Extrapolating Vapor-Pressure Curves

Below 0°C there are two curves, one for subcooled water and one for ice. The subcooled water curve (practically a straight line) is almost a linear extension of the liquid line, as logic suggests it should be. The ice curve (also practically a straight line) falls below the subcooled water line and has 

Between the critical and triple points, the line is a true equilibrium vapor pressure. 

Above the critical point the line is an extrapolation, whose meaning is discussed in Ch 9. 

FIGURE 5.7 Extrapolation of the vapor-pressure curve on Inp vs. 1/r coordinates, to temperatures above the critical and below the triple points. 

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Use of the Wagner equations requires relatively extensive and internally consistent data they may produce unphysical slopes of the vapor-pressure curve if fitted to inconsistent data. Neither the Wagner nor the Antoine equation should be extrapolated far below the temperature range in which it was fitted. Vapor-pressure data at low temperatures (where the vapor pressures are small) are scarce. Often, better estimates of the vapor pressure at low temperatures may be obtained from extrapolation techniques that make use of heat-capacity data, as discussed in The Properties of Gases and Liquids [15]. 

The range of validity of the present equation of state is for all fluid states at densities to the triple-point liquid density, temperatures from the triple-point to infinity, and pressures to at least 700 bar. Vapor pressures of the solid at temperatures below the triple point are replaced by an extrapolation of the vapor-pressure curve to absolute zero temperature, and a completely new type of formulation is presented for densities of saturated vapor from absolute zero to the critical-point temperature. 

Iteration for Coexisting Densities. Orthobaric densities near the critical point generally cannot be obtained accurately from isochoric PpT data by extrapolation to the vapor-pressure curve because the isochore curvatures become extremely large near the critical point. The present, nonanalytic equation of state, however, can be used to estimate these densities by a simple, iterative procedure. Assume that nonlinear parameters in the equation of state have been estimated in preliminary work. For data along a given experimental isochore (density), it is necessary merely to find the coexistence temperature, Ta(p), by trial (iteration) for a best, least-squares fit of these data. 

The best device, however, is the quartz coil manometer, the coil of which can be heatedto500°C(in special cases to 600-700°C). In all cases the null point of the instrument must be checked after each measurement. Therefore the manometer should be provided with a heating coil, which doe snot need to be at the test temperature but must nevertheless be at a sufficiently high temperature to prevent condensation in the coil and in the capillary connections (which are likewise provided with a heating coil). With compensation to zero, the pressure is read off on the Hg manometer. In those cases where it cannot be ascertained by the usual method (with a thermometer and distillation flask) the boiling point is determined more accurately by extrapolation of the vapor pressure curve. 

Figure 12.19 Effect of temperature on fugacity of a pure saturated liquid. Vapor-phase nonidealities (cpf) lower from the pure vapor-pressure curve, but the variation of /j-"with 1/T remains roughly linear. At supercritical temperatures, jnue vapor pressures do not exist nevertheless, for (0.9 < r /T < 1), we may choose the hypothetical pure liquid for the standard state and obtain a value of f° by extrapolation. These values were comjnited for pure water using data from steam tables [14].

The vapor pressure of the hypothetical liquid can be obtained by extrapolation of the vapor pressure curve to temperatures... 

The following values were obtained by direct observation and/or extrapolations from vapor pressure curves or by other calculations without the source of data being clear in all cases ... 

Fig. 1. Cox chart for extrapolating vapor-pressure temperature curves. Lewis, McAdams, and Gilliland, " Principles of Chemical Engineering, 3d ed UUl Book Company, Inc., New York, 1937.)...

The resulting (H3Si)2CHCl is a colorless liquid of bp. 63 °C (extrapolated from the vapor pressure curve) and can be obtained in a 70% yield [93]. 

Interpolation beriveen data For such common liquids as water, many refrigerants, and others, the vapor-pressure-temperature curve has been established at many points. For most liquids, however, only relatively few data are available, so that it is necessary frequently to interpolate between, or extrapolate beyond, the measurements. The curve on arithmetic coordinates (Fig. 7.1) is very inconvenient for this because of the curvature, and some method of linearizing the curve is needed. Most of the common methods stem from the Clausius-Clapeyron equation, which relates the slope of the vapor-pressure curve to the latent heat of vaporization... 

Beebe et al. [175] recommend the use of krypton at liquid nitrogen temperature which, due to its low saturation vapor pressure, reduces the amount of unadsorbed gas in the gas phase. Beebe s value of 0.185 nm2, for the area occupied by a krypton molecule is preferred by most investigators [176-178] but 0.195 nm2, has also been quoted [179] There is also disagreement over the correct saturation vapor pressure to use. The use of the solid saturation vapor pressure of 1.76 torr at 77.5 K usually results in the production of markedly curved plots [180]. Later investigators [181] tended to use the extrapolated vapor pressure of 2.63 torr. 

We might try to estimate the Henry s law constant by extrapolating the vapor-pressure curve from the critical point, but the extrapolation is so large and thus uncertain that instead we normally use the experimental concentration values for dissolved gases, plotted as in Figure 3.9 this topic is discussed again in Chapter 9. 

A true equilibrium vapor-pressure curve extends only from the triple point (the lowest temperature at which vapor and liquid can be in equilibrium) to the critical point (the highest temperature at which vapor and liquid have separate existence). What would happen if we extrapolated the vapor-pressure curves The logical coordinates on which to do this are In p vs. l/T, which correspond to the C-C equation, as shown in Figure 5.7. 

For solids that dissolve without molecular interactions with their solvents, we can make tolerable estimates of the solubility from the extrapolated liquid-phase vapor-pressure curve. Many common solids, such as NaCl and sucrose, do interact with water as a solvent, thus producing much greater solubility than we would estimate this way. 

Hoffman, Welker, Felt, and Weber (1962) pointed out that satisfactory values for the hypothetical vapor pressure may be obtained by extrapolating the vapor pressure curve into the supercritical region. Methane, ethane, propane, n-pentane, and n-octane were examined. 

Triple Point. Boiling Point. The triple point temperature of HN3 is 193 K, where the vapor pressure is 1 Torr. The normal boiling point of 308.9 K was extrapolated from the vapor pressure curve [3]. 

Caution should be used in extrapolating curves to very low pressures because of the possibility of curvature in the vapor pressure lines over a manyfold range of pressures. 

Below 500 K heating of the solid salt results primarily in the vaporization of the covalent molecule as a monomer. In this temperature range the only thermal decomposition, into NOz and 02, is exhibited by the solid. The vapor is more stable. The vapor pressure of Cu(N03)2 was determined by Addison and Hathaway48 by extrapolating pressure-time curves to zero time in order to subtract the pressures of N02 and 02. These vapor pressures increased from 0.32 torr at 430 K to 3.59 torr at 405 K. A plot of log P vs. 1/T is linear and yields a sublimation enthalpy of 67.0 kJ. Above 500 K both the solid and the vapor phase decompose to N02 + 02. 

Finally, the activity coefficient with respect to molality may be determined in precisely the same manner as sketched above. One equates (3.5.21c) with (3.1.4a) to obtain the analogue of (3.11.8), with c replaced by m. One then chooses as a reference solution the hypothetical Henry s Law case in which the straight line region of the P versus m, curve at low mi is extrapolated to m,- = 1. Let the corresponding hypothetical vapor pressure be P . This leads to an equation of the form (3.11.10), (3.11.11) and to the relation... 

The vapor-pressure measurements were carried out with hexa sublimed in high vacuum. Pressures of 10 to 10 mm. Hg, corresponding to 20 to 85°, were measured with a quartz filament manometer 2, S). At temperatures of 120 to 210° a simple mercury manometer was used, and the vapor pressures were obtained by extrapolating the pressure-time curves plotted in Fig. 1 to time zero. 

Comprehensive tables of vapor-pressure data of common liquids, such as water, common refrigerants, and others, may be found in Refs. [2,3]. For most liquids, the vapor-pressure data are obtained at a few discrete temperatures, and it might frequently be necessary to interpolate between or extrapolate beyond these measurement points. At a constant pressure, the Clausius-Clapeyron equation relates the slope of the vapor pressure-temperature curve to the latent heat of vaporization through the relation... 

Questim by C. E. Taylor, Lawrence Radiation Laboratory Is the curve of vapor pressure for hydrogen extrapolated below 10°K ... 

The trial-and-error, discussed in detail elsewhere [14], consisted essentially of cross-plotting PV -products vs. temperature and entropy with extrapolation of YendalPs cuiwes until agreement was reached which satisfied the critical values, the vapor pressure and temperature relationships and the PK-products on both plots. The resulting information was then plotted on the T S chart and reexamined for slope and for smoothness. The entropy of the saturated liquid curve resulting from the above procedure was checked by graphical integration. The agreement... 

Here is the vapor pressure of pure liquid solute at the same temperature and total pressure as the solution. If the pressure is too low for pure B to exist as a liquid at this temperature, we can with little error replace with the saturation vapor pressure of liquid B at the same temperature, because the effect of total pressure on the vapor pressure of a liquid is usually negligible (Sec. 12.8.1). If the temperature is above the critical temperature of pure B, we can estimate a hypothetical vapor pressure by extrapolating the liquid-vapor coexistence curve beyond the critical point. 

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