P-V-T Relations of
GASEOUS MIXTURES E. R. GILLILAND Massachusetts Institute of Technology, Cambridge, Mass.
The slope of the isometric of a mixture can be calculated by the molal average of the slopes of the isometrics of the pure constituent gases, all measured at the same molal concentration. This rule, together with a method for estimating the internal pressure, furnishes a means of predicting the P-V-Tproperties of the mixture. For the three gaseous mixtures given, the agreement with * the experimental data is better than either Amagat’s or Dalton’s laws, and is probably within the experimental error of the data.
HE increased use of high pressure in chemical and mechanical industries makes it imperative for the engineer to have simple and accurate means of predicting the properties of the various gaseous mixtures involved. Recently, methods of correlation have been developed for pure gases and vapors which allow the prediction of their properties over wide limits with but a limited amount of data (11, 12, I S ) . While a knowledge of the pressure-volume-temperature relations of pure vapors is very desirable, more often in highpressure calculations mixtures must be handled instead of pure vapors. Although the P-V-T data on pure gases are meager, they are abundant in comparison to the experimental data available on gaseous mixtures. This fact, together with the methods of correlation which have been developed for Pure gases, makes it highly desirable to predict the properties of a mixture from the properties of its components. Two general methods of dealing with gaseous mixtures have been proposed. The first assumes additivity, direct or modified, of certain physical properties of the mixture in terms of those of the components; the second averages the constants of an equation of state applicable to the pure constituents. The familiar illustrations of the first method are Dalton’s (6) and Amagat’s (f) laws. The former assumes that the pressure of a mixture is the sum of the individual pressures exerted by each pure component, in the same volume a t the Same temperature. If this law holds for suecessive additions of a pure component to a given volume, it is obvious that Boyle’s law must apply quantitatively throughout the pressure range in which Dalton’s law is applicable. The latter assumes that gases mix a t constant temperature and total pressure mrithout change in volume. This relationship is obviously valid for the mixing of successive partions of a given pure gas at constant temperature and pressure regardless of its P-V-T rehtionship. Gillespie (7) has shown that the Lewis and Randall fugacity rule is applicable only when this relationship holds. A more complicated rule of the same type is that of van Lerberghe ( I O ) the utility of which however is slight, because of the difficulty of evaluating his terms. It has been generally assumed that Amagat’s law reproduced the experimental data better than Dalton’s
The attempt to average constants in an equation of state goes back a t least to van der Waals (17). The method has been most successfully developed by Beattie (S), but is complicated for engineering work and limited by lack of the necessary constants for most substances, One of the most useful correlations of the P-V-T properties of pure gases has been the graphical representation of the isometrics-i. e., a graph of pressurevs. temperature a t constant volume. I n the range above the critical temperature the isometrics are essentially straight lines for a large number of substances. Lewis and Luke ( I S ) have expressed the equation of the isometric in the form of
P = @ I T - +a (1) For straight isometrics and 4 j z are functions of volume only; however, if the isometrics are not straight, a tangent to the isometric can be expressed in the above form, but in such cases @I and @pz are not unique functions of the volume.
ONE of the classical approaches to the study of the bezib havior of gases is the assumption that the total volume occupied by the gas may be considered as made UP of the effective volume of the molecules themselves and of the free volume available for molecular motion, and similarly that the total pressure is the sum of the actual pressure externally exerted upon the gas and the so-called internal Pressure equivalent to the mutual attraction of the molecules. The reciprocal of the slope of the isometric (1/*1) is a measure of the free volume and the negative intercept of the tangent to the isometric (@2) is generally defined as the internal pressure. If we visualize the effective volume of the molecule as a constant quantity, independent of the conditions, we would expect the effectivemolecular volume in the mixture to be the weighted mean of the effective volume of the constituents: where
+
+
(VrniX- gmIx)= a’ (V’ - $’) z”(V” - $”) i ~ = ) effectivemolal volume of molecules
(v-
(2)
However, it is well recognized that the effective volume changes considerably with the volume or, what is equivalent to the same thing, the imposed pressure, even a t constant temperature. If we assume that the effective volume of each constituent is constant a t constant total molal concentration,
(8)‘ 212
INDUSTRIAL AND ENGINEERING CHEMISTRY
FEBRUARY, 1936
irrespective of the character of the other molal constituents in the mixture, the effective molecular volume of the mixture would obviously be given by Equation 2, with the added consideration that V,i, = V’ = V” = V”’ = . . . ., and Equation 2 then becomes: =
x/$,’+ X”q5”
+ ...
