II
J. I. A p s e and R. W. Missen Deportment of
Chemical Engineering University of Toronto Toronto, Canada
I
Natural Equilibrium Distribution of Methane and the Deuteromethanes
The dctcrmination of the distribution of methane and the four deuteromethanes at equilibrium and with the natural H t,o D abundance is an interesting example of a moderatrly complex problem in chemical equilibrium which docs not require special computational techniques for ils solution. I t also emphasizes the great disparity in thc nat,ural occurrence of these species, and t.his is relatively insensitive to temperature. The free energy data required have been put in the form of equilibrium constants for the following equetions by Jones and l\l&owelll : CHI
+ I)?
=
CHsl)
+ HI)
+ I)%= CHD. + HD CHJ)? + I)* CHDa + HD CHJ)
=
(1)
(2) (3
and CHDI
+ Ih = CDI + HI)
14)
The eonsttlut,~an? based OII statistical ralculatiot~si n the ideal-gas state, and are given at seven temperatures over the range, - 180"- 1000" C in Table I.' For a sysloni cout,aining the five species CHI, CH& CHZD1,CHDr and (Xh, it can he shown, using the methods described by Denbigh,2 that three equations are required to represent the stoichiomet,ry. These can be chosen as
and CHI
+ Clh
=
CHID
+ CHDs
(7)
The equilibrium constants for these equations van be calculated from those for eqns. (1)-(4). Thus, sinrr eqn. ( 5 ) is ohtainrd by subtrarting (2) from (I), Ks = K , / K .
(8)
Similarly,
where the suhscript to the equilibrium constant refers J. I. Apse is the holder of an Applied Science and Engineering Graduate Fellowshio. ' JONES,L. H.. I'VD MC~)OWELL, R . S., J . Mol. Spectroscopy,
to the number of the stoichiometric equation. These three equilibrium constants in turn lead to three equilibrium expressions in terms of t,he mole fractions ( X ) of the five species:
and K7 = X I X a / X o X n
113)
u-here the subscript to the mole fraction refers: to the number of hydrogen atoms in the molecule (X4is mole fraction of CHI, etc.) To these three equilibrium equations in the five unknowns must be added two conservation equations. Of the three equat,ions representing the elemental and isotopic balances, only two are linearly independent, since whew nx, no, and nc are the numbers of moles of hydrogen, deuterium, and carbon, respectively, present in the (closed) system. We arbitrarily set nc = 1 so that the mole numbers of species present becomc mole fractions. The carbon balance then is which is the same as saying the sum of the mole fractio~~s is one. The other conservation equation chosen is the deuterium balance, since this does not involve methane. If we take the natural abundance rat,io3of H to D as 6410 to 1 , then the deut,erium balance, usingeqn. (14), is Equations ( l l ) , (12), (13), (15), and (16) are to be solved for the fivc unknown mole fractions, with the equilibrium constants calculated from eqns. (8)-(10) and the data in Table 1. An iterative procedure can be used which converges so rapidly that only one iteration is required for three significant figures. This
Table 1.
Equilibri~m Constanta -180
-
Equilibrium Constants for Equotions ( 1 ) to (4)
Temperature, 'C -100
25
100
300
500
1000
.
3. 632 11959). ,
' DENBIGH, K. ti.,
"The Principles of Chemical Eqoilihrium," Cambridge, 1955, Chap. 4 . HODGMIN,C. D., (Editor), "Handbook of Chemistry and Physics," 44th ed., Chemical Rubber Pub. Co., Cleveland, 1962-63, p. 450.
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Journal o f Chemical Education
" Subscript to equilibrium constant refers to equation number in text.
+
+
assumes that Xa >> 2X2 3x1 4x0. Then from eqn. (16), Xs = 6.24 X lo-' and from eqn. (15), X4 = 0.9994. Then X2,XI, and Xo can he calculated in turn from eqns. ( l l ) , (12), and (13), respectively. The calculated values of these last three confirm the original assumption, and a second iteration is unnecessary. The calculated mole fractions are given in Tahle 2, together with the values of the equilibrium constants of eqns. 5 ) 7 ) for the temperature range -180'1000°C. The calculations confirm that virtually all the deuterium is present in the form of CHaD. The concentration of 624 ppm is about the same as ohtained by analysis of natural gas sample^,^ if we assume that all the deuterium is present in this form. The relative amounts of CH4and CHsD, being much greater than the others, are governed by the H to D ratio and not by cquilibrium considerations. As such, they are independent of temperature, as shown in Tahle 2. The relative amounts of the other three species increase rapidly as the temperature decreases below 25' C, although they are still negligibly small at -180" C. The relation X4 >> Xa >> Xp>>XI >> Xo is maintained, independent of temperature.
Table 2.
a
Equilibrium Constants and Mole Fractions of Methane and the Deuteromethanes
Subscript to mole fraction refers to number of H etams in
rnebhane species.
These calculations refer to the distribution dictated by the natural abundance of H to D. Higher concentrations of the deuteromethanes can be obtained if an artificial or enriched D to H ratio is used, as shown, for example, by the kinetics results of McKeeVor catalytic exchange reactions starting with CHI and DZ in a molar ratio of 2 to 1. "ASMADJIAN. orivate communi. D... Universit~of Toronto.. . cation. "'ICKEE, D. W., J . Phys. C h a . , 70, 525 (1966).
Volume 44, Number I, Jonuary 1967
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