The viscosity of xenon was measured along ten isochores and the viscosity of ethane was measured along five isochores in the region of the gas-liquid critical point. This change in resonant properties can be related to the viscosity-density product of the fluid. The resonant properties of the quartz crystal were measured by connecting the crystal to the unknown arm of a Wheatstone bridge circuit, modified to measure parallel resistance and capacitance.
Drawing of a cell with sapphire windows Photograph of a cell with sapphire windows Critical xenon opalescence. XIII Xenon viscosity with a sapphire window cell XIV Ethane viscosity with a sapphire window cell. In recent years, there has been much interest, both theoretical and experimental, in the study of phenomena near critical points.
THEORY OF METHOD
The change in these electrical properties can be related to the viscosity of the medium. Far from resonance, the impedance of the series arm is extremely high; thus the total impedance of the crystal is the parallel capacitance C'. The logarithmic decrease in Eq. 30) represents the damping of the crystal due to viscous losses.
The inviscid losses can be represented by 6, which is assumed to be the case. are equal to the logarithmic decrease of the crystal in a vacuum, given by. where the subscript o indicates vacuum conditions. The surface area and volume V of the crystal are given by respectively. where r is the radius of the crystal. An impedance bridge, a frequency generator and a null detector were used to determine the resonance properties of the crystal.
EXPERIMENTAL METHOD A. Apparatus
This made it possible to take the cell and weighing bomb 1 out of the temperature bath when necessary. The temperature coefficients of all the above equipment are negligible for the conditions in the laboratory, i.e. ± 1°c. The output from the recorder is fed to the quartz heating element in the primary temperature bath.
The results of the calibration are shown in Table IV (for the masses) and Figure 19 (for the chain). The volume of the weighing bomb was determined by calibration with distilled water as follows. The pressure was measured with the precision manometer 2. The PVT data of Michels, et al. for xenon, Ruimer, et al. ethane, and Michels and Michels[421. for carbon dioxide were used to determine the pressure required to obtain the desired density for the run.
After completion of the isochore, the liquid in the cell was transferred to weighing bomb 2 by condensing with liquid nitrogen. During the measurements the cell was open to the manifold and the pressure was measured on gauge #1. The densities needed to calculate the vis- cosities were determined from the PVT data of Reamer, et al.
DISCUSSION
Excess viscosity is often assumed to be a function of density only and independent of temperature [SOJ. The critical point of the xenon used in this investigation was determined by visual observation (see Appendix II). The maximum value for improper viscosity at the nearest critical is about 15% of the ideal viscosity.
In Figure 35, the logarithm of the anomalous viscosity is plotted as a function of the logarithm of the temperature change reduced from critical. In Figure 36, the anomalous viscosity is plotted as a function of the logarithm of the reduced temperature difference. The straight line fit depends only slightly on the nature of the anomalous viscosity determination and a straight line.
The results of the present investigation should not be interpreted as excluding the possibility of a finite limit or small exponential divergence for the anomalous viscosity. This is taken as the "critical" isochore and used in the determination of the anomalous viscosity. Figures 38 and 39 show the logarithm of the anomalous viscosity and the anomalous viscosity plotted as a function of the logarithm of reduced temperature difference.
Close to the critical point, where the isothermal compressibility becomes very large, this phenomenon results in large density gradients in the direction of the field. First, close to the critical temperature, the density can vary significantly over the vertical component of the crystal, even though the crystal is aligned parallel to the horizontal and the diameter is only 0.3 cm. There will probably be some sort of "averaging" of the viscosity-density product, possibly masking the anomalous.
Since the average density value is used in calculating viscosity based on the resonance properties of the crystal, the result is a.
CONCLUSIONS AND RECOMMENDATIONS
Another possibility is to use a crystal with two electrical leads attached at the ends to the axis of the crystal, instead of the four radially connected leads used in the present investigation. Thus, the average density in the crystal would be very similar to the average density in the cell. The crystal used in this study is as small (0.3 cm) as is commonly used, although it may of course be possible to produce smaller diameter crystals.
