Thermodynamic scale of temperature

In order to define the thermodynamic scale uniquely, we must determine the numerical value of the additive constant. Taking the constant as zero, we have coincidence with the ordinary temperature scale. Thus the freezing point of water (T=273° absolute) becomes M73 on the thermodynamic scale, and we have 6 — lnT or T — e. It follows from the equation 0 = lnT that 0 becomes equal to — oo at the absolute zero (T = 0). This harmonises with the fact that we are unable in practice 

The simphcity of the relationship between the thermodynamic scale and the gas thermometer scale is due principally to the simple properties of rarefied gases, and also to the fortunate choice of mercury as thermometric substance by Celsius and Reaumur before the discovery of the gas laws. The coefficient of expansion of mercury happens to be almost exactly proportional to the coefficient of expansion of rarefied gases. All our thermodynamical relationships would have been very much more comphcated had water or alcohol, for example, or the resistance of a metal, been used for the definition of the practical scale of temperature. Their strict validity, however, would not have been affected. 

At the beginning of this chapter (p. 129) we put the question Under what conditions can we predict the direction in which any particular process will go In answer we may now say  

The process will take place in the direction which involves an increase in the entropy of the system. It must therefore be one of the objects of science to determine the entropy of any given system as a function of its variables of condition. On p. 143 we have shown how this may be done for a perfect gas. In other cases the problem is not so simple, but the calculation is always possible if we know the equation of condition, e.g. van der Waals equation for real gases. Yet even when it is not possible to obtain an exphcit expression for the entropy, the entropy law can lead us to important conclusions, just as the law of the conservation of energy is important in many cases in which we are unable to give a numerical or analytical value for the energy of the system. 

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It was Lord Kelvin who recognized that Carnot s hypothetical engine was of fundamental importance, and used it to define a thermodynamic scale of temperature that has become known as the Kelvin temperature. He set the thermodynamic temperature T of the reservoirs proportional to the amount of heat exchanged at each that is. 

The Carnot cycle forms the basis for a thermodynamic scale of temperature. Because e = 1 — Ti/Tf, the Carnot efficiencies determine temperature ratios and thereby establish a temperature scale. The difficulty of operating real engines close to the reversible limit makes this procedure impractical. Instead, real gases at low pressures are used to define and determine temperatures (see Section 9.2). 

From the characteristics of a particularly simple kind of heat engine, the Carnot engine, and from universal experience that certain kinds of engine cannot be constructed, we concluded that all reversible heat engines operating between the same two heat reservoirs have the same efficiency, which depends only on the temperatures of the reservoirs. Thus it was possible to establish the thermodynamic scale of temperature, which is independent of the properties of any individual substance, and to relate the efficiency of the engine to the temperatures on this scale ... 

The temperature scale used in kinetic theory is the absolute or thermodynamic (Kelvin) scale. The thermodynamic scale of temperature uses units of kelvins (K), which have the same size as the more familiar degrees Celsius (°C) but whose zero is absolute zero (-273.15 C)... 

The Celsius and Fahrenheit scales are based on the physical properties of water but the absolute or thermodynamic scale of temperature is not based on the physical properties of any substance. The kelvin is the SI unit of temperature and is based on absolute zero (approximately -273 °C)... 

Kelvin then replotted his data, this time extrapolating each graph till the volume of the gas was zero, which he found to occur at a temperature of -273.15 °C see Figure 1.5. He then devised a new temperature scale in which this, the coldest of temperatures, was the zero. He called it absolute zero, and each subsequent degree was equal to 1 °C. This new scale of temperature is now called the thermodynamic (or absolute) scale of temperature, and is also sometimes called the Kelvin scale. 

DEGREE CELSIUS (°C). One unit of temperature on the Celsius temperature scale, which is derived from the thermodynamic of Kelvin scale of temperature and related by Temperature (degrees Celsius) equals Temperature (Kelvin units) minus 273.15. See Temperature. 

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 a reversal of the argument of Chap. II, Sec. 6, where we used Joule s law as an experimental fact to prove that the gas scale of temperature was identical with the thermodynamic temperature. Here instead we assume the temperature T in Eq. (1.1) to be the thermodynamic temperature, and then Joule s law follows as a thermodynamic consequence of the equation of state. 

At equilibrium the two gases are at the same temperature (the zeroth law of thermodynamics), so we could, if we wanted, define the temperature to be T = (3/2). However, because the scale of temperature we use is degrees Kelvin, we need a constant of proportionality. This is k (Boltzmann s constant) when n is the number of molecules [JcT = (3 2)) and R, the gas constant, when n is the number of moles (RT - (3 2)]. If you stop and think for a moment, this result immediately gives a physical meaning to the absolute temperature as the point where molecular motion ceases. 

In this equation, /3 is an arbitrary frictional drag parameter (inverse time constant), chosen as the coupling parameter which determines the time scale of temperature fluctuations, T0 is the mean temperature, and T(t) is the temperature at time t (see Eq. 8). Constant-pressure conditions are enforced with a proportional scaling of all coordinates and the box length by a factor related to the isothermal compressibility for the system.94 In principle, simulations carried out in all of these ensembles should yield the same results for equilibrium properties in systems of sufficient size of course, when differing ensembles are employed, appropriate corrections (e.g., a PV correction to compare the NVT and NPT ensembles) must be introduced. So far, little work has been done to determine quantitatively the system size needed to reach the thermodynamic limit. 

Initial concepts of temperature came from the physical sensation of the relative hotness or coldness of bodies. This sensation of warmth or cold is so subjective relative to our immediate prior exposure that it is difficult to use for anything but simple qualitative comparison. The need to assign a quantitative value to temperature leads to the definition of a temperature scale. The concept of fixed points of temperature arises from the observation that there exist some systems in nature that always exhibit the same temperatures. The scientific or thermodynamic definition of temperature comes from Kelvin, who defined the ratio of the thermodynamic or absolute temperatures of two systems as being equal to the ratio of the heat added to the heat rejected for a reversible heat engine operated between the systems. This unique temperature scale requires only one fixed point, the triple point of water, for its definition. 

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. 

The current definition of the temperature scale is based on one fixed point, the triple point of water. The absolute temperature of that point is defined arbitrarily as 273.16 K exactly. (The triple point of water is that temperature at which pure liquid water is in equilibrium with ice and water vapor.) This definition fixes the size of the kelvin, the degree on the thermodynamic scale. The size of the Celsius degree is defined to be equal to one kelvin exactly and the origin of the Celsius scale of temperature is defined as 273.15 K exactly. 

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 of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy, temperature, and the second law. Clausius introduced the word entropy. The second law is a statement of existence of stable equilibrium states and distinguishes thermodynamics from mechanics and other fields of physics. The many stable equilibrium states and various other equilibrium and nonequilibrium states contemplated in thermodynamics are not contemplated in mechanics (Gyftopoulos and Beretta, 2005). The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of any material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

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