# W 15 “Isotherms of Real Gases”

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Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum

W 15 “Isotherms of Real

1. Measure the isotherms of a substance for eight . Describe the processes that occur close to the critical point.

2. Plot the isotherms and determine the saturation vapour ps in the region of the Maxwell line (coexistence of 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 of 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 . Plot the 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 , M. Alonso and E. J. Finn, 15.6

Accessories

Instrument for investigation of the critical point, thermostat

Keywords for preparation:

- Ideal and real , kinetic theory, molar - Isothermal and adiabatic state transformation - Isothermal, isobaric, isochoric diagrams in p,V,T-space - Isotherms of an ideal and in the p-V diagram - after van der Waals, description in the p-V diagram - Maxwell construction, critical point - factor z (, 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 . The thermostat with the open 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 , n the amount of substance and Vm = V/n the . In the kinetic theory of the ideal gas the are regarded as point without an own volume. These particles collide elastically ( transfer) on contact; any other, especially long distance (cohesion forces) between the molecules are neglected. These approximations are justified as long as the gas is far away from the 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 , 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=0.8810 m mol (1 %) ethane (W 15/2) a=0.56 N m4 mol-2 (1 %), b=0.6510-4 m3 mol-1 (1 %)

The 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 by P. W. Atkins) with more than two parameters reach better agreement with the experimentally measured equations of state.

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A clear demonstration of the various changes is obtained by plotting p, V, T diagrams. In case of an , 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 coexist.

At T = Tk this region degenerates into a point (critical point). For temperatures larger than Tk of the gas is impossible even at very high . 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: dp Q s 23 . (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 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 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

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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. At a given temperature Q23 might be calculated from the Clausius-Clapeyron equation (7): dp QTV s (12) 23 m dT The volumes VG and VF might be estimated from the corresponding p-V diagram, see Fig. 1. If the amount of substance n is known (see below), the molar volume difference is given by

Vm()/. V G V F n The slope of the vapour-pressure curve dps / dT might be obtained by fitting Eq.

(10) to the measured data of the saturation vapour-pressure ps, followed by a numerical differentiation of the fitting curve.

Determination of the amount of substance n

The determination of the amount of substance n is based on the idea that in a good approximation a diluted gas might be regarded as an ideal gas. Therefore one finds

limV (pV ) nRT . (13) If the product (p V) is plotted as a function of the reciprocal volume (1/V) for the isotherm measured at the highest temperature, then the extrapolation (1/V ) 0 yields the value ()pV V ; from this the amount of substance n is calculated.

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Experimental setup (see Fig. 2)

The high pressure capillary made from thick-walled (1) contains the gas under study. On its outer sheath a scale is fixed allowing for a direct reading of the volume of the capillary. The capillary is enclosed by a water bath (2) that allows for the regulation of the temperature using an external thermostat. At the same time the water sheath serves as a protective cover in case the high is damaged. The pressure is generated by pressing mercury that is located in a chamber below the capillary, upwards by slow turns the hand wheel. The pressure is measured by a manometer firmly connected to the mercury chamber. The temperature is measured in the water bath of the thermostat by a digital thermometer. After a change of pressure, volume or temperature one has to wait long enough for the equilibrium state to be recovered. Measure the isotherms at the following temperatures: SF6: room temperature, 33, 36, 39, 42, 45, 47, 50°C.

Fig. 2 Instrument for the study of the critical point 1 Glass capillary with substance under study 2 Plexiglas tube for temperature control 3 Pipe connection for thermostat fluid 5 Hand wheel for pressure regulation 6 Manometer

SECURITY ADVICE The pressure must not exceed 50x105 Pa (50 bar) and the temperature not exceed 55°C in any circumstance, otherwise mercury might be set free and/or the capillary might be damaged.

Measurements at the critical point (Fig. 3)

Approaching the critical point it will be more and more difficult to distinguish between liquid and gas phase, since the pressure differences between the two phases become smaller and smaller. Under normal conditions the ratio between gas and liquid is of the order of 1:1000. At the critical point both are the same. From this extraordinary fact a series of interesting effects follow that can be demonstrated experimentally by changing temperature and volume in small steps in the immediate vicinity of the critical point.

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o Start in the state (V1, T1) with V1 < Vk and T1 < Tk o Temperature increase to T2 > Tk o Volume increase to V2 > Vk o Temperature decrease to T1

Fig. 3 Circumnavigating the critical point (schematically)

Of interest for the observation are especially the liquid/gas fraction, the visibility of the interface as well as the occurrence of fog and chords (critical opalescence).

References

[1] D. Koutsoyiannis, Eur. J. Phys. 33, 295–305 (2012) [2] K. M. Watson, Ind. Eng. Chem. 35, 398–406 (1943) [3] G. Narsimhan, J. Phys. Chem. 67, 2238 (1963) [4] M. Funke, R. Kleinrahm und W. Wagner, J. Chem. 34, 735–754 (2001)

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