(3)
The fact that the free volume is considered to be proportional to the reciprocal of the slope of the isometrics leads to the conclusion that we should average the reciprocals of these slopes : 1 G
x’ = x
xN +-//
91’
a1
+ ...
(4)
CLEARLY, the assumption of constant molecular volume *of a given species a t constant total molecular concentration, irrespective of the nature of the other molecular species present in the mixture, is only a rough approximation. From the behavior of pure compounds it is well known that the effective molecular volume tends to increase with total volume. Certainly, however, this tendency must be due primarily to the increase in the free volume around the individual molecule, rather than to the increase in the total volume as such. When a molecule of large effective volume is introduced into a mixture with other molecules of small effective volume, the free volume around the larger molecule is larger than would be the case in a mass of the first molecular species alone, at the same total molal concentration. Conversely, in such a mixture the free volume available to the smaller molecules is reduced in comparison with that in a mass of the pure constituent at the same total molal concentration. The effective volume of a given species is certainly a complex function of the free volume, but since in a mixture a t constant total molal concentration the change in free volume in comparison with that in the pure component is small, as an approximation it is suggested to call the effective molecular volume of the component in the mixture inversely proportional to the free volume:
1
x/
x“
*yx = 7+ p + This assumption imposes a correction on the effective molecular volume which is in the right direction and which may well be satisfactorily accurate because of the extremely small changes in free volume involved in the specific method of application here recommended.’ This method of approach is obviously equivalent to assuming the slopes of the isometrics additive arithmetically. (Ql)mlx =
X’@1‘
+
2” @ I ”
+ .. .
213
while Berthelot (4) and Galitzine (6) used Cmix =
(2’
dF+2‘’ d/Cx)2+. . .
Either Equation 8 or 9 may be used in combination with 4 or 7, givingfour possible combinations. Equation 2 can be rewritten, 1
pmm.
+
2’
(*z)rmr
P‘
+
+ + . ..
2” ___
~ 1 + ’
P”
+
P,,, (@2)mlx = 2’ (P’ P,,, = ZIP‘ + x’/P’/+ 1
Qz’)
2
4’ XIt + C” 2”
+ ...
(10)
”
Q,”)
~
It should be emphasized in the use of these equations that all values for the component gases are measured a t the molal concentration and temperature of the mixture. Equation 10 is best used with either Equation 8 or 9 without further rearrangement; however, Equations 11 and 8 result in
P,,,
=
X’P’+ X“PV+ . . .
(12)
and Equations 11and 9 give P,,,
=
X‘P’ + 2”P”+ 2’2” (4%-
d Q i ) Z
+
, ,
(i3)
The combined use of Equations 9 and 10 will be termed the “additivity of the reciprocals of intrinsic pressures;” and the use of Equations 9 and 11 will be termed the “additivity of intrinsic pressure.’’ *SINCE several assumptions have been made in arriving a t these methods, their use and application are justified only if they reproduce the experimental data on gaseous mixtures satisfactorily. A real test of the applicability of the methods can be obtained only in cases in which the properties of the mixtures vary appreciably from Amagat’s and Dalton’s laws. Wide deviations from these two laws are usually experienced when one of the gases in the mixture is much more imperfect than the other. Data of this type are furnished by the argonethylene and oxygen-ethylene results of Masson and Dolley (14). In addition the results of the intrinsic pressure methods are compared with those of Amagat’s and Dalton’s laws for Scott’s data (16) on hydrogen and carbon monoxide, and Keyes and Burks’ data (9) on nitrogen and methane. Whenever it was possible, data for the pure component gases were taken from the results of the same authors who obtained the data on the mixture. Thus the data of Masson and Dolley (14) were used for argon and oxygen, and Amagat’s results (8) were taken for ethylene. For nitrogen the
the slope of the isometric of the mixture at the molal concentration in question, they do not predict its height. To determine this height, the point chosen is the intercept on the pressure axis a t T = 0. Several methods of estimating such terms from the corresponding terms for the component gases have been given. Berthelot (4)and van der Waals (17) used =
2
++ 2~~+ 2”- (a/z)m,xl (P“+ + .. + , . . (ii)
THESE rules, however interesting, are of themselves Y completely without practical utility in predicting the P-V-T relation of the mixture, because, although they predict
c,,
~
and likewise for Equation 7,
(7)
(8)
Such a correction applied over wide range8 of free volumes would of courw be completely untenable.
