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W 15 “Isotherms of Real Gases” Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum W 15 “Isotherms of Real Gases” Tasks 1. Measure the isotherms of a substance for eight temperatures. Describe the processes that occur close to the critical point. 2. Plot the isotherms and determine the saturation vapour pressure ps in the region of the Maxwell line (coexistence of fluid and vapour). 3. Plot ln(ps) as a function of (1/T). Fit the vapour-pressure equation to the data and determine the average molar latent heat of vaporization of the substance under study. 4. Determine the amount of substance of the substance under study. 5. Use the Clausius-Clapeyron equation to determine the molar heat of vaporization as a function of temperature. Plot the latent heat as a function of the reduced temperature T/TK and fit a power law to the data. Literature Physikalisches Praktikum, Hrsg. W. Schenk, F. Kremer, 13. Auflage, Wärmelehre 2.0.1, 2.0.3 Physics, M. Alonso and E. J. Finn, 15.6 Accessories Instrument for investigation of the critical point, thermostat Keywords for preparation: - Ideal and real gas, kinetic theory, molar volume - Isothermal and adiabatic state transformation - Isothermal, isobaric, isochoric diagrams in p,V,T-space - Isotherms of an ideal and real gas in the p-V diagram - Equation of state after van der Waals, description in the p-V diagram - Maxwell construction, critical point - Compressibility factor z (ideal gas, real gas) - Vapour-pressure curve, Clausius-Clapeyron equation Remarks The “instrument for the investigation of the critical point” (company PHYWE) is already prepared for the measurement of a p-V diagram. At all times follow the guidelines for temperature and pressure settings available in the laboratory. If the temperature (pressure) is too high, there is the danger of environmental contamination with mercury. The thermostat with the open water bath should only be operated under surveillance. After thermal equilibrium has been reached a minimum of ten measurements should be made for one isotherm with the last two values recorded with a nearly liquefied gas. 1 Theoretical background As a first approximation for a real gas one starts from the simpler case of the ideal gas. The equation of state of an ideal gas containing an amount of substance of n moles is: pV nRT . (1) The equation of state for one mol of an ideal gas is then pVm RT . (2) In Eqs. (1) and (2) p denotes the pressure, V the volume, T the absolute temperature, R the molar gas constant, n the amount of substance and Vm = V/n the molar volume. In the kinetic theory of the ideal gas the molecules are regarded as point particles without an own volume. These particles collide elastically (momentum transfer) on contact; any other, especially long distance forces (cohesion forces) between the molecules are neglected. These approximations are justified as long as the gas is far away from the condensation point, e.g. air at room temperature and atmospheric pressure. Otherwise correction terms have to be added to the equation of state. The equation of state after van der Waals is given by a n2 V p2 () Vm b p a b RT . (3) Vm V n 2 The correction terms take into account the cohesion pressure (/)aVm as well as the volume of the molecules b. Equation (3) is an algebraic equation of third order in Vm (cubic equation). Using the appropriate mathematical relations Vm might be calculated. In order to find the relationship between the critical parameters pk, Vk and Tk and the van der Waals constants a and b, one considers the shape of the critical isotherm that has an inflection point at the critical point. At this point the critical isotherm has both a horizontal tangent pV/0m T and a vanishing second derivative 22pV/0 . The partial derivatives are: m T paRTk 2 230 VmT () V mk b V mk . (4) 2 pa2RTk 6 2 3 4 0 VmT () V mk b V mk With the van der Waals state equation the critical point can be evaluated as: aa8 p, T , V 3 b (5) k27b2 k 27 Rb mk or 27R22 T RT abkk, . (6) 64ppkk 8 Calculate for the substances sulfur hexafluoride (SF6) and ethane, which are used in the experiment, the values of the critical parameters Tk and pk using the well-known van der Waals constants a und b. 4 -2 -4 3 -1 sulfur hexafluoride SF6 (W 15/1) a=0.79 N m mol (1 %), b=0.8810 m mol (1 %) ethane (W 15/2) a=0.56 N m4 mol-2 (1 %), b=0.6510-4 m3 mol-1 (1 %) The van der Waals equation beside other equations of state for the real gas is only an approximation that – due to its simplicity – has only two parameters. Other equations of state (e.g. virial equations, see e.g. Physical Chemistry by P. W. Atkins) with more than two parameters reach better agreement with the experimentally measured equations of state. 