CORRESPONDENCE
ON THE REDUCED FROST-KALKWARF VAPOR PRESSURE EQUATION
Their final equation is log P, =
( Z C
-
);
r;(
- 1)
+
SIR: In recent papers (4, 5), Reynes and Thodos discussed the reduced Frost-Kalkwarf vapor pressure equation
c
(2)
results from evaluating Equation 1 a t the critical poini. Upon analyzing the constants b and c determined from experimental data for a number of polar and nonpolar licpids, they found the correlation:
where L = 2.302585. -bL
a = - + T,
(3)
Table 1.
Compound
Acetic acid Acetone Ammonia Aniline Bromine n-Butyl alcohol Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorine Chlorobenzene Chloroform Ethyl alcohol Ethyl chloride Fluorobenzene Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen sulfide Methyl alcohol Methyl chloride Nitric oxide Nitrogen Nitrous oxide Oxygen Phenol Phosgene n-Propyl alcohol Sulfur dioxide Sulfur trioxide Water Helium Neon Argon Krypton Xenon Standard error of estimate Computed from Equation 9.
78
-b
4,6809 4,6353 4.2495 4.7922 4.2775 7,4769 3.3830 3.3191 4.2540 3.5588 4,5229 4.7141 5.6093 4,0888 4,6482 1.4025 4.1177 3.5213 3.6204 4,5644 3.8771 6,9531 3.1637 4.0335 2,9556 6,5956 4.1952 6,6099 4,5570 7,8764 4,6673 0,6800 2,5833 2,9636 2,9636 2.9636
($2
- I)
From Equation 1 we obtain
( dL E)
dP,
2dLPp
d T , - y
+ C
Vapor Pressure Constants of Substances Investigated -611
4.0116 3.9384 4.0959 4,2262 5,2362 7,6222 3.2848 3.3732 4.3161 3.6623 4.5378 5,0959 4.3398 4.0520 4.7480 1 ,6852 4.9349 3.4308 3.6799 3.0159 3.8991 7.1151 3.1380 4.0519 2.8522 7.4849 4.2833 6,1066 4,5720 9.0609 4,4620 0.6593 2.4343 2,9696 2.9696 2.9696 0.520
Computed from Equation 12B.
l&EC FUNDAMENTALS
+ 0.1832
ac
- bc
aa4
7.94 7.30 7 .OO 7 , 92c 6.25c 8.9lC 6.32 6.04 6.75 6.18 7.04 6.86
4.3380 4.1201 4.1678 4.5007 4,8709 7,4040 3,3630 3,3618 4.2772 3.6229 4.5233 4.9144 4 8556 4,0658 4.6826 1.7158 4.5952 3,4948 3.6694 3.6952 3,9010 7.0001 3,1629 4.0305 2.9244 7,0406 4,2490 6,2042 4.5526 8,3506 4,5520 0.7305 2.5324 2,9760 2.9836 2.9936 0.296
7,968 7.292 7.098 7.721 6.430 8.967 6,338 6.020 6,767 6,167 7.060 6.896 8.944 6.759 7,081 4.561 6.189 6.419 6.346 8.529 6.594 8.864 5.957 6.635 5.888 8,048 6.836 8.838 7.118 8.267 7,562 3.909 5,549 5.707 5.751 5.784
8 98
6 . 73c 7.02 4.74 6.09 6.39 6 . 3lC 8.48 6.55 8.88 5.98 6.59 5.92 7.89" 6.74 8.85 7.05c 8 . 2OC 7.39 4.01 5.66 5.76 5.79 5.83 c
(4)
from which G may be obtained with the knowledge of the critical temperature and critical pressure plus one other vapor pressure-e.g., the normal boiling point. A somewhat better correlation for b and c which uses Riedel's ( 6 ) a, is described below. Riedel's CY is defined (6) as
where d = 0.1832 and Lvhere the relation ~ + b + d = O
log T ,
Evaluated at boiling point using Riedel's equation ( 6 ) .