(9)
TEMPERATURE -*C,
1
FIGURE1
INDUSTRIAL AND ENGINEERING CHEMISTRY
2 14
VOL. 28, NO. 2
+
TABLEI. CALCULATIONS ON 59.86 PER CENTETHYLENE)40.14 PER CENTARGONAT 25' C. Amagat--Masson and DolleyDifference Vobsvd. Pobsvd. Poaied. in P L%ter/mole Atm. Aim. %
--Dalton-
7--
0.743 0.638 0.415 0.334 0.276 0,233 0,2005 0,1743 0.1546 0.1388 0.1320
30 40 50 60 70 80 90 100 110 120 125
29.1 38.5 46.8 54.0 60.2 66.0 73.0 85.1 97.2 108.0 117.9
-
3.0 3.7 6.4 -10.0 -14.0 -17.5 -18.9 -14.9 -11.6 -10.0 5.7
-
+
-Equations
Poalod.
Difference in P
Atm.
% '
30.8 41.3 52.2 62.9 73.8 84.8 95.3 105.9 115.9 125.2 129.8
$2 5 +3.3 +4.3 +4 8 t6.4 +5.9
-Equation
8 10Difference Poaicd. in P Atm. %
29.0 38.5 47.7 56.3 65.0 73.3 81.3 89.0 96.7 104.8 108.8
+6.9
+5.9 +5.3 +4.3 +3.8
- 3.3
Pcalcd.
12Difference in P
Atm.
%
65.8
74.3 82.7 91.3 99.3 107.5 111.8
-11.0 -12.1 -12.6 -13.0
__.
.3 - 23.3 -- 4.0 - 5.0 6.0 - 7.1 -- 8.7 8.1 - 9.7
29.3 38.7 48.0 57.0
- 3.8 - 4.6 - 6.2 -- 78.4. 1 - 9.7
-Additivity of-Reciprocals of -Additivity of-Intrinsic Pressure Intrinsic Presaure Difference Difference Pcslcd in P Pd .d in P .~ __. .-. Atm. S, Atm. % 29.98 -0.07 30.03 +o 1 39.9 49.8 59.6 69.2 78.8 88.4 97.7 107.3 116.3 120.6
-10.4 -10.6
-0.23 -0.34 -0.67 -1.1 -1.5 -1.8 -2.3 -2.5 -3.1 -3.5
+Os 15 f0.24 +0.05 +0.05 -0.01 -0.24
40.06 50.12 60.03 70.03 79.99 89.78 100.1 109.6 118.8 123.8
f0.13 -0.43 -1.0 -1.0
TABLE11. CALCULATIONS ON 51.7 PER CENT HYDROGEN f 48.3 PER CENT CARBON MONOXIDE AT 25" C.
The values of (%)E are then read from Figure 1 at the values of V Rcalculated: ( @ P ) R = ~ ~1.~3 ~
2.452 0.6196 0.3586 0.2647 0.1991 0.1647
10 40 70
100 130 160
10.0 40.3 70.5 102 0 134.8 165.0
0 +0.8 fO.7
+2.0
4-3.7 4-3.1
9.98 39.6 68.7 97.0 124.5 151.3
-0.2 -1.0 -2.07 -3.0 -4.2 -5.4
10.001 40.08 70.15 100.2 130.05 160.1
0
+0.2 f0.2 +0.2 +0.04 +0.07
9.98 40.1 70.2 100.3 129.9 169.7
-0.2 +O.ZS $0.29 +0.3 -0.08 -0.19
( $ 2 ) ~ ~=
and
.-Additivity of-. -Additivity ofReciprocals of FAmagat- -Dalton-Intrinsic Pressure Intrinsic Pressure --Keyes andDifferDifferDifferDifferBurka ence ence ence ence Pcalpd. Vobsvd. Pobrvd. Pcalod. in P Poalcd. in P Poalod. in P in P (+*)mix Liter/moZeAtm. Atm. % Atm. % % Atm. % Atm. Atm. 37.76 46.81 61.68 76.35 90.95
38.3 47.0 62.7 78.0 93.0
+1.4 t-0.4 4-1.6 +2.2
+2.3
38.4 47.8 63.2 78.4 93.4
4-1,s 37.66 4-2.1 4-2.5 +2.7 4-2.7
46.7 61.4 76.2 90.3
-0.27 -0.24 -0.45 -0.20 -0.72
37.8 46.7 61.7 76.2 90.1
+0.1 -0.24 4-0.03 -0.20 -0.94
5.8 8.9 15.3 24.2 33.1
AS AN ILLUSTRATION, a sample calculation for the argon-ethylene mixture a t a volume of 0.276 liter per mole will be made. This mixture contained 59.86 per cent ethylene and 40.14 per cent argon; at 25' C. Masson and Dolley found the pressure to be 70 atmospheres. Calculating the reduced volumes,
%
2
)
~
=~
~
~
~
(50.9) (1.3) = 66.3 (48) (0.325) 15.6
At a molal volume of 0.276 liter per mole and 25' C., Masson and Dolley found the pressure of pure argon to be 85.1 atmospheres, and Amagat's data for ethylene gives 52.7 atmospheres under these conditions: P A 85.1 P O ~ H , 52.7 =i
E:
Calculating (%)mix
results of Verschole (16) were used; hydrogen and carbon monoxide data were taken from Scott (16), and the methane results are those of Keyes and Burks (9). To obtain ax, the internal pressure of the various gases, the data were plotted as isometrics in Figure 1, and the curves extrapolated from the region involved to the intercept on the pressure axis a t T = 0. As pointed out above, for straight isometrics these intercepts are a function of volume only, and the results of such extrapolations are given in Figure 2. In Figure 2 the results are plotted as reduced internal pressure ( 4 % ) ~(i. e., the internal pressure divided by the critical pressure) us. the reduced volume (i. e., the volume divided by the critical volume). This method of plotting was adopted since it brings the curves for the different gases more closely together. Tables I, 11, and I11 give the results of the calculations on an argon-ethylene, a hydrogen-carbon monoxide, and a methane-nitrogen mixture, respectively. Table IV summarizes the physical constants used in these calculations.
0.325
(Pc)c,H~ ( $
=
($1)~5
TABLE111. CALCULATIONS ON 56.65 PERCENTNITROGEN 443.3 PERCENT METHANEAT 0" C .
0.57 0.456 0.342 0.274 0.228
($I)CIH~
by Equation 8,
+ 0.4014 (15.6) +(66.3) 6.26
= 0.5986 = 39.7 = 46.0
by Equation 9, (@z)mix
[ZO?H4 ('%CPH~''~)
++ (0.4014) (15.6)1'z]a Z A (@2A"2)1z
= I(0.5986) (66.3)"z 5: 41.8
Now using Equation 10,
- 0.5986 0.4014 - 52.7 4-66.3 + 85.1 +- 15.6 = -0.5986 + - 04014
1
+
Pmix
119 0.00503
100.7
+ 0.00399 = 0.00902 =
+
Pmi, (+z)rnix = 111 Pmi*= 111 - (@&nix
by:Equations 8 and 10,
- 46.0 70 65 X 100 = = 70 -
Pmix= 111 = 65
%error
-7.1%
by Equations 9 . a n d 10, PdX
111
- 41.8
= 69.2
%error = 70
-7069.2 x
100 = -1.1%
FEBRUARY, 1936
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLEIV. Po Atm. 50.9 48 45.8
Ethylene Argon Methane
PHYSICAL CONSTANTS Po Atm. Nitrogen 33.5 Hydrogen 12.8 Carbon monoxide 35
vo
Liter/mole 0,1275 0.0737 0.0989
VC Lilet/mole 0.0901 0 06975 0.09
OF VALUES FROM FIGURE 3 TABLE V. COMPARISON AND FROM KEYESAND BURR 1000 c.---------. --------200~ c.------7--
Vobrvd.
Pabsvd.
Poalod.
Difference
Pobsvd.
Podad.
Difference
Litar/mole 0.57 0.456 0.342 0.274 0.228
Atm. 53.6 67.2 89.9 113.0 136.6
Atm. 53.6 67.1 89.5 113.0 135.7
%
Atm. 69.4 87.3 117.7 149.2 181.7
Atm. 69.3 87.5 117.5 149.8 181.0
-0.15 $0.23 -0.17 +o.40 -0.39
0 -0.15 -0.45 0 -0.66
%
The same conditio& will be used to illustrate Equation 11. The values of P A , PC,=, (ai)+,( @ 2 ) 0 z ~ , will be unchanged. Using Equation 12 (Equation 8 Equation 11)’
+
P,i,
= 0.5986 (52.7) = 31.6 $. 34.2 =
+ 0.4014 (85.1)
65.8
2 15
latter data do not indicate much choice between the two intrinsic pressure methods, but in view of the results of Table I, the additivity of intrinsic pressure is preferred to the additivity of the reciprocals of int r i n s i c pressure. The agreement obtained by the m e t h o d of additivity of intrinsic pressures is probably within the exp e r i m e n t a l and computation errors. I I l o 100 200 300 4 ~ ) 500 Table I11 also TEMPERATURE -*K. gives the internal pressures, FIGURE 3 and these together with the pressure calculated by the additivity of intrinsic pressures furnish a means of constructing a set of isometrics for this nitrogen-methane mixture. Figure 3 shows the isometrics constructed in this manner, and they can be checked against the data of Keyes and Burks who have measured the P-V-T properties of this mixture a t temperatures ranging from 0” C. (the temperature for which Table I11 was constructed) to 200” C. Table V gives a comparison of the values taken from Figure 3 with those obtained experimentally by Keyes and Burks. The agreement is very good, even though there is almost a twofold extrapolation of the isometrics which would magnify the computation errors; even better agreement would probably have been obtained if the values of the pressure had been calculated a t 200” instead of 0” C. and the other values had been obtained by interpolation instead of extrapolation.
-
Using Equation 13 (Equation 9
+ Equation l l ) , +
Pmix= 0.5986 (52.7) $. 0.4014 (85.1) (0.5986) (0.4014) [(66.3)”* - (15.6)”Zla = 31.6 34.2 4.23 = 70.03 70‘0 x 100 = 0.045% yoerror = 70
+
+
-
T H E ethylene-argon data given in Table I furnish the 4 best test of the various methods because these data show appreciable differences from Amagat’s and Dalton’s laws. This table indicates that the intrinsic pressure methods give the most satisfactory reproduction of the experimental data, and the additivity of intrinsic pressure is in better agreement than the additivity of the reciprocals of intrinsic pressure. The internal pressure of the mixture as calculated by Equation 8 fails to reproduce the experimental data, as the columns headed “Equation 8 10” and “Equation 12” indicate. Thus on the basis of these data the additivity of intrinsic pressures is the best of the methods tested. In the other tables only the two intrinsic pressure methods are compared with Amagat’s and Dalton’s laws as well as the experimental data; the other two methods are discarded because of their failure to check in Table I. The data of Tables I1 and I11 do not give as crucial a test of the methods as do the ethylene-argon data, because in these cases Amagat’s and Dalton’s laws represent the data reasonably well. However, the intrinsic pressure methods do show better agreement in both cases than Amagat’s or Dalton’s laws. These
+
10 0 8.0
6.0 4.0
Nomenclature
pressure critical pressure absolute temperature molal volume critical volume = slope of the isometrics = interce t of isometrics on P axis at T r/. = molal gee volume C = consiant P = P, = T = V = Vc =
=
0
Literature Cited Amagat, Ann. chim. phys., [5] 19, 384 (1880); Compt. rend., 127,88 (1898).
Amagat, Ann. chim. phys., [6]29,68 (1893). Beattie and Ikehara, Proc. Am. Acad. Arts Sci., 64, 127 (1930). Berthelot, Compt. rend., 126,1703,1857 (1898). Dalton, Gilherts Ann. Physik, 12,385(1802); 15,1 (1803). Galitzine, Wied. Ann. Physik, 41,770 (1890). Gillespie, Phys. Rev., 34, 352,1605 (1929). Holborn and Otto, 2. Physik, 23, 77 (1924); Leduc, Compt. rend., 126,218(1898).
P 20
10 0.8 0.6 0.08 0.1
0.2
0.4
2.0
0.6 0.8 I O (O$R
FIGURE 2
4.0
(0
ea io0
Keyes and Burks, J . Am. Chem. SOC., 50,1100(1928). Lerberghe, van, Bull. mad. TO^. Belg., [5]15,No. 6 (1929). Lewis, Trans. Am. Inst. Mining Met. Engrs., 107,11 (1934). Lewis and Kay, Oil Gas J . , 32,N o . 45,40,114(1934). Lewis and Luke, IND. ENQ.CHEM.,25,725(1933). Masson and Dolley, Proc. Roy. SOC.(London) 103A,524 (1923). Scott, Ibid., 125A,330 (1929). Verschole, Proc. Roy. SOC.(London),IllA,552 (1926). Waals, van der, “Die Continuitat.” Vol. 1, 2nd ed., Leipzig, Barth, 1899. RECEIVED July 6. 1935.
