CORRESPONDENCE
The Intersection of the Inversion Curve and the Unit Compressibility Line
The compressibility factor Z may be defined in terms of the reduced temperature, molar density, and pressure
8 = TIT,; 6
= d/d,;
ir
=
P/doRTB
(1)
as
z= TO6
=
(bZ/dO),d6'
+ (bZ/b6),d6
(3)
=
( b Z / d B ) ,d0
+ (dZ/bir)H dir
(4)
and for Z ( 0 , x )
dZ
If we apply the restraint of unit compressibility (UC), so that Z = 1 and T = 86, then the number of independent variables is reduced to one, and some relationship exists between any two of the three variables a, 8, 6 under this condition. It has been shown (Holleran, 1967, 1970; Holleran and Gerardi, 1968; Morsy, 1963) from experimental data that for many fluids this UC relation is
8
+6
E
1; ir
=
8(1 - 8) = 6(1 - 6)
(5)
From these equations we see that along the UC line
(dOld6)Z = -1; (dir/dO)Z = 1-28
(6)
which combine with eq 3 and 4 to give
(bZ/b8),= (dZ/db)#
(7)
as was noted previously (Holleran and Jacobs, 1972), and
(bZ/dO), = (28
- IXdZ/dir)B
(8)
as characteristic of points on the UC line. Equation 8 is of interest in the present context because it allows an easy identification of the intersection of the unit compressibility line with the Joule-Thompson inversion curve (IC). The IC may be defined as the locus of points for which
(b6/d8,,
=
- s/e
(9)
+
From eq 2 we find (dZ/d0), = - ( 2 / 6 ) [ ( 8 / d O ) . 6/01, so that eq 9 is equivalent, except at zero density and pressure, to the requirement that
(az/de), = 0
(10)
on the IC. Thus, eq 8 and 10 locate the intersection of the UC line and the IC as the point of nonzero pressure on these lines where
(28
- 1,caz/airx
z = 1 + Bir + Cir? + ...
(2)
where do and the Boyle temperature T B are the reducing constants which experimentally satisfy eq 5 below. Z is a function of any two of the three variables, 8 , 6 , a, and in particular for Z ( 0 , 6 )
dZ
There is one point on the UC line for which (az/dn)~is zero but this is a t zero pressure and density, and 0 = 1. This temperature is the Boyle temperature where the second virial coefficient B is zero, and Z us. n, expressed in the virial equation as
=
0
(11)
This occurs a t 0 = l/~,and eq 5 then identifies the remaining coordinates as 6 = YZ and n = %.
(12)
is seen to have a zero initial slope. However, eq 11 does not hold a t x = 6 = 0. In fact, there is no intersection here, because a t zero pressure the Joule-Thompson inversion occurs a t a higher temperature, where T = B/(dB/ dT) . For the Lennard-Jones 6,12 intermolecular potential this occurs at 0 r 1.9 (Hirschfelder, et al., 1964). Bursik (1973) has recently concluded (incorrectly) that the condition (dZ/an)s = 0 must hold a t the intersection. He treated this result as an anomaly because it does not agree with observed behavior. Experimentally, the Boyletemperature isotherm of Z us. n is the only one whose minimum lies on the UC line. At higher temperatures the isotherms have no minimum, and at lower temperatures, where B is negative, Z us. n begins at zero n with the value of unity and with a negative initial slope. After reaching a minimum, Z increases again, with the isotherm passing through the IC and the UC line (in this order if 0 < yZ, in reverse order if 0 > to Z values greater than unity at high pressures. Thus the condition of zero isotherm slope, (aZ/dn)~= 0, is not experimentally associated with either the IC or the UC line (except for the latt e r a t 0 = 1,a = 6 = 0 ) . The error in Bursik's treatment arises from his simultaneous solution of eq 13, which, as he shows, holds on the UC line
6
+ (dir/bS)y = 8 + (dir/dB)i,
(13)
and eq 14, which holds on the IC
6(dir/dS)H= d(d'iT/bO)d
(14)
Bursik's solution, that at the intersection
(bK/bS)H
8
(15)
(bir/bB),= 6
(16)
=
and then leads to the requirement that (dZ/aa), = 0. This solution would be valid if the intersection were located at any other point except where it is, i.e., where 0 = 6. Not knowing this location of the intersection, it was easy to overlook the fact that at this point any value of ( d a / a 6 ) , = (dx/aO)a will satisfy eq 13 and 14. Actual values of ( a a / d d ) ~ = (aa/aO)b depend on the gas and can be determined from the observation (Holleran and Sinka, 1971) that on the UC line, (dZ/a0)8 = ( d Z / d 6 ) , = k B 6 / 8 ' , where k B is a dimensionless constant characteristic of the gas. With eq 2, this gives ( a a / d 6 ) ~= (aa/aO), = (1 + k B ) / 2 a t the intersection point. Values of k B range frorr;1.2 to over 3 for common gases. Ind. Eng. Chem., Fundam., Vol. 13, No. 3,1974
297
It may be noted that the intersection point is the mid6 uc line, and that at this point the point on the 0 pressure has its maximum value ( P = d&T5/4). This is the highest pressure a t which 2 us. P isotherms cross the 2 = 1 axis, and the pressure a t which the envelope of these isotherms, which represents the IC (Bursik, 1971), CrOSseS this axis. The isotherm which at this point is the one for T = T5/2.