4,3450 4,1181 4,1924 4 4507 4.9162 7 4184 3,3676 3,3569 4.2815 3.6197 4.5284 4,9235 4.8465 4,0729 4,6979 1 ,6710 4,6201 3,5021 3.6784 3.7075 3.9122 6,9962 3 1571 4.0418 2.9164 7 ,0804 4.2732 6,2012 4,5697 8,3672 4,5952 0.7053 2,5045 2,9628 2,9738 2,9819 0,302
6.46 5.15 4.58 6.42 3.31 8.75 3 142 2.99 4.13 3.20 4.65 4.33 8.93 3.50 4.62 1.38 3.06 3.53 3.31 7.60 3.79 8.67 2.90 3.86 2.82 6.36 4.12 8.60 5.28 7.86 5.32 0.74 2.45 2.59 2.84 2.69
and dPr
bL
(\ - E2 -
pr
-
dT,
(1
2dLP, T,3
~
-k
17,)
- d$)
(7)
By substituting Equation 7 into 6, evaluating the result a t the critical point (where T r = P, = I ) , and solving for b , we obtain the desired result: 1 b = -
[C
L
-2 -
(ec -
2)(1 - d L ) ]
(8)
For many compounds, a , may be obtained from the table of Lydersen, Greenkorn, and Hougen (2). In other cases, ac may be obtained from the following purely empirical expression: (9) This expression \vas (obtained by looking for a correlation between a, and AH,,,IRT,; the second term in parentheses is the Giacolone approximation (3) for the latter. T h e calculated a , agree very well with tabulated values (2) or those calculated from Riedel.’~equation ( 6 ) evaluated a t the boiling point. Only eight of the 85 substances considered deviate by more than 0.1; only H2, HlO, Br2, phenol, and aniline exceed 0.15: none deviate more than 0.2. Figure 1 shows the correlation. Table I shows a comparison of values of - b found by applying the different procedures to the 36 compounds selected by Hamrin, Thodos, and Reynes ( 7 , 5). The first column lists the value obtained from the experimental data. T h e second column, headed - b,, shows values computed from the experimental values of c using Equation 3 ; the column headed -b, was obtained with Equation 8 and tabulated values of aC (2) ; the values of -- 6, were obtained using Equations 8 and 9. A glance at the individual values and their standard errors of estimatr, shown a t the bottom of the respective columns, indicates the superiority of Equation 8. From Equations 8 and 1 and the numerical value of d, the vapor pressure expression becomes log P , = 0.4343
[C
-2
- 0.57817 (a,- 2)] X
I
I
I
1
I
4
5
6
7
8
9
a Figure 1.
Correlation of aC with a = T,blnP,/(l
- T,b)
Solid curve (Equation 9 ) is least squares fit
Putting Equations 12 and the known value of d in Equation 1, we obtain
(
log Pv = 0.1832 (ac- 2)* 1 -
9+
-
- 0.42183 (e,- 2)
(ac-
l ) ] log T ,
0.1832
+
(g2-
1)
(13)
The b’s computed from Equation 12B in general are in anly fair agreement with experimental values ; the results for alcohols and acids are very poor (see Table I). Because P, from Equation 13 is very sensitive to a,, the use of tabulated or estimated ac is unsatisfactory. Moreover calculating b or c from such a i s using 12B or 12C,respectively, and then adjusting c o r 6 , respectively, in Equation 1 to fit at the boiling point gave results at low pressures somewhat inferior to those from Equations 4 or 10. Similar unsatisfactory results \vere obtained from Equation 13 when a , was adjusted to fit at the boiling point. Acknowledgment
T o obtain a value of c, only one other vapor pressure (such as the boiling point) is required together with a tabulated or estimated a,. Surprisingly, in spite of the superiority of Equation 8 over Equa.tion 3, sample calculations using Stull’s 10 mm., do not show any d a t a (7) for low pressures-i.e., noticeable superiority of Equation 10 over Equation 4. We next briefly consider imposing Riedel’s empirical relation ( 6 ) dtu = 0 at T , d T7
=
1
on Equation 1. By (differentiating Equation 6, substituting in the derivative of Equation 7, setting P, = T , = 1, and setting the result equal to zero, we find that Q
= d [ ( a c-
2)’ - 11
b = - d(a, c = a,
-- dL(a,
- 2)’
- 2)(a, - 1 )
This work was performed under the auspices of the U. A4tomicEnergy Commission. literature Cited
(1) Hamrin, C. E., Jr., Thodos, G., J . Chem. Phys. 35, 899 (1961). (2) Lydersen, A , , Greenkorn, R., Hougen, 0. A., “Generalized Thermodynamic Properties of Pure Fluids,” LVisconsin Univ., Eng. Expt. Sta., Rept. No. 4, Madison, Wis., 1955. (3) Reid, R., Sherwood, T., “The Properties of Gases and Liquids,” p. 93, McGraw-Hill, New York, 1958. (4) Revnes. E. G.. Thodos. G.. A.I.Ch.E. J . 8. 357 (1962). (5j Reines; E. G:, Thodos, G., IND.ENG. CHEM.FUNDAMENTALS 1, 127 (1962). (6) Riedel, L., Chem. Zngr.-Tech. 26, 83 (1954). (7) Stull, D. R., Ind. Eng. Chem. 39, 517 (194’).