2 A clear demonstration of the various phase changes is obtained by plotting p, V, T diagrams. In case of an isothermal process, a p-V diagram (Clapeyron diagram, Fig. 1) is chosen. Of specific importance is the isotherm at the critical temperature Tk. For all temperatures smaller than the critical temperature one observes a region with a horizontal isotherm (Maxwell line, Maxwell construction) on which gas (vapour) and liquid coexist. At T = Tk this region degenerates into a point (critical point). For temperatures larger than Tk liquefaction of the gas is impossible even at very high pressures. If the temperature is raised further, the isotherms approach those of the ideal gas. In case of a homogeneous substance three regions are distinguished: Fluid + Vapour (coexistence): within the hatched area in Fig. 1 Pure Liquid: below Tk and to the left of the hatched area Pure Gas (vapour): above Tk and to the right of the hatched area Fig. 1 Isotherms of a real gas Clausius-Clapeyron equation and latent heat The critical point (Tk, pk, Vk) is characteristic for a substance; its location can be experimentally determined by recording a set of isotherms. From the temperature dependence of the saturation pressure ps, where the isotherm is practically horizontal, the molar heat of vaporization can be determined. Starting point of this analysis is the Clausius-Clapeyron equation: dps Q23 . (7) dT T Vm In this equation dps / dT denotes the slope of the vapour-pressure curve (phase line between gaseous and liquid phases) in the p-T-diagram, Q23 the molar latent heat at the transition from liquid to gas phase and VVVm mG mF the difference between the molar volumes of the gaseous and liquid phases. The Clausius-Clapeyron equation follows directly from the Gibbs-Duhem equation SdT Vdp ndµ 0 , (8) if the latter is used for both gaseous and liquid phase, if these two equations are subtracted from each other, if the relation TSQm 23 is used and if the equality of the chemical potentials at the phase transition line is taken into account. For a fit of the vapour-pressure curve ps(T) the differential equation (7) has to be integrated. Here one has to bear in mind that Q23 and Vm will in general be temperature dependent. In the following two approximations will be discussed. 1.T « Tk: Q23 = constant. At temperatures far below the critical temperature Tk the latent heat Q23 is constant to a good approximation; see [1] for a discussion of correction terms. Further, the liquid volume is much smaller than the gas volume and can be neglected; if the volume in the gas phase close to the phase boundary is approximated by the ideal gas equation, Vms RT/ p , one obtains the simple differential equation dp p Q ss23 (9) dt RT 2 3 that can be readily integrated yielding the vapour-pressure equation in the form Q ppexp 23 . (10) ss0 RT ps0 denotes the integration constant. Therefore, in the low temperature region the molar latent heat might be determined from a fit of the vapour-pressure equation (10) to the measured values of the saturation vapour-pressure ps . 2. T Tk: Temperatures below, but close to the critical point. Close to the critical point both the molar latent heat Q23 as well as the molar volume difference Vm are strongly temperature dependent. To a good approximation one finds 3/8 QQTT23 23(0)(1 /k ) (11) [2,3] and also Vm vanishes at the critical point [4]. An integration of the Clausius-Clapeyron equation is not straightforward in this case. Phenomenologically it is observed, however, that the approximation (10) still fits the measured vapour-pressure curve rather well. This might be due to the fact that the ratio Q23/Vm is nearly temperature independent and that also the temperature variation in the considered temperature range is rather small; therefore, Eq. (9) is still a good approximation. However, the meaning of the parameter Q23 is modified: Q23 only has the meaning of an effective, at best averaged, molar latent heat; the fit of Eq. (10) to the vapour-pressure data is inappropriate to yield reliable values of the latent heat in the temperature region close to the critical point. Although much more elaborate series expansions for ps(T) exist, see [4], the use of these does not lead to a significantly better description of the data in our case, especially in view of the appreciable measurement uncertainties in the undergraduate physics laboratory experiment and the small number of measurement points. Therefore, for the determination of the latent heat Q23 one should proceed as follows.
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