The Thermodynamic Temperature Scale

The temperature scales that have been discussed earlier are quite arbitrary and depend upon the properties of a particular substance. Kelvin was the first to observe that the efficiency of a reversible heat engine operating between two temperatures is dependent only upon the two temperatures and not at all upon the working substance. Therefore, a temperature scale could be defined that is independent of the properties of any substance. 

The second law is independent of the first law. Historically, the second law was generally accepted and understood before acceptance of the first law. Therefore, we base the discussion on the efficiency of a reversible heat engine used in a Carnot cycle. If the efficiency of a reversible heat engine 

Temperature scales can be defined by assigning specific functions to F(9) and the derivatives. Many scales could be defined, but the choice of a suitable scale has depended upon utility and past history. Only the functions used in defining the present thermodynamic or Kelvin scale are discussed here. The value of the derivative is assigned a constant value, k, and F(9) is defined as 1/9. Then 

The ratio of the two temperatures is related to the efficiency and the ratio of the heats absorbed along the isotherms by 

The Kelvin scale is thus defined in terms of an ideal reversible heat engine. At first such a scale does not appear to be practical, because all natural processes are irreversible. In a few cases, particularly at very low temperatures, a reversible process can be approximated and a temperature actually measured. However, in most cases this method of measuring temperatures is extremely inconvenient. Fortunately, as is proved in Section 3.7, the Kelvin scale is identical to the ideal gas temperature scale. In actual practice we use the International Practical Temperature Scale, which is defined to be as identical as possible to the ideal gas scale. Thus, the thermodynamic scale, the ideal gas scale, and the International Practical Temperature Scale are all consistent scales. Henceforth, we use the symbol T for each of these three scales and reserve the symbol 9 for any other thermodynamic scale. 

Equation (6.16), which includes Equation (6.6), is a mathematical statement of Carnot s theorem  

As Qi and Q2 have opposite signs, their ratio is opposite in sign to the ratio of their absolute values. Thus, 

Now consider a group of three heat reservoirs at temperatures fj 2 h and a reversible Carnot engine that operates successively between any pair of reservoirs. According to Equation (6.19) 

Equating the right sides of Equation (6.20) and Equation (6.23), we obtain 

As the quantity on the left side of Equation (6.23) depends only on ti and the quantity on the right side also must be a function only of fj and t2- Therefore, 3 must appear in the numerator and denominator of the right side in such a way as to cancel, which thereby gives 

For a reversible engine, both the efficiency and the ratio QJQi can be calculated directly from the measurable quantities of work and heat flowing to the surroundings. Therefore we have measurable properties that depend on temperatures only and are independent of the properties of any special kind of substance. Consequently, it is possible to establish a scale of temperature independent of the properties of any individual substance. This overcomes the difficulty associated with empirical scales of temperature described in Section 6.5. This scale is the absolute, or the thermodynamic, temperature scale. 

The work produced in the cycle kW = Q + go which, using Eq. (8.16), becomes 

Now if the high-temperature reservoir is cooled until it reaches do, the temperature of the cold reservoir, then the cycle becomes an isothermal cycle, and no work can be produced. Since it is a reversible cycle, W = 0, and so 0 = aOo + Qohence, go = —  

Since there is nothing special about the temperature of the cold reservoir, except that d do, Eqs. (8.18) and (8.19) apply to any reversible heat engine operating between any two thermodynamic temperatures d and do - Equation (8.18) shows that the work produced in a reversible heat engine is directly proportional to the difference in temperatures on the thermodynamic scale, while the efficiency is equal to the ratio of the difference in temperature to the temperature of the hot reservoir. The Carnot formula, Eq. (8.19), which relates the efficiency of a reversible engine to the temperatures of the reservoirs is probably the most celebrated formula in all of thermodynamics. 

