THE EQUATION OF STATE OF A MIXTURE DETERMINED FROM THE EQUATIONS OF STATE OF ITS CONSTITUENTS. BY W. EDWARDS
DEMING
AND
L O L A E. S H U P E ,
Fertilizer and Fixed Nitrogen Investigations, Bureau of Chemistry and Soils U. S. Department of Agriculture.
THE determination of the equation of state for a mixture from the equations of state for the pure component gases has received attention since the time of van der Waals. In a recent paper Dr. Kleeman i has given a method by which the equation of state of a mixture m a y be derived from those of its constituents. Let the equation of state P i = F ( m j v , T, ai, bi, ci, . . . )
(I)
for rn~ grams of the pure gas i of unvarying molecular species, when expressed in powers of I/v, be p,=~_,A. .
[m'Y ' v
/
"
(2)
The A i , are functions of temperature and of the particular gas. The most general equation for a mixture of two gases r and s showing no chemical combination, could then be written p = (A~lm~ + A ares)T
+
A~2m~2 + As:m8 2 + f2(T)m~m8
V
V2
(3)
+ Ar3mr ~ + As~m8 3 + fs(T)mr2rn8 + f~l(T)mrm8 2 + . . . . V~
We shall assume t h a t F has the same form for all components. If f2(T) = 2 4--A,~Aa2, f~(T) = 3 ~ 83, f31(T) = 3 ~-Ar3A 832, • ' ' , (3) becomes P = Y~ (~A-7~mr + ~ A4-AT,,m,)"v-".
(4)
n
x R. D. K l e e m a n , J. FRANKLIN INSTITUTE, 206, 51I (I928). 389
390
W.
EDWARDS DEMING AND LOLA E . SHUPE.
[J. F. I.
This is K l e e m a n ' s equation ( I I ) o n page 515 of the reference cited. T h e equations for a mixture m u s t reduce to (I) and (2) if either mr or ms becomes zero. (3) satisfies this condition. So does (4). T h e a r g u m e n t s t h a t led K l e e m a n to equation (4) could have led him to equation (3). If Dalton's law holds, f 2 ( T ) = fa(T) = fal(T) . . . . . o, b u t equation (4) does not allow this. T h e evaluation of the p r o d u c t terms (interaction terms) has been the inspiration for several pieces of work in kinetic theory, e.g. the recent one b y L e n n a r d - J o n e s and C o o k y A good history is given b y Beattie. a In (4) the interaction terms are fixed b y the A ;,, b u t obviously this can not be p e r m i t t e d . K l e e m a n e v i d e n t l y considered all the A i, > o. T h e situation is easily handled, however, w h e n a n y A i, < o. In the figure a case is shown where one coefficient (At,) is negative and the other (A~) is positive. T h e ordinate to a curve n -- I, 2, 3, "'" at a n y abscissa is the proper n t h coefficient for the b i n a r y m i x t u r e composed of x weight fraction of gas s and I - - X o f gas r. T o c o n s t r u c t such a curve, write A r, = ~r~lAr~l, As~ = ~ I A ~ I , where ~ and ~,~ are + I or - I as the case m a y be, lay off the distance ~ ~ l~lr~ ] at x - o, and the distance e,~/IAs~ j at x = I, and join b y a straight line. Let x, e l y ] be the co6rdinates of a point on this line. T h e n x, e[yl ~ will be the co6rdinates of the corresponding point on the curve. ( e = + I or - I.) This idea is easily extended to a mixture of a n y n u m b e r of components. K l e e m a n ' s scheme (our equation (4)) can be described as " c o m b i n a t i o n of coefficients." A n o t h e r scheme is to derive the parameters for the equation of state of the mixture from the p a r a m e t e r s for the pure components, e.g. by linear combination, as has often been done. Let us refer to it as " c o m b i nation of p a r a m e t e r s . " This idea has recently been used by Beattie 3 on mixtures of nitrogen and m e t h a n e using the B e a t t i e - B r i d g e m a n equation of state. Kinetic t h e o r y indicates t h a t " c o m b i n a t i o n of p a r a m e t e r s " rests on a firmer 2 Lennard-Jones and Cook, Proc. Roy. Soc., xI5A, 334 (1927). 3 James A. Beattie, J. Am. Chem. Soc., 5 I, 19 (1929).
Sept., ~929.]
EQUATION OF STATE OF A MIXTURE.
391
theoretical basis t h a n does " c o m b i n a t i o n of coefficients." Lorentz showed t h a t for the first order constants, kinetic theory leads to a quadratic rule for combination of constants, and general considerations indicate t h a t this m a y be true of any of the constants. Under certain conditions quadratic combination becomes linear combination. Although all equations of state itre more or less empirical, yet the various parameters have some p h y s i c a l s i g n i f i c a n c e , whereas the FIG. I.
X_x: we!qhl;
/%°=-4 -
fra.otion o,! gas
~
Graph showing how the ~irst four coefficieni:5 vary with the composition oF a binary mixture.
