 

                                                                                                               (3.68)

 

Let the temperature difference (θ1 – θ2) be equal to an exactly selected temperature interval n°, i.e.

 

                                                                                                                           (3.69)

 

(for instance, the difference between the boiling point of water and the melting point of ice).

Subtracting (3.68) from (3.67), we get:

 

                                                                                                               (3.70)

 

whence, with account taken of equation (3.69),

 

                                                                                                                            (3.71)

 

Substituting this expression for b in Eq. (3.66), we obtain the thermodynamic temperature θ:

 

                                                                                                                 (3.72)

 

 

 

The quantities Q, Q0, Q1 and Q2 are the heats of isothermal expansion of a working medium between two arbitrary adiabats of a reversible Carnot cycle (Fig. 3.11). In other words, for a reversible Carnot cycle realized between the temperature interval θ – θ0, the quantity Q is the amount of heat added to the working medium from the high-temperature source with a temperature θ and Q0 is the amount of heat rejected from the working medium to the low-temperature source with a temperature θ0. Similarly, Q1 and Q2 represent the amounts of heat added and removed in a reversible Carnot cycle realized within the temperature difference θ1 – θ2. In general, the magnitude of Q1, Q2, Q0 and Q can be determined experimentally. Knowing the magnitudes of Q1, Q2, Q0 and Q and making use of Eq. (2.73), we can find the value of θ on this scale for any value of Q.

 

 

Fig. 3.11

 

The thermodynamic temperature scale established with the aid of Eq. (3.72) is called the logarithmic scale.

If in evaluating the magnitude of one degree of this scale the main interval or distance from the melting point of ice to the boiling point of water must be assigned 100°, the temperature θ1 should be assigned to the melting point of ice, and temperature θ2 to the boiling point of water, and the difference between these temperatures should be assumed equal to 100°:

 

                                                         

 

As regards the selection of the constant quantity θ0, Kelvin assumed that

 

                                                                   

 

i.e. equal to the melting point of ice. Thus relation (3.72) takes the form

 

                                                                                                                       (3.73)

 

It is clear that at a temperature θ < θ0, Q becomes smaller than Q0. Then the quantity  and, consequently, θ become negative. At Q → 0, θ → - ∞; at Q → ∞, θ → ∞.

In this way, on the logarithmic scale temperature (denoted by °L) can vary within the following limits:

 

                                                                                                                       (3.74)

 

Table 3.1 gives the relation between the most commonly used uniform thermodynamic temperature scale established by Kelvin (described in detail below) and the logarithmic thermodynamic scale. As can be seen from the data given in the table, the logarithmic scale fails to coincide numerically with the previously accepted conventional scale; that is why this scale has found no recognition.

 

Table 3.1 The absolute thermodynamic scale and the logarithmic thermodynamic scale

 

θ

(K)

θ

(oL)

θ

(K)

θ

(oL)

∞

∞

100

-322

1000000

2630

10

-1060

100000

1892

1

-1798

10000

1154

0.1

-2536

1000

416

0.01

-3274

373.15

100

0.001

-4012

273.15

0

0

- ∞

 

 

A more convenient scale can be established by choosing the temperature function F (θ) in the form

 

                                                                                                                     (3.75)

 

where θ0 is a constant temperature.

Substituting this function in Eq. (3.64) and integrating, we obtain:

 

                                                                                                                   (3.76)

 

whence

 

                                                                                                                         (3.77)

 

From relationship (3.77) it follows that

 

                                                                                                                    (3.78)

 

or, which is the same,

 

                                                                                                                (3.79)

 

For temperatures θ1 and θ2, the corresponding amounts of heat being Q1 and Q2, we obtain from Eq. (3.77)[5]:

 

                                                                                                                     (3.80)

 

and

 

                                                                                                                       (3.81)

 

Subtracting (3.81) from (3.80), we obtain:

 

                                                                                                           (3.82)

 

Let, as before, the temperature difference, θ1 – θ2, be equal to the temperature interval selected, n°, i.e.