Lord Kelvin was the first to define the thermodynamic temperature scale, named in his honor, from the properties of reversible engines. If we choose the same size of the degree for both the Kelvin scale and the ideal gas scale, and adjust the proportionality constant a in Eq. (8.16) to conform to the ordinary definition of one mole of an ideal gas, then the 

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The way, that the gas temperature scale and the thermodynamic temperature scale are shown to be identical, is based on the microscopic interpretation of temperature, which postulates that the macroscopic measurable quantity called temperature, is a result of the random motions of the microscopic particles that make up a system. 

We have already shown that the absolute temperature is an integrating denominator for an ideal gas. Given the universality of T 9) that we have just established, we argue that this temperature scale can serve as the thermodynamic temperature scale for all systems, regardless of their microscopic condition. Therefore, we define T, the ideal gas temperature scale that we express in degrees absolute, to be equal to T 9), the thermodynamic temperature scale that we express in Kelvins. That this temperature scale, defined on the basis of the simplest of systems, should function equally well as an integrating denominator for the most complex of systems is a most remarkable occurrence. 

See W. F. Giauque and D. P. MacDougall, "Experiments Establishing the Thermodynamic Temperature Scale below 1 =K. The Magnetic and Thermodynamic Properties of Gadolinium Phosphomolybdate as a Function of Field and Temperature". J. Am. Chem. Soc., 60, 376-388 (1938). 

In general, a thermometer is called primary if a theoretical reliable relation exists between a measured quantity (e.g. p in constant volume gas thermometer) and the temperature T. The realization and use of a primary thermometer are extremely difficult tasks reserved to metrological institutes. These difficulties have led to the definition of a practical temperature scale, mainly based on reference fixed points, which mimics, as well as possible, the thermodynamic temperature scale, but is easier to realize and disseminate. The main characteristics of a practical temperature scale are both a good reproducibility and a deviation from the thermodynamic temperature T which can be represented by a smooth function of T. In fact, if the deviation function is not smooth, the use of the practical scale would produce steps in the measured quantities as function of T, using the practical scale. The latter is based on ... 

The relationship between the thermodynamic temperature scale and the ideal gas temperature scale can be derived by calculating the thermodynamic quantities for a Camot cycle with an ideal gas as the working substance. Eor this purpose, we shall use 0 to represent the ideal gas temperamre. 

The second law also defines both S and the thermodynamic temperature scale as follows ... 

The thermodynamic temperature scale T is defined by the second law of thermodynamics. It can be shown that the thermodynamic temperature scale is identical with the perfect-gas temperature scale defined as follows ... 

The relation between the international temperature scale and the thermodynamic temperature scale must be determined empirically with the aid of careful measurements involving gas thermometers. 

In 1954, the thermodynamic temperature scale (i.e., the absolute Kelvin scale was redefined by setting the triple point temperature for water equal to exactly 273,16 K). 

An absolute scale of temperature can be designed by reference to the Second Law of Thermodynamics, viz. the thermodynamic temperature scale, and is independent of any material property. This is based on the Carnot cycle and defines a temperature ratio as ... 

This is the form we have already used to describe the linear responses which define the properties of materials, but in some cases, notably for the temperature T, it is inconvenient to set the initial value To to zero (this would require redefining the thermodynamic temperature scale), and so eq. (3) is used instead (see Table 15.7). In the particular example of a change in temperature, the conjugate response is... 

The ideal gas temperature scale is of especial interest, since it can be directly related to the thermodynamic temperature scale (see Sect. 3.7). The typical constant-volume gas thermometer conforms to the thermodynamic temperature scale within about 0.01 K or less at agreed fixed points such as the triple point of oxygen and the freezing points of metals such as silver and gold. The thermodynamic temperature scale requires only one fixed point and is independent of the nature of the substance used in the defining Carnot cycle. This is the triple point of water, which has an assigned value of 273.16 K with the use of a gas thermometer as the instrument of measurement. 

Gas Thermometers. These are expansion thermometers that depend on the coefficient of thermal expansion. They use, for example, helium gas and have helped to establish the thermodynamic temperature scale, and also for measurements at very low temperatures. 