DONALD G. MILLER
(12‘4) (12B) (12C)
S.
Lawrence Radiation Laboratory University of California Livermore, Calif.
VOL. 2
NO. 1
FEBRUARY 1963
79
CORRESPONDENCE
ON THE REDUCED FROST-KALKWARF VAPOR PRESSURE EQUATION SIR: hliller offers an interesting approach for the relationship between constants b and c of the Frost-Kalkwarf vapor pressure equation. This is accomplished through the introduction of Riedel’s parameter (7) a t the critical point, cyc = (dlnP,dlnT),. The work of Reynes and Thodos (5. 6) for nonpolar substances. including hydrocarbons of all types, recognized the direct linear relationship between constants b and c
Greenkorn, and Hougen (4)report for argon the value cyc = 5.76. T h e equivalence a t Equation 3 and Miller‘s Equation 8 has been reached by tkvo independent investigators, and credit should be given to Miller for bringing out into the open the relationship between constants 6. G, and a ; . Also. current interest in the Frost-Kalk\varf vapor pressure equation is expressed in a recent publication by- Bondi and McConaughy ( 7 ) ) in which the vapor pressures of paraffinic. naphthenic, and aromatic hydrocarbons u p to 12 carbon atoms per molecule have been investigated. I n that study, these investigators correlate constants b and G with the molecular structure sufficiently well to permit the calculation of these
-
constants in terms of AH; and X p , respectively, where
-
I n the Sliller stud!. a relationship is presented in Equation
8 between these t\\o constants and the Riedel parameter at the critical point. cy,. Equation 8 was recognized by Damasius and Thodos ( 2 )i n a recent study in which the Enskog modulus a t the critical point is: ( b p s ) , = a,:, - 1
(2)
I n the course of that investigation, the Frost-Kalkwaif equation \%‘asapplied a t the critical point to produce the relationship.
(5) = 1.7298 c - 3.9837 b ~ T R
- 1.4596
(3)
I n Equation 3. the partial derivative is equal to cy,, and therefore Equation 3 and Miller‘s Equation 8 become identical. Miller reverses the procedure and selects to relate b in terms of c and cyc. Equation 3 was applied to argon, using the values of b = - 2.9636 and c = - 2.6786 reported by Hamrin and Thodos (3) to produce the value cyc = 5.71. Lydersen,
80
I&EC FUNDAMENTALS
AH8 = AH; - X , T . literature Cited
(1) Bondi, A., McConaughy, R. B., “Estimation of Vapor Pressures for Pure Hydrocarbons with 5 to 30 Carbon Atoms.” American Petroleum Institute, Fairmont Hotel, San Francisco. Calif., May 1962. (2) Damasius, Gediminas, Thodos, George, IND.ENG. CHEhf. FUNDAMENTALS 2, 7 3 (1963). (3) Hamrin, C. E.! Jr.: Thodos, George, J . Chem. Phys. 35, 899 (1961). (4) Lydersen, A , , Greenkorn, R., Hougen, 0. A,, “Generalized Thermodynamic Properties of Pure Fluids,” bl’isconsin Univ.. Eng. Expt. Sta., Rept. No. 4, Madison, Wis., 1955. (5) Reynes, E. G., Thodos, George, IND. ENG. CHEbf. FUNDAM E N T A L S 1,127 (1962). (6) Reynes, E. G., Thodos, George, A.Z.Ch.E. J . 8, 357 (1962). (7) Riedel, L., Chem. Zngr.-Tech. 26, 83 (1954).
GEORGE THODOS nhrthwestern Uninersity Evanston, 111.