 

                                                              

 

Then, from Eq. (3.82) it follows that

 

 

                                                                                                                     (3.83)

 

Substituting this quantity in Eq. (3.79), for the thermodynamic temperature we get:

 

                                                                                                             (3.84)

 

The thermodynamic temperature scale, established by Eq. (3.84), is called the Kelvin thermodynamic temperature scale.

From the expression for the maximum thermal efficiency of the Carnot cycle accomplished within the temperature difference θ – θ0,

 

                                                                                                                       (3.85)

 

it follows that the thermal efficiency of the Carnot cycle reaches its maximum value (η = 1) when no heat is transferred to the low-temperature source, i.e. when Q0 = 0. Since the thermal efficiency of a cycle cannot exceed unity[6], the heat transferred to the low-temperature source in a reversible Carnot cycle cannot be less than zero. Hence, no isotherm can exist below the isotherm corresponding to Q0 = 0. The temperature corresponding to this isotherm is called the absolute zero temperature or simply absolute zero:

 

                                                                                                                                       (3.86)

 

The quantity θ0 can be assigned any positive values, whereas, in accordance with the foregoing, the values θ0 < 0 have no physical meaning. If we assume θ0 = 0, Eq. (3.84) takes the form

 

                                                                                                                     (3.87)

 

Assuming, as before, that n° = θ1 – θ2 = 100°, from Eq. (3.87) we obtain:

 

                                                                                                                (3.88)

 

The Kelvin thermodynamic scale established on the basis of this equation is called the absolute thermodynamic scale (Kelvin scale, K).

It is clear from Eq. (3.88) that for the temperature θ2,

 

                                                                                                               (3.89)

 

From Eqs. (3.88) and (3.89) we get:

 

                                                                                                                       (3.90)

 

i.e. in order to measure any temperature θ on the absolute scale, the magnitude of the temperature θ2 must be known, i.e. we must determine by experiment the distance from temperature θ2 to the absolute zero temperature, θ0 = 0 K.

The melting point of ice θ2 has been shown experimentally to be equal to 273.15 K on the absolute temperature scale.[7]

The temperature determined on the absolute thermodynamic scale will be denoted below by T. From the foregoing it follows that the quantity T can be positive or equal to zero.

Just like the absolute scale, the 100-degree thermodynamic scale (the Celsius scale, °C) can be easily defined: we assume that θ0 = θ2 = 0. Then, since Q0 = Q2 in this case, from Eq. (3.84) we get:

 

                                                                                                              (3.91)

 

If θ = θl in Eq. (3.91) Q = Q1 and θ = 100 °C as could be expected (inasmuch as the interval between the temperatures θ1 and θ2 is divided into 100 units, or degrees).

It is clear from the foregoing that

 

                                                        

 

The temperature of the Celsius scale will be denoted by t.

Let us now compare the temperature on the thermodynamic scale established on the basis of Eq. (3.88) with the temperature on the ideal-gas scale. Denoting in the further reasonings the temperature on the ideal-gas scale by T*, in accordance with Clapeyron's equation (which, as mentioned above, was used to establish the ideal-gas temperature scale) we have:

 

                                                                                                                             (3.92)

 

On the other hand, using the thermodynamic temperature T, we can write for an ideal gas

 

                                                                                                                            (3.93)

 

In accordance with Eq. (2.34), for an ideal gas we have:

 

                                                               

 

Below, in Chapter 4, the following differential equation will be derived:

 

                                                      

 

for the time being, this should be taken for granted.

With account taken of Eq. (2.34) this reduces to

 

                                                                                                                              (3.94)

 

Differentiating Eq. (3.93) with respect to temperature, provided that v = const, we obtain:

 

                                                                                                                 (3.95)

 

Substituting (dp / dT)v from Eq. (3.94), we get:

 

                                                                                                                           (3.96)

 

Comparing Eqs. (3.93) and (3.96), we obtain:

 

                                                                

 

Integration of this differential equation yields:

 

                                                                                                               (3.97)

 

or

 

                                                                                                                                (3.98)

 

where χ is a constant.