The torsion crystal method seems particularly suitable for viscosity measurements of binary liquid mixtures in the region of their consolute points. These techniques appear questionable in the consolute field and could be improved by the torsion crystal viscometer. In developing the equation relating fluid viscosity to crystal properties, it was necessary to establish that the term V'P was negligible relative to the other terms in the equation of motion, Eq.
Assuming the above assumption is valid, it is possible to estimate the maximum magnitude of the pressure variation, to see what effect this would have in the critical region, where the isothermal compressibility is very large. To do this it is necessary to determine the velocity on the crystal surface v. w. This can be done by equating the energy dissipated in the viscous wave to the electrical energy supplied to the crystal. The maximum amplitude of a point at the end of the crystal is given by v /w and is equal to approximately.
For the usual conditions, the maximum Reynolds number is about 0.3, safely in the laminar region. It is also important to consider whether the viscous heating of the liquid results in a significant temperature gradient in the liquid. In the critical region, the isobaric volumetric expansion becomes very large, and a small temperature gradient will cause a large density gradient.
The approximate value of thermal conductivity of xenon in the critical region is 40.2 x10-6. with the usual values for n and v w. 100).
APPENDIX II
The position of the spurious peak (at 40.5 °C) appears to be the same, but the maximum value is slightly smaller. Except in areas of false peaks, the use of a second cell did not appear to affect the viscosity measurements. This was, of course, to be expected, since the ripple produced by the torsional oscillation of the crystal weakens very quickly.
Normally, as it cools until the coexistence curve is reached, either a gas phase will form at the top of the cell or a liquid phase will form at the bottom of the cell, depending on the density of the charge. At densities close to critical, the meniscus separating the two phases would appear within the visible region of the cell. The meniscus forms in the cell at the level where the local density is equal to the critical density.
Whenever the meniscus appears and disappears within the cell, the transition temperature is always the same, and is in fact the critical temperature. In the present investigation, the temperature and position of the meniscus were determined (if visible) at the transition point for different mean densities. Assuming that the meniscus should appear near the center of the cell for an average density equal to the critical, it was possible to bracket the critical density along with its measurement.
Very close to the critical point, the fluid became dark under normal illumination and could not be seen in the cell. The change in appearance within the cell is indicative of the density gradient caused by gravity. The temperatures of meniscus appearance and disappearance appeared to differ by several thousandths of a degree.
The values above are not so much best estimates of the exact values as they are midpoints of the range in which the exact values are thought to lie.
OSCILLOSCOPE
FREQUENCY GENERATOR
Actual Pressure (psia}
Actual Pressure (psia)
Null Detector
Potentiometer
Standard_/
Resistor
TEMPERATURE (°C)
T-T) log Tc c
T- Tc) log Tc
Density (g/cm 3 )
It is proposed that the predictions made by information theory about the distribution of winners in horse races are a good representation of the actual results. The heart of the theory, known as Shannon's postulate, gives a quantity that is a measure of uncertainty. A good discussion of the basics of information theory can be found in Information Science and Theory by Leon.
If nothing is known about the system, the maximum uncertainty assigns equal a priori probabilities to all states of the system. The density of states function indicates where the states of the system are located, but says nothing about how these states are filled. To determine the expected value, < f(x) > data was collected for the first 120 racing days of the 1967 season at the Aqueduct circuit in New York City.
It can be seen from eq. 17), that when calculating the probabilities, this constant cancels out. The first step in determining the probabilities is to determine the multiplier µ from equations. An iteration was performed on the IBM 7094 computer, where the value of the integrand of Eq. was calculated.
The results were plotted as the value of the integrand I versus y for different values of µ. Integration was performed by counting the squares under each curve. The other curve in Figure 4 is a graph of the distribution of winners from the original sample of 1075 races. In the limit, the probability of being in the interval f(x) +df(x) is given by pf(x)df(x), where pf(x) is the ordinate of the graph.
Recently, Rowlinson [S] criticized the indiscriminate use of the maximum entropy postulate to predict probability distributions. However, Rawlinson seems to view the application of information theory somewhat narrowly. It is only claimed that the maximum entropy postulate gives the least biased guess of the probability distribution for given information.
