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Diffusion of nitrous oxide in He, Ne, Ar and Kr
Physica 75 (1974) 573-582 © North-Holland Publishing Co.
D I F F U S I O N OF N I T R O U S O X I D E I N He, Ne, Ar A N D Kr A.S.M. WAHBY, A.J.H. BOERBOOM and J. LOS FOM-lnstituut voor Atoom- en Molecuulfysica, Amsterdam, Nederland
Received 13 May 1974
Synopsis The relative diffusion coefficients of NO in noble gases have been measured as a function of temperature. The mixtures investigated were 4He-NO, Ne-NO, Ar-NO and Kr-NO. The temperature range is from 237 K through 420 K. The accuracy of the measurements is about one or two per mill. It is found that the combining rules eua6j = (etla,ejjajj) 6 6 are not valid for these mixtures. The deviation is attributed to rotational inelastic collisions which influence the temperature dependence of the diffusion coefficient. Seemingly, good consistency could be obtained by applying these combining rules on the systems Ne-NO and Ar-NO only as in these mixtures Zu, rot is approximately equal. Effective potential parameters for the various mixtures have been obtained from the measui'ed temperature behaviour of the diffusion coefficients. 1. Introduction. The transport properties of molecular gases can be described
quite well by the monatomic Chapman-Enskog theory. Theoretical considerations indicate that corrections for rotational inelastic collisions and nonsphericity of the intermolecular potentials only amount to a few percent. This concerns viscosity, diffusion as well as thermal conductivity if, in the latter case the Eucken factor for the transport of internal energy is applied. This means that in general transport measurements are not very sensitive to rotational inelastic collisions. There is, however, an exception. The thermal diffusion factor in mixtures where at least one of the constituents is a diatomie molecule is very sensitive to isotope substitution. The formal theoretical explanation for this phenomenon has been given by Monchick, Sandier and Mason 1) and independently by Van de Ree2), who applied the theory of thermodynamics of irreversible processes. These theoretical considerations, based on the Wang Chang, Uhlenbeck and De Boer 3) approach, in which spin polarization is neglected, show the appearance of new terms in the expression for the thermal diffusion factor. These terms, which are linked with the partial internal heat conductivities of the components can explain the anomalies observed in thermal diffusion of molecular gas mixtures. 573
574
A.S.M. WAHBY, A.J.H. BOERBOOM A N D J. LOS
The coupling coefficients (6E u - 5) are collision integrals depending on the rotational quantum numbers of the colliding molecules. A theoretical evaluation of these collision integrals has been performed for simple models, like for example the loaded-sphere model. As, however, (6C~j - 5) also appears in the temperature derivative of the diffusion coefficient, measurements on diffusion as a function of temperature for different isotopic molecules in noble gases should be consistent with thermal diffusion measurements on the same mixtures. The theoretical relation between the thermal diffusion factor and the thermal dependence of the diffusion coefficient indeed was verified for helium-hydrogen and isotopic hydrogen mixtures4). For lorentzian mixtures in which the heavy abundant component is monatomic this relation reduces to the simple expression aL=2_
(tglnD'] . \ O In T i p
(1)
Very good agreement was obtained between thermal diffusion factors measured for the almost lorentzian mixtures A r - H 2 , HD, D2 and K r - H 2 , HD, D2 5) and the values for ~ obtained from diffusion-coefficient measurements as ,a function of temperature for these same mixtures6). On the other hand calculation of (6(7u - 5) for the simple case of loaded sphere - degenerate loaded sphere of equal mass and size gives an approximate relation with Z, ot (6(7,~ - 5) "~ 0.75/Zu, rot.