Let us now find the magnitude of the constant χ. Substituting the term f (T) from Eq. (3.98) in (3.93), we obtain for an ideal gas:

 

                                                                                                                               (3.99)

 

This relationship is the equation of state for an ideal gas, in which the temperature is measured on the thermodynamic scale.

On the other hand, as can be seen from Eq. (3.92), the same equation of state, incorporating an ideal-gas temperature, has the following appearance:

 

                                                                

 

It is clear from Eqs. (3.92) and (3.99) that

 

                                                                                                                           (3.100)

 

In accordance with (3.100), for the two main temperatures, the melting point of ice and the boiling point of water, we have:

 

                                                                                                                           (3.101)

 

and

 

                                                                                                                                (3.102)

 

In accordance with the conditions specified, on both scales

 

                                                                                                     (3.103)

 

On the other hand, it follows from Eqs. (3.101) and (3.102) that

 

                                                                                                         (3.104)

 

Comparing (3.103) and (3.104), we see that

 

                                                                                                                                 (3.105)

 

and thus

 

                                                                                                                                    (3.106)

 

i.e. the temperature measured on the ideal-gas temperature scale coincides with the absolute thermodynamic temperature measured on the scale established on the basis of Eq. (3.88).[8]

This conclusion is of paramount importance; all the relationships which were previously derived for the temperatures measured on the ideal-gas scale will obviously be also true for thermodynamic temperatures. That is why the thermodynamic temperature will be denoted by the same letter T, used thus far to designate the ideal-gas scale temperature. Employing such notation, it should be borne in mind that only thermodynamic temperature will be dealt with below (although the old notation is retained). As it will be shown below, it is exactly the thermodynamic temperature which is present in the so-called combined mathematical formulation of the first and second laws of thermodynamics, and also in all other relationships based on this equation.

It should be stressed once more that although the thermodynamic and the ideal-gas scales have been shown to be absolutely identical numerically, from the qualitative point of view they differ from each other essentially: the thermodynamic scale is the only temperature scale which does not depend on the properties of the thermometric substance involved, as distinguished from other temperature scales, among them the ideal-gas temperature scale.

                                                  

An exact reproduction of the thermodynamic temperature scale, just like an exact reproduction of the ideal-gas scale, entails a number of serious experimental difficulties.

The establishment of the thermodynamic scale, based directly on Eqs. (3.90) or (3.91), would be practically inexact since the measurement of thermodynamic temperature would then resolve in measuring the amount of heat added or removed (rejected) in isothermal processes; measurements of this kind are rather inexact.

The thermodynamic temperature scale can also be established in principle with the aid of other techniques, using to this end various thermodynamic regularities.

For the temperature interval from 3 K to 1235 K, in particular, use is made of the ideal-gas thermometer method (i.e. an ideal-gas scale similar to the thermodynamic scale is accomplished). For temperatures below 3 K and above 1235 K (the hardening point of silver) other methods are used, whose consideration is beyond the scope of this book.

The measurement of thermodynamic temperature with each of these methods involves many difficulties. Indeed, the gas-filled thermometers, used to measure temperature on the ideal-gas scale, are cumbrous and complicated devices,[9] which are extremely hard to handle in experimental work, the more so, as it was already mentioned above, it is necessary to introduce numerous corrections (for gas nonideality, for instance) into the readings of such thermometers. Taking these difficulties into account, the 7th General Conference on Weights and Measures, 1927, adopted the so-called International Tempe­rature Scale, which can be easily applied to experimental work.