-
coefficients after the term in I/v a m u s t be entirely empirical, or complicated functions of simpler terms. T h u s "combination of p a r a m e t e r s " would appear to be more rational than " c o m b i n a t i o n of coefficients." The n u m b e r m of independent coefficients A ,., in equation (2) will not be greater t h a n the n u m b e r of parameters ai, bi, c~, • • .. In forming the equation for a mixture by Kleeman's scheme, the coefficients after the ruth would be d e p e n d e n t on the first m coefficients. This is doubtless what Kleeman had in mind when he said these coefficients " . . . should initially be looked upon as being entirely independent of each other. If relations exist between them, they would be expressed by separate equations whose existence we need not consider at present." The expansion in powers of I/V of the BeattieVOL. 208, No. I245--28
392
W. EDWARDS D E M I N G AND LOLA E . S H U P E .
[J. F. I.
B r i d g e m a n e q u a t i o n is finite, containing four i n d e p e n d e n t terms. T h u s no coefficient d e p e n d s on a n y other and it is v e r y e a s y to a p p l y b o t h of the a b o v e methods. T h e " c o m b i TABLE
I.
Comparison of the Observed Pressures with those Calculated by (a) Linear Combination of the Parameters a, b, ~ o , B0, c in the Beattie-Bridgeman Equation of State (as already published by Beattie); (b) Combination of the Coefficients. 69.556% nitrogen, 30.444% methane, by weight. Temp., °C.
o°
25
Obs. Pobs.-P~al~. (a) .... (b)
31.64 0.04 --0.04
Obs. Pobs,-Peale. (a)
37.76 0.07 --0.04
-o.15
46.8I
57.00
....
20
IO
1500
200 °
38.17 0.08 --0.I0
45.71 O.II
44.66 --o.17
5I.I3 o,II 0.25
57.59 o.I3 0.59
53.63 o.15 -0.25
61.51 o.17 0.36
69.38 o.19 0.86
67.I5 0.26 --0.37
77.22 0.28 0.58
87.28 0.32
0.I0
(b)
--0.06
O. I 8 -- 0.22
Pob8-Pc~le (a) (b)
61.68 0.24 --O.08
75.8I 0.33 -0.39
89.89 0.47 --0.64
lO3.82 0.52 1.o4
117.74 0.59 2.44
Obs. Pobs.-Pe~Ic. (a) .... (b)
76.35 O.40 --0.09
94.72 0.54 -0.59
113.oi 0.76 --0.98
I3I.II
149.23
Obs.
90.95 0.65 --0.07
II3.86 0.84 --o.79
I36.64 I.I3 -I.37
159.25
112.93 1.19 0.06
I43.25 1.46
I73.48 2.04
203.31 2.44 4.26
233 .o7 2.9 ° 9.39
I5o.93 2.88 0.87
I95.53 3.59 --o.95
239.85 4.77
283.62 5.83 9.o6
327.44 7.20 I8.72
Obs.
....
I2
(b)
Obs. Pobs.-Peale. (a)
....
I5
IOO °
Pressures, atmospheres
Volume 30
50 °
Pob~.-P~l~. (a) .... (b) Obs.
Obs.
Po .
I;I I;I
0.I2
--i.o8
--1.88
--2,20
0.88 1.69
1.36
1.07
3.96 I81.66
1.39
1.53
2.56
5.69
nation of parameters" has the advantage, usually, of requiring less computation than the other, because most equations of state develop into an infinite power series in I/V, and it might be necessary to combine more than just a few coefficients to
Sept., I929.]
393
E Q U A T I O N OF S T A T E OF A M I X T U R E .
a t t a i n the desired accuracy. I t is interesting to see which gives the b e t t e r results. T h e following tables are Beattie's s Tables IV, V, VI, into which we have inserted pressures as TABLE II.
Comparison of the Observed Pressures with those Calculated by (a) Linear Combination of the Parameters a, b, ~ o , Bo, c in the Beattie-Bridgeman Equation of State (as already published by Beatlie); (b) Combination of the Coefficients. 3I.OI4% nitrogen, 68.986 % methane, by weight. T e m p . , °C.
0°
50 °
Volume 29.9456
Obs. Pobs.-Pcalc. (a)
9.9456
--o.I 7 54.81
(b)
--o.13
--0.28
54.82
67.95
Pobs.'Poale. (a)
--O.OI
O.OI
(b)
--o.I 9
-0.40
Pobs.-Peale. (a)
71.11 0.03 --0.31
Obs. Obs. (b)
Obs.
Obs. Pobs.-Pcalc. (a) (b)
Obs.
--
200 °
70.59
65,o4 o.14 --o.51
74.92 --
84.9 I -0.05 o.14
0,02
-0.35
O.OO
o.I3
lO6.68 --0.08
89.48 -o.05 --0.79
lO7.71 --O.O5
I25.79 -o.09
--1.84
--I.OO
I43.7 o --0.23 o.3I
86.97 0.29 --0.23
11o.9o --0.09
I34.73
158.57 --o.16 1.58
I82.o 3 --0.36 0.48
IO2.36 0.52
I32.46
I62.65 --o.I 4 --4.I6
192.51 --0.20
222.07
206.42
2 8 6 . IO --O.I8
--6.20
246.51 0.07 --3.13
239.94
288.11
335.68 0.57 3.06
--0.22
- - 1 . 2 5
--O,II --
1.79
(b)
0.66
Obs. Pobs.-Pcale. (a)
143.52 3.28 1.74
19I.o2 0.74 --2.69
(b)
62.42 0.02 --0.21
93.82 -0.05 --0.55
165.87 o.19 --2.44
....
54.23 o.o3 --o.42
O.O1 --I.OI
126.o4
Pobs.-Pe~le. (a)
....
6.9456
--0.08
--O.OI
....
7.9456
O.O1
O.OI
....
I1.9456
45.99
0.00
44.66
Pobs.-Pealc. (a)
....
14.9456
(b)
37.70
Obs. ....
I9.9456
I50 °
Pressures, atmos )heres
"
24.94561
I00:'
1.85
80.92 --
--0,23
--3.02
O.I2
0.8I
--7.45
--2.24
0.79 --3.38
0,22
--0.40 0.83
1.71
calculated by K l e e m a n ' s scheme. T h e observed pressures were t a k e n by K e y e s and Burks at the M a s s a c h u s e t t s I n s t i t u t e of Technology.
394
W.
EDWARDS D E M I N G AND LOLA E .
SHUPE.
[J. F. I.
TABLE III.
Comparison of the Observed Pressures with those Calculated by (a) Linear Combination of the Parameters a, b, "¢Ao, Bo, c in the Beattie-Bridgeman Equation of State (as already published by Beattie); (b) Combination of the Coe~cients. 29.69 % nitrogen, 7o.31% methane, by weight. Temp., °C.
o°
Volume
50°
IOO°
15o°
200°
Pressure, atmospheres Obs.
I;I
28.87 O.OO -- 0.04
34.97 0.03 --0.07
4I .04 0.05 -o.19
--
47.o8 0.06 o.o6
53.08 0.04 o.Ii
Obs. Pobs.-Pealc. (a)
32.75 O.OO -0.05
39.81 0.04 --0.09
46.82 0.06 -0.25
53.8o o.o8 --o.o8
60.74 0.06 o.16
Obs. Pobs.-Peale. (a)
(b)
37.84 o.oi -0.07
46.22 o.o6 --0.II
54.51 o.o8 -0.35
62.77 o.Io --o.13
70.99 0.09 o.2I
25
Obs. Pob~.-P~al¢. (a) .... (b)
44.82 0.02 --O.IO
55.09 o.08 -o.17
65.26 0.1:2 --O.50
75.36 o.13 --o.19
85.41 O.lI 0.29
20
Obs. Pob~.-Pcal¢ (a) .... (b)
55.00 0.06 -o.12
68.26 o.14 --0.26
81.35 o.18 --o.80
94.37 0.20 --0,30
IO7.3I o.19 0.46
15
Obs. Pobs.-Peal¢. (a)
(b)
71.3o o.i6 --0.16
89.91 0.25 --0.47
lO8.28 o.32 -- 1.42
126.48 0,33 --0.57
I44.6I o.34 o.83
12
Obs. Pobs.-Pcal¢. (a) .... (b)
87.Ol o.37 --o.13
111.45 0.42 --0.70
135.61 0.54 --2.17
159.52 o.61 --0.79
183.31 0.67 1.43
IO
Obs.
lO2.36 0.70 --0.02
133.2o 0.74 --0.88
163.66 o.92 --2.99
193.78 i .05 --0.96
223.80
40
Pob.* ,o 35
....
30
....
....
(b)
Pob~-P¢~1¢.(a) .... (b)
1.26
2.36
I n general, the tables show t h a t m e t h o d (a) gives the b e t t e r results. T h e exceptions consistently occur where b o t h the t e m p e r a t u r e is low and the pressure high, t h a t is, m e t h o d (b) a p p a r e n t l y gives smaller deviations in the lower left h a n d corners of the tables. These are the points where one usually expects the widest deviations. M e t h o d (b) has less rational theoretical basis, is usually more laborious to use and reproduces the d a t a on mixtures of m e t h a n e and nitrogen less
Sept., 1929.] EQUATION OF STATE OF A MIXTURE.
395
accurately than the combination of parameters in the BeattieBridgeman equation of state. It is interesting to note that Kleeman, even after deriving our equation (4), gives an equation of state for a mixture of two gases based on van der Waals' equation for a pure gas by combining linearly ~-a, b. "Combination of coefficients" possesses the advantage that it can be used when isotherms for the pure components have been determined at a single temperature only. Professor James A. Beattie of the Massachusetts Institute of Technology and Dr. Albert R. Merz of the Bureau of Chemistry and Soils have kindly made several suggestions that have been incorporated into this paper.