(2)
Although for more realistic models this relation won't apply, it is still expected that (6(7 - 5) will depend on the rotational collision numbers. As in a moleculargas-noble-gas mixture Zu. rot will depend on the mass of the noble gas atom it is expected that the thermal dependence of the diffusion coefficient in these mixtures also will be a function of the mass of the noble-gas atom. The most appropriate method seems to be the determination of the potential parameters e12 and al z for these mixtures in which the molecular component is the same and to test if the combining rules are still valid. Van Heijningen et al. 7) have measured the diffusion coefficients for ten binary mixtures of the noble gases 4He, Ne, Ar, Kr and Xe as a function of temperature and concentration. The temperature range was from 65 K through 400 K. From these measurements they concluded the validity of the combining rule for the dispersion force constant; not, however, the normal combining rules for el z and a~ 2, e.g.
i?tj(T6j =
( ~ . ~ . ~6 j j ~ j j )6
"} ,
(3)
where e is the depth of the potential well and tr is the separation at zero energy. In this formula the subscribts i, j refer to different species of molecules.
DIFFUSION OF NITROUS OXIDE IN He, Ne, Ar AND Kr
575
Our main purpose in the present investigation was to check the validity of formula (3) for a mixture containing one diatomic constituent. We have chosen the diatomic molecule NO together with the noble gases 4He, Ne, Ar and Kr. With these combinations a very suitable range of T* can be reached through which the mixture parameters can be determined accurately. It is relevant here to point out that Hirschfelder, Curtiss and Bird s) have derived that the effective total energy of interaction between a polar and a nonpolar molecule can be expressed in the same form as that between two nonpolar molecules. They have given combining rules similar to those applied for nonpolar molecules, containing a correction factor for the polar molecules. However, the correction factors in these combining rules for e12 , respectively, a12 are very small, due to the small dipole moment of the NO molecule. These corrections are of the order of 1 x 10 -5 (for 4He-NO: 1 × 10 -4) with respect to unity, which is much smaller than the experimental accuracy*. The relative diffusion coefficients of NO against the noble gases 4He, Ne, Ar and Kr have been measured as a function of temperature. The experiments were carried out in the temperature range of 237 K up to 420 K.
2. Experimentalprocedure and results. The details of the experimental method and procedure of the apparatus employed for the measurements of the relative diffusion coefficients have been described fully in an earlier paperg). Again the efficiency of the cycling action of the thermal syphon pump was checked by reversing its direction. Also, the absence of mass discrimination in both the inlet system of the diffusion apparatus and the mass spectrometer has been checked. Our experimental data of the diffusion coefficients for NO against the noble gases 4He, Ne, Ar and Kr are given in tables I and II. We have given in the third column the ratios of the diffusion coefficients at room temperature To and at T, which is varied from 237 K to 420 K, Dlz(To)/D12(T). Each ratio of diffusion coefficients is the mean value of four experimental runs. The room temperature To was kept at 299.14 K during the experimental runs for all cases; the pressure was about 340 torr. The fourth column gives the first-order values of the diffusion coefficients [D]I. These were obtained by applying the second-order Chapman and Cowling1°) correction fD2 on the experimental values. The correction function fD2 deviates only slightly from unity, hence this correction is minimal. In column five the ratio (A In [Dh/A In T)p is recorded as a function of temperature. In the last column of tables I and II we have given the deviation in percent of the ratio of the diffusion coefficients calculated by applying formulae (4) and (6), from the experimental points. In our temperature range, 237 K to 420 K, the corresponding T* interval ranges from 6 to 11 for the system 4He-NO. In this range this function is nearly temperature independent for a Lennard-Jones (12-6) potential as well as for a Bucking* See Note added in proof.