In principle, any practical temperature scale is a totality of so-called fixed points (i.e. of easily realizable states of this or another substance, whose temperature is known) and interpolation formulas, giving the magnitude of temperature based on thermometer readings. So, for instance, two fixed points are used to establish the conventional uniform centigrade scale of the mercury-in-glass thermometer, the melting point of ice (assigned 0 °C) and the boiling point of water (assigned 100 CC); the interpolation formula relating the height of the column of mercury in this thermometer and the temperature being mea­sured is simple:

 

                                                             

 

where h0 and ht, are the heights of the column of mercury at 0 CC and at the temperature t being measured, respectively.

This equation has two constant factors, h0 and A, which are determined with the aid of the fixed points. The number of fixed points which must be available to establish this or another empirical temperature scale is determined by the number of constant factors present in the interpolation formula. In particular, for the uniform centigrade scale of the above-mentioned mercury-in-glass thermometer, the constant h0 is determined by the fixed point 0 °C, and the constant A is determined, as it follows from the relation ht = f (t) from the known quantities h0 and h100 (the height of the column of mercury at the 100 °C fixed point) in the following way:

 

                                                             

 

In establishing various temperature scales, the so-called triple point of water, the freezing points of antimony, sulphur, zinc, gold, and other points were or are being used as fixed points, in addition to the above-mentioned melting point of ice and the boiling point of water at atmospheric pressure. The numerical values of the temperatures corresponding to each fixed point are strictly determined with the aid of a gas thermometer (as was already mentioned, the thermodynamic temperature scale was shown by Kelvin to require only one fixed point).

The International Temperature Scale adopted in 1927 is a convenient scale, since it can be easily realized in experimental work. In particular, for the temperature interval from -182.97 °C (the boiling point of liquid oxygen at atmospheric pressure) to 660 °C the scale was established using the readings of a standard platinum resistance thermometer[10]. The International Temperature Scale was established so (i.e. the empirical formulas for the dependence of the electrical resistance of a platinum thermometer on temperatures were so selected)[11] to coincide as close as possible with the centigrade thermodynamic scale (the accuracy being on the level of the measurement accuracy of a gas-filled thermometer reached at that time, i.e. by 1927).

Persistent and thorough investigations and the development of a corresponding measuring technique permitted metrologists to raise the accuracy of establishing the thermodynamic temperature scale and on this basis determine the deviation of the readings on the International Temperature Scale (Tint) from the readings on the thermodynamic scale (T). In particular, at the 9th General Conference on Weights and Measures, 1948, an equation was proposed, relating the temperature readings taken on the International Temperature Scale and on the centigrade thermodynamic scale in the temperature interval from 0°C to 444.6 °C:

 

                              

 

In the last few decades the magnitudes of these deviations have been somewhat corrected, compared with the values calculated from this equation, the order of magnitudes, however, remaining unchanged.

The temperature scale currently in common usage is The International Practical Temperature Scale of 1990 (IPTS-90) adopted by the International Committee on Weights and Measures, 1989. This scale is so selected that the temperature readings on it are close to the thermodynamic temperature and the difference between the two temperatures stays within the up-to-date measurement errors.

It must be emphasized that thermodynamic temperature appears in all thermodynamic relationships, and in all precision experimental investigations temperature is measured with the aid of instruments that are calibrated following the international scale. In this connection, it should be borne in mind that when experimental data are substituted in thermodynamic equations used in calculation, then, strictly speaking, not the experimentally measured temperature (international scale) but the thermodynamic temperature must be substituted in these equations. This thermodynamic temperature T can be calculated on the basis of the experimentally measured temperature Tint from the relation

 

                                                               

 

where the correction Δ is determined from the foregoing correction formula. As can be seen from the equation T - Tint = f (Tint), the correction Δ is rather small. Therefore, in applications in which we handle relatively (compared with the magnitude of Δ) inexact experimental data (which is usually the case), the difference between T and Tint can be ignored, since its magnitude will certainly fall beyond the limit of accuracy of such experimental data.

However, when dealing with extra-accurate experimental data (such as, for instance, the most accurate data on the specific volumes of water vapor in equilibrium with boiling water) and, in particular, with accurate thermodynamic calculations involving the derivatives of various quantities in respect to temperature, the neglect of the difference between T and Tint, when using thermodynamic equations for practical purposes, may lead to errors comparable with the error of the experimental data involved. This important fact must always be taken into consideration while carrying out experimental studies (especially in the future, as the accuracy of experimental investigations rises).

In conclusion it should be noted that the application of various thermodynamic regularities has made it possible to elaborate various methods of introducing corrections to any empirical temperature scale, so as to reduce them to the thermodynamic scale, i.e. to establish a thermodynamic scale based on this or another empirical scale (for instance, on the scale of a gas-filled thermometer).

 

 

 


[1] Celsius assigned the temperature 100 °C to the melting point of ice and the temperature of 0 °C to the boiling point of water; later on the presently accepted values were assigned to these scale marks.

[2] Since Clapeyron's equation is used to establish an ideal-gas temperature scale, to determine whether a given gas is close to an ideal one use should be made of another characteristic of ideal gases not related to Clapeyron's equation. Such a characteristic is the independence of the internal energy of an ideal gas on volume (Joule's law), established in Chapter 2.

[3] The ideal-gas temperature scale can be divided into any number of intervals (or degrees) since the number of intervals does not affect the properties of the scale, i.e. an ideal-gas scale can be made similar to the Celsius scale, Fahrenheit scale, Reaumur scale and other uniformly divided (linear) temperature scale.

[4] The value Q/Qo present in Eq. (3.64) is, evidently, related to the thermal efficiency of a reversible Carnot cycle accomplished between the temperatures θ and θ0 in the following way: by definition

 

                                                                                 

 

whence,

 

                                                                                         

 

[5] The meaning of the quantities Q, Q0, Q1 and Q2 is, of course, the same as in the reasonings for the logarithmic scale.

[6] This would violate the first law of thermodynamics.

[7] It can be shown that if the magnitude of the interval no = 100° is given, the temperature θ2, cannot be chosen arbitrarily. On the contrary, if an arbitrary value is assigned to the temperature θ2, the temperature interval θ1 — θ2 will not necessarily be equal to 100° at all.

[8] On the ideal-gas scale temperature can be measured not only in degrees Celsius or Kelvin but also in other units; the properties of this scale do not depend on the value of the scale unit. In some countries the Fahrenheit (°F), Rankine (°Ra), or Reaumur (°R) scales are applied, which was already mentioned in Sec. 1.1; temperatures of one scale may be converted into temperatures of another scale. As distinct from the Celsius scale, on the Fahrenheit and Reaumur scales the interval between the melting point of ice and the boiling point of water is not divided into 100 units but into 180 and 80 units, respectively. In addition, on the Fahrenheit scale the melting point of ice is assigned to the temperature of 32 °F. The Rankine scale is an absolute scale, like the Kelvin scale; the temperature on the Rankine scale is 9/5 of that on the Kelvin scale. Thus, on the Rankine scale, just as on the Fahrenheit scale, the main temperature interval is divided into 180 units. The conversion data for Fahrenheit, Rankine, and Reaumur temperature scales, in relation to temperatures on the Celsius scale, are given in Table 1.1.

[9] The fact is that for the gas used in a thermometer of this kind, whose properties make it a real gas, to be as close as possible to an ideal gas, its pressure must be low, and its specific volume is thus large.

[10] We can see that the properties of this non-thermodynamic scale are but again "tied" to the properties of a concrete thermometric substance, platinum.

[11] In accordance with the statute for the International Temperature Scale of 1990, this scale is established by means of a platinum resistance thermometer for a somewhat different temperature interval, from -259.347 °C (triple point of hydrogen) to 961.78 °C (the hardening point of silver). At higher temperatures the International Temperature Scale is based on the Planck’s law. At temperatures below the hydrogen triple point the establishment of the temperature scale is more involved.