The general concept of temperature scales is discussed briefly in Exp. 1. The thermodynamic temperature scale, based on the second law of thermodynamics, embraces the Kelvin (absolute) scale and the Celsins scale, the latter being defined by the equation... 

The International Practical Temperature Scale of 1968 (IPTS-68) has been replaced by the International Temperature Scale of 1990 (ITS-90). The ITS-90 scale is basically arbitraiy in its definition but is intended to approximate closely the thermodynamic temperature scale. It is based on assigned values of the temperatures of a number of defining fixed points and on interpolation formulas for standard instruments (practical thermometers) that have been cahbrated at those fixed points. The fixed points of ITS-90 are given in Table 1. 

Since the triple point of water is defined to be exactly 273.16 K on the thermodynamic temperature scale, this is an especially important fixed point. It is also a point that can be reproduced with exceptionally high accuracy. If the procedure of inner melting (described below) is used, the temperature of the triple point is reproducible within the accuracy of current techniques (about 0.00008 K). This precision is achieved by using the triple-point cellshov n in Fig. 1. This cell, which is about 7.5 cm in outer diameter and 40 cm in overall length, has a well of sufficient size to hold all thermometers that are likely to be calibrated. ... 

We attempt to correlate an empirical temperature scale t to the thermodynamic temperature scale T. For this purpose we rewrite Eq. (1.13.16) in the form... 

As in thermodynamics, the thermodynamic temperature T is used in heat transfer. However with the exception of radiative heat transfer the zero point of the thermodynamic temperature scale is not needed, usually only temperature differences are important. For this reason a thermodynamic temperature with an adjusted zero point, an example being the Celsius temperature, is used. These thermodynamic temperature differences are indicated by the symbol i), defined as... 

Precision thermometry based on the thermodynamic temperature scale had its beginnings with the work of P. Chappuis and of H. L. Callendar during the period from the late 1880s to the early 1900s. Chappuis transferred the hydrogen scale in the range from 0 to 100°C, provided by his constant-volume hydrogen gas thermometer, to several carefully made mercury thermometers (Chappuis, 1888). These were then used to calibrate many other mercury thermometers which in turn were to be used in many countries to put temperature measurements on the same scale. The probable uncertainty of those thermometers was stated to be 0.002°C. 

The thermodynamic temperature scale is called the Rankine scale, in which temperatures are denoted by degrees Rankine and defined by... 

Herzfeld, C.M., 1962, The thermodynamic temperature scale, its definition and realization Chapter 6, pp. 41-50 in Temperature, Its Measurement and Control in Science and Industry, C.M. Herzfeld, ed. Vol. 1, Basic Standards and methods, F.G. Brickwedde, ed. New York, Rheinhold Publishing Co., 848 pp. 

Equation (2.6) defines a new temperature scale, called a gas scale of temperature or, more exactly, an ideal gas scale of temperature. The importance of this scale lies in the fact that the limiting value of Kq, and consequently I/kq, has the same value for all gases. On the other hand, o does depend on the scale of temperature used originally for t. If t is in degrees Celsius (symbol °C), then 1/ao = 273.15 °C. The resulting T-scale is numerically identical to the thermodynamic temperature scale, which we will discuss in detail in Chapter 8. The SI unit of thermodynamic temperature is the kelvin (symbol K). Temperatures on the thermodynamic scale are frequently called absolute temperatures or kelvin temperatures. According to Eq. (2.6) (see also Appendix III, Sect. A-III-6),... 

Fortunately, there is a way out of this predicament. Using the second law of thermodynamics it is possible to establish a temperature scale that is independent of the particular properties of any substance, real or hypothetical. This scale is the absolute, or the thermodynamic, temperature scale, also called the Kelvin scale after Lord Kelvin, who first demonstrated the possibility of establishing such a scale. By choosing the same size degree, and with the usual definition of the mole of substance, the Kelvin scale and the ideal gas scale become numerically identical. The fact of this identity does not destroy the more fundamental character of the Kelvin scale. We establish this identity because of the convenience of the ideal gas scale compared with other possible scales of temperature. 

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