A Graphical Representation of the Equation of State for an Ideal Gas JORGE GUERRA Williams Industri=d Products, S. A,,Havana, Cuba
Tm.4iwm.m charts are widely used for plotting properties of three-component systems as a function of composition. The property of equilateral triangles permitting their adaptability is that the sum of perpendiculars from any point in the triangle t o the sides is a constant. I n this type of graph the altitude is usually taken to represent 100 per cent (or unity) ; hence, the sum of the perpendiculars from the three sides to a point in the triangle must always be equal to 100 per cent (or unity). Thus, that particular geometrical property of the equilateral triangle, taken in conjunction with three suitable arithmetical scales directed along the three altitudes of the triangle, furnishes a basic triangular coordinate plane each and every point of which represents, uniquely, a definite value of the composition of a ternary system in terms of its three components. The property being plotted, like melting point, boiling point, density, index of refraction, viscosity, etc., can then be shown on the basic coordinate plane in the form of contour lines. If logarithmic scales are used instead of arithmetical
scales on such a diagram, a triangular logarithmic coordinate plane results. This has the property that the product of the three coordinates of any point in the triangle is a constant. The logarithmic transformation of the scales brings about a special situation in that the property in question is found to obtain not only for points within the triangle or on the sides of the trhngle, but also for points outside the triangle. Each of the three scales may be extended logarithmically without limit in both directions outside the triangle so that an infinitely extended plane is the result. For each and every point of this plane it is found that the product of the three point-coordinates is a constant. The figure shows the application of this type of triangular logarithmic chart t o the graphical representation of the pressure-volume-temperature relationships, or equation of state, of one mole of an ideal gas for which PVT-I
=
a constant = 62.36 liter-mm. Hg X (OK.)-' X
(gram-mol)-'
For convenience and ready reference the two coordinate lines corresponding to standard conditions, i. e., pressure 760 mm. Hg and temperature 273.16'K., as well as the line corresponding t o the molar volume under standard conditions (22.414 liters) have been emphasized (made heavier). I n particular, the chart has been so constructed as to locate one mole of ideal gas under standard conditions in the geometrical center of the figure. The lines corresponding to a pressure of 600 mm. Hg, a temperature of 346"K., and avolume of 17.695 liters, originally employed as reference or guide lines during the construction of the chart, have also been emvhasized. The coordinate lines have been extended over and beyond this original triangle, however, in order to stress and help convey the idea that the coordinate scales may be extended equally well into the plane outside the triangle. It is obvious that other units of pressure, volume, and temperature may be used, as well as any volume
VOLUME 33, NO, 10, OCTOBER, 19S6
other than the molar volume, provided the corresponding gas constant is adopted. For any point in the plane of the figure the values of the three coordinates of pressure, temperature, and molar volume satisfy the above equation. It follows that by locating in the plane the point of intersection of any two of the coordinates we thereby locate, uniquely, the corresponding third coordinate; i. e., given P (mm. Hg) and T ("K.) we can read directly off the chart the corresponding value of the molar volume (liters). Also, since all lines in the chart are iso-lines (constant pressure or isopiestics, constant volume or isochores, and constant pressure or isotherms) it follows that there may be read off directly from the chart, if needed, the values assumed jointly by any pair of variable gas
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properties as we move along the third one at constant value. The chart may be found useful in the classroom as an adjunct t o the discussion of the combined formula of Boyle's and Charles' laws and for acquainting the students with graphical methods. I n the physical chemistry laboratory the chart, enlarged and with the scales subdivided, may save time spent in routine calculations concerning gas volumes as required in molecular-weight determinations by vapor-density methods. The assumptions and approximations involved in the molecular-weight determination by the Victor Meyer method are such that the use of the chart for this purpose is not a t all likely t o introduce errors greater than those inherent in the method itself.
