Slopes, isochoric

An isochoric equation of state, applicable to pure components, is proposed based upon values of pressure and temperature taken at the vapor-liquid coexistence curve. Its validity, especially in the critical region, depends upon correlation of the two leading terms the isochoric slope and the isochoric curvature. The proposed equation of state utilizes power law behavior for the difference between vapor and liquid isochoric slopes issuing from the same point on the coexistence cruve, and rectilinear behavior for the mean values. The curvature is a skewed sinusoidal curve as a function of density which approaches zero at zero density and twice the critical density and becomes zero slightly below the critical density. Values of properties for ethylene and water calculated from this equation of state compare favorably with data. 

In 1976, Hall and Eubank (12,13) published two papers which have direct bearing upon the present equation of state. In the first paper, they noted the rectilinear behavior for the mean of the vapor and liquid isochoric slopes issuing from the same point on the vapor pressure curve near the critical point and the power law behavior for the difference in these slopes. The second paper presented an empirical description of the critical region which generally agreed with the scaling model but differed in one significant way—the curvature of the vapor pressure curve. 

Figure 1. Coexistence curve with isochores and the isochoric slope, j/0. At points A and B the curvature of the isochores is zero as well as along the locus indicated by the dotted line (the locus of the isochoric heat...

The isochoric slopes, 4, obey a reasonably simple relationship in the critical region ... 

Figure 3. Mean reduced isochoric slopes at the coexistence curve for ethylene as a function of the reduced...

Greatest experimental deviations from the uniform isochore slopes given in (4) occur near 0.7 times the critical volume of 67 cc/g mol and correspond to about minus 3 in these slopes. The "adjusted" isochore intercept data of column 5 in Table I are the result of passing the computed slopes of column 3 through the individual data points of Stewart and Johnson in the arbitrary range 45 5 atm. These adjusted intercepts are consistent with the calculated slopes. 

Using Van der Waals isochore slope from (2), modified by (4), together with (5), the wide-range equation of state for hydrogen in units of Table I is... 

We consider first the behavior of yy. With experimental values of the isochore slopes and the corresponding volumes, it is straightforward to calculate effective hard-sphere diameters as a function of temperature or density using the first term on the right-hand side of Eq. 

These effective diameters are simply the values of cr needed to account for the isochore slope at each point along the coexistence curve. 

Figure 5.17 The solubility s of a partially soluble salt is related to the equilibrium constant (partition) and obeys the van t Hoff isochore, so a plot of In s (as y ) against 1 IT (as x ) should be linear, with a slope of AF lution -t- R . Note how the temperature is expressed in kelvin a graph drawn with temperatures expressed in Celsius would have produced a curved plot. The label KIT on the x-axis comes from l/T -t- 1/K...

Both slopes are necessarily positive. With Cp > Cy, isochores are steeper. 

Show tlrat isobars and isochores have positive slopes in the single-phase regions of a TS diagram. Suppose tlrat Cp = a bT, where a aird b are positive constants. Show tlrat tire curvature of an isobar is also positive. For specified T and S, wliich is steeper an isobar or an isochore Why Note tlrat Cp > Cy. 

Real locus of the system in pressure-composition space during an isochoric step from the /f-1th to the kth point on the recorded isotherm (Eqns 7.10, 7.11). The excursion of the pressure in the hydrogenator to pfys,o) occurs when the valve S (Fig. 7.4) is opened instantaneously. The system then approaches equilibrium according to the kinetics of the sample and the thermal relaxation time of the sample/cell sub-system. The pressure excursion is lessened if S is opened slowly, so that absorption commences while Psys is still rising. The slope of the isochore is constant only if the compressibility is constant. 

Figure 28 shows the temperature-dependent D and D data in the Arrhenius plot for the 256-particle Gay-Beme system GB(3, 5, 2, 1) along two isochors [161]. D and D are obtained from the slopes at long times of the respective... 

In our experience, a necessary but insufficient condition for a well-behaved critical isotherm is that, at the critical point, the slope of the critical isochore from the equation of state be equal to the slope of the vapor-pressure equation, 6P/6T = dP /dT. This constraint always is applied in the following work via the least-squares program (7). 

Johnston, Keller, and Friedman represented experimental data on the density of normal hydrogen in the form of straight isochores with a slope given by Bv = —7.11 + 437 (1 fV). The agreement with the calculated yv is shown in Fig. 4. 

When molar volume data are not available for at least one high pressure value, as for liquid oxygen, the assumption of the linearity of the p T isochores may be used in order to make reasonable estimates. In this case the entire p-F-T diagram can be described by a series of straight lines determined each by the V value at saturated vapor pressure and by the slope yV. At high pressure the assumed linearity is not valid the error on />, however, remains smaller than 1 % and the accuracy on is then of the same order of magnitude. 

In Figure 11.17 we show Arrhenius plots of the isobaric [t (7)]/> (F = 0.1 MPa) and isochoric [t (7)] v/ (vy = 20 to 30 mm /g) dielectric relaxation times of PMPhS, where the free volume Vf was obtained from the S-S equation of state. The slopes at the intersection of the isochoric and isobaric curves yield the respective activation enthalpies for ambient pressure. 

It was mentioned earlier that the enthalpy is finite at the critical point, however, the slope represented by Equation (A6) is infinite and thus the isochoric heat capacity is infinite at the critical point. And in the region near the critical point the heat capacity becomes very large. This is true not only for carbon dioxide but for all pure substances. 

The very nearly linear behavior of isochores or isometrics, which are lines of constant molal volume or density upon P vs. T coordinates, provides motivation for seeking a simple, albeit approximate equation of state. Figure 1 illustrates isochores for normal hydrogen. Those identified by V are from Ref. 2 by p are from Ref. 1. Experimental data usually are in the form of isotherms, from which it is not possible to make an isochore plot directly. Tedious graphical or mathematical interpolation procedures are required. The compendium of Woolley, Scott, and Brickwedde [1] and the linear isochores of Stewart and Johnson [2] provide the necessary interpolations. Such data, therefore, are not directly experimental. A linear isochore may be expressed in terms of its intercept and slope by (1). This may be compared with Van der Waals (2), wherein v is molal volume,... 

The isochores of the alkali metals, or any other liquid for that matter, are nonlinear in principle. If one could measure with sufficient accuracy or obtain data over a sufficiently wide range of temperature and pressure, the isochores would always exhibit curvature. The curvature (temperature dependence of the slope ( dpldT)y) is a consequence of the state-dependence of the interatomic interaction. It reflects, in particular, softening of the repulsive part of the potential as the temperature is increased. If we choose to represent the repulsive potential by a hard sphere diameter, the softening appears as a reduction of the diameter at higher temperatures. 

The phase boundaries of the Pd-X (X = H,D,T) system were determined from pressure concentration temperature data because of the high risk of handling PdT samples outside our tritium loading equipment. Pd forms no stable oxide layers as is the case for or Nb that prevent the tritium to leave the sample. The boundaries between the miscibility gap and the 3-phase were obtained from the shape of the desorption isotherms. The values of concentration and temperature of the solvus line between the a- and the two phase regions a+3 were obtained by quasi isochoric measurements. A PdX sample with the concentration x slightly in the miscibility gap was heated in small temperature steps so that the concentration of the sample decreased and finally belonged to the pure a-phase. The change of slope in the equilibrium pressure as a function of the inverse temperature is interpreted as the intersection with the solvus line. 

---