Bursik, J. W., Ind. Eng. Chem., Fundam., 12, 256 (1973). Hirschfelder. J. O., Curtiss, C. F., Bird, R. B., "Molecular Theory of Gases and Liquids," pp 173-175, Wiley, New York, N. Y., 1964. Holleran, E. M., J. Chem. Phys., 47, 5318 (1967).
~~~~~~~~;E: ;: ~ ~ ~ ~ C ~ 72, ; '3559P(1968), ;."~~~~~
Holleran. E. M., Jacobs, R . s.. hd. Eng. Chem., Fundam., 11, 272 (1972). Holleran. E. M.3 Sinka, J. v.9 J. Chem. PhYS., 55,4260 (1971). Morsy, T. E.,Dissertation, Technische Hochschule, Karlsruhe, West &rmany, 1963.
Chemistry Department S t . John's University Jamaica, New York-11439
Literature Cited Bursik, J . W., Ind. Eng. Chem., Fundam., 10, 644 (1971).
Eugene Holleran
Boyle Points, UC and Compressibility Curves
In the graph that follows (Figure l),a study is made of the prediction of the linear t us. d UC theory which requires the temperature a t the maximum pressure point of the UC curve to have the value Tl(max P = %TB, where TB is the Boyle temperature. The substance is helium-4 and the data are from McCarty (1972). The curve labeled I in this 2 us. P (atm) plot is a segment of the inversion curve. Curves A and B are respectively 12.58 and 165°K isothermal segments. The A isotherm corresponds to the temperature at which the 2 us. T K plot of the inversion curve data of McCarty crosses 2 = 1. The B isotherm corresponds to Tl(max P ) = VZTBwhere the Boyle temperature is taken from Glasstone (1946) as 33°K. It is obvious that the isotherm labeled B does not pass through the point of maximum pressure as required by the linear UC theory. If, however, the Boyle temperature is taken as 24.06"K (Keyes, 1941), one-half of this value is 12.03"K and this is close to the isothermal A value of 12.58"K. Obviously, the second choice of Boyle temperature gives a much better agreement with the linear t us. d UC theory. The scatter in reported Boyle temperatures is not confined io helium. Nitrogen has a t least two values, 324 (Steiner, 1948) and 335 (Guggenheim, 1957); methane, 508 (Douslin, et al., 1964) and 491 (Guggenheim, 1957); and carbon dioxide, 721.7 (Holleran, 1967) and 710.7 (Vukalovich and Altunin, 1968). There are additional problems in mixing theoretical and experimental values. For argon, various Boyle temperatures are: 396.4 (Sze and Hsu, 1966), 410 (Steiner, 1948), 411.5 (Guggenheim, 1957), and 408.2 (Sze and Hsu, 1966). The two by Sze and Hsu are obtained theoretically for Lennard-Jones (6, m ) gases with m respectively taken as 16 and 12. All of the above temperatures are Kelvin. As with the helium example, 2-P plots of isothermal and inversion curve segments for any of these substances
I
/
16
20
24
28
32
p-
Figure 1.
may lead to ambiguity in attempting to interpret whether or not the YzT5 isotherm passes through the maximum pressure point of the unit compressibility curve, depending on which value of TB is used for a particular substance. Literature Cited Douslin, D. R . , et ai., J. Chem. Eng. Data, 9, 358 (1964). Glasstone, S.. "Textbook of Physical Chemistry," 2nd ed, p 247, Van Nostrand, Princeton. N. J., 1946. Guggenheim, E. A,, "Thermodynamics," 3rd ed, pp 116, 167, North-Holland Publishing Co., Amsterdam, 1957. Holleran, E. M., J. Chem. Phys., 47, 5318 (1967) Keyes, F. G.. "Temperature, Its Measurement and Control in Science and Industry," p 59, Reinhold. New York, N. Y., 1941. McCarty, R. D., "Thermophysical Properties of Helium-4 from 2 to 1500 K with Pressures to 1000 Atmospheres," National Bureau of Standards. Technical Note 631, 1972. Steiner. L. E.. Introduction to Chemical Thermodynamics," 2nd ed, p 75, McGraw-Hill. New York. N.Y., 1948. Sze, M. M., Hsu, H. W., J. Chem. Eng. Dara, 11, 77 (1966). Vukalovich, M. P.. Altunin, V. V., "Thermophysical Properties of Carbon Dioxide," p 185, Collet's, LTD., London, 1968.
School of Engineering Rensselaer Polytechnic Institute Troy, New York 12i81
Joseph W. Bursik
CORRECTION T H E CRITICAL PROPERTIES OF BINARY HYDROCARBON SYSTEMS In this article by Sung C. Pak and Webster B. Kay [Ind. Eng. Chem., Fundam., 11, 255 (1972)J there is a typographical error in eq 1, which appears on p 260. The
298
I n d . Eng. Chem., Fundam., Vol. 13,
No. 3, 1974
equation should read: log P H = ~ 5.92822 - 3037.6/T, where T = "K and P H =~partial pressure of mercury in poundspersquareinchabsolute.