576
A . S . M . W A H B Y , A . J . H . B O E R B O O M A N D J. L O S TABLE I
T h e experimental ratios o f the diffusion coefficients as a f u n c t i o n of t e m p e r a t u r e for N O in 4He and Ne
1
System
exp
1
{AIn[D]I]
[D1-~-SJI
~ A " 7 ] ~ Tn ] v
Deviation in percent f r o m eqs. (4) a n d (6)
'*He-NO
420.36 398.22 373.50 348.32 324.35 271.25 255.50 237.28
0.56848 0.62193 0.69159 0.77596 0.87369 1.17700 1.30070 1.47240
0.56855 0.62200 0.69166 0.77602 0.87373 1.17700 1.30060 1.47220
1.6598 1.6597 1.6606 1.6660 1.6683 1.6651 1.6667 1.6694
+2.10 -2.2 -2.9 +3.3 +1.4 +0.5 + 1.0 --0.6
x x x x x x x x
10 - ¢ 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4
Ne-NO
420.51 398.38 373.33 348.28 324.58 271.55 255.67 237.42
0.56155 0.61527 0.68655 0.77309 0.87088 1.17970 1.30820 1.48580
0.56192 0.61564 0.68689 0.77333 0.87101 1.17940 1.30760 1.48480
1.6850 1.6877 1.6909 1.6943 1.6977 1.7064 1.7093 1.7130
+1.4 +0.3 +2.4 -1.8 -2.1 + 3.2 +0.5 --2.9
x x x x × × × ×
10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4
TABLE II T h e experimental ratios o f the diffusion coefficients as a function o f t e m p e r a t u r e for N O in A r and Kr [D12(To)]
[Da2(To)]
/ d In [Dla]
D e v i a t i o n in percent
(K)
[ D 1 - - - - ~ J e,p
[ DI---'~'~J,
\ A"~n T ] v
f r o m eqs. (4) a n d (6)
Ar-NO
420.49 398.45 373.27 348.62 324.59 271.50 255.47 237.33
0.54722 0.60139 0.67445 0.76123 0.86417 1.19110 1.33040 1.52190
0.54853 0.60262 0.67550 0.76212 0.86473 1.19010 1.32900 1.51940
1.7383 1.7463 1.7560 1.7661 1.7768 1.8033 1.8124 1.8233
+ 1.3 -0.9 +0.3 - 2.1 -0.2 + 1.1 -0.3 -0.7
x x x x x × × x
10 - 4 10 -'~ 10 - 4 10 - 4 10 -'~ 10 - 4 10 - 4 10 - 4
Kr-NO
420.52 398.38 373.56 348.62 324.40 271.68 255.44 237.42
0.54251 0.59712 0.66974 0.75772 0.86293 1.19270 1.33550 1.53010
0.54503 0.59953 0.67184 0.75929 0.86388 1.19130 1.33300 1.52610
1.7528 1.7618 1.7726 1.7841 1.7962 1.8258 1.8361 1.8483
+0.5 -0.2 - 1.9 + 2.2 +0.7 -- 1.7 + 3.2 -- 1.2
x x x x x × x ×
10-* 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4
System
T
•
577
D I F F U S I O N O F N I T R O U S O X I D E I N He, Ne, A r A N D K r
ham exp-6 potential. This allows one to fit the data with a straight line of the form A In [D]~ = A + B A In T.
(4)
However, according to theory, the logarithm of the diffusion coefficient is strongly temperature dependent when T* is below 6. So the experimental points of the mixtures Ne-NO, Ar-NO and Kr-NO (1.5 < T* < 6), are fitted according to the formula A In [Dh = A + B A In T + C ( A In T ) 2.
(5)
TABLE I I I The temperature dependence of the diffusion coefficient for the different mixtures Gas mixture 4He-NO Ne-NO Ar-NO Kr-NO
A
--2.35 +4.47 +1.67 -1.01
x x x x
10 - 4 10 -4 10 - 4 10 - 4
B
C
1.666 1.702 1.789 1.810
+ 2 . 4 5 × 10 -2 +7.43 x 10 -2 + 8 . 3 5 x 10 -2
Standard deviation
+2.19 +__1.38 +0.30 +0.64
x x x x
10 - 4 10 - 4 10 - 4 10 - 4
The constants A, B and C for the different mixtures obtained by least square analysis are tabulated in table III together with the root mean square deviation of the experimental points with respect to the empirical fit. The function
(~lnT
/p
can be calculated for the mixture 4He-NO from the value of B in formula (4) while for the other mixtures it is calculated according to OIn [Dh = B + 2C In (To~T). dlnT
(6)
Absolute values have been obtained by normalization of our value for the mixture Kr-NO on the value of Singh, Saran and Srivastava 11) for the same system at 288.8 K. Our absolute values for the various mixtures at 299.14 K and at one atmosphere along with the potential parameters are recorded in table IV. In a binary gas mixture the first-order approximation of the diffusion coