The Ideal Gas Temperature Scale

The next problem is the assignment of numerical values to this property. Evaluation of the products Pj Vj, P V, . .. and Pj Vj, P2 V, . .. indicates that, at low pressures, the values are very close independently of the gases involved. More important, if the products are plotted against the correspondingpressures and then extrapolated to zero pressure, i.e. to the ideal gas state, the lines for the two gases converge to a common value. 

If the procedure is repeated using a bath of different hotness, the lines again converge to a common, but different from the previous case, value of(PV).  

We will demonstrate next how these ideal gas values, obtained by ex-trtq)olating the product (PV) to P = 0, are used to establish a temperature scale that, in contrast to liquid-phase thermometers (mercury, ethanol, etc.) is independent of the fluid used. 

The proc ure, which leads to the ideal gas temperature scale (T), involves the following two arbitrary steps  

The value of T at the triple point of water is set equal to 273.16 K. Experimental determination of the product PV) for several gases at the triple point of water has resulted in an accepted value of 22711.6bar cm mol and, consequently  

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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. 

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. 

From this observation, we are motivated to define temperature T on the ideal gas temperature scale as... 

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. 

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. 

The state of a gas at the limiting condition where P - 0 deserves some discussion. As the pressure on a gas is decreased, the individual molecules become more and more widely separated. The volume of the molecules themselves becomes a smaller and smaller fraction of the total volume occupied by the gas. Furthermore, the forces of attraction between molecules become ever smaller because of the increasing distances between them. In the limit, as the pressure approaches zero, the molecules are separated by infinite distances. Their volumes become negligible compared with the total volume of the gas, and the inter-molecular forces approach zero. A gas which meets these conditions is said to be ideal, and the temperature scale established by Eq. (3.9) is known as the ideal-gas temperature scale. 

The answer is the ideal-gas temperature scale defined as follows Measure the volume V of a fixed quantity of gas or gas mixture at a sequence of low pressures p at the temperature of the... 

Let ns nse an ideal gas in a Carnot cycle and find the efficiency of the cycle by using ideal-gas properties in ennmerating the changes in the four steps of the cycle. Let us designate the intial state of step 1 with the subscript A, the initial state of step 2 with B, and so on. The high temperature at step 1, which is i on the thermodynamic temperatnre scale, will be Tj on the ideal-gas temperature scale. The low temperature of step 3 will be T2, corresponding to 02. The work and heat terms of a step will be designated with subscripts 1, 2, 3, or 4. 

Thus, the thermodynamic and the ideal-gas temperature scales become the same if the values are selected to be identical at one finite temperature. 

Since neither the absolute pressure nor the molar volume of a gas can ever be negative, the temperature defined in this way must always be positive, and therefore the ideal gas temperature scale of Eq. 1.4-3 is an absolute scale (i.e., T > 0). 

The existence of the ideal gas temperature scale and the fact that it is the same for all gases are consequences of two laws which express conclusions drawn from a large number of experimental observations. One of these laws is Boyle s law which, in a modified form, states that for any given gas... 

Two particular temperature scales are used extensively. The ideal-gas temperature scale is defined by gas thermometry measurements, as described on page 42. The thermodynamic temperature scale is defined by the behavior of a theoretical Carnot engine, as explained in Sec. 4.3.4. These temperature scales correspond to the physical quantities called ideal-gas temperature and thermodynamic temperature, respectively. Although the two scales have different definitions, the two temperatures turn out (Sec. 4.3.4) to be proportional to one another. Their values become identical when the same unit of temperature is used for both. Thus, the kelvin is defined by specifying that a system containing the solid, liquid, and gaseous phases of H2O coexisting at equilibrium with one another (the triple point of water) has a thermodynamic temperature of exactly 273.16 kelvins. We... 

Thus, the efficiency of a Camot engine must depend only on the values of Tc and Th and not on the properties of the working substance. Since the efficiency is given by e = 1 + Qc/qh, the ratio c/ h must be a unique function of Tc and Ti, only. To find this function for temperatures on the ideal-gas temperature scale, it is simplest to choose as the working substance an ideal gas. 

Just as measurements with a gas thermometer in the limit of zero pressure establish the ideal-gas temperature scale (Sec. 2.3.5), the behavior of a heat engine in the reversible limit establishes the thermodynantic temperature scale. Note, however, that a reversible Carnot engine used as a thermometer to measure thermodynamic temperature is only a... 

It is now possible to justify the statement in Sec. 2.3.5 that the ideal-gas temperature scale is proportional to the thermodynamic temperature scale. Both Eq. 4.3.13 and Eq. 4.3.15 equate the ratio TJT to —qc/qh, but whereas Tc and Th refer in Eq. 4.3.13 to the ideal-gas temperatures of the heat reservoirs, in Eq. 4.3.15 they refer to the thermodynamic temperatures. This means that the ratio of the ideal-gas temperatures of two bodies is equal to the ratio of the thermod5mamic temperatures of the same bodies, and therefore the two scales are proportional to one another. The proportionality factor is arbitrary, but must be unity if the same unit (e.g., kelvins) is used in both scales. Thus, as stated on page 41, the two scales expressed in kelvins are identical. 

The thermodynamic temperature scale is not related to any particular kind of substance and is therefore more fundamental than the ideal gas temperature scale. We now show that the thermodynamic temperature scale can coincide with the ideal gas temperature scale. Assume that the working fluid of a Carnot engine is an ideal gas with a constant heat capacity. For the first step of the Carnot cycle, from Ekj. (2.4-10)... 

We will start with the statement of the zeroth law, use it to establish temperature as a state property, and proceed to develop the ideal gas temperature scale. 

The ideal gas temperature scale is thus established and it is the same, as we will see in the next Chapter, with the absolute thermodynamic tem-perature scale, arrived at through the second law of thermodynamics. 

Use of the ideal gas behavior exhibited by real gases in the limit of zero pressure leads to the establishment of the ideal gas temperature scale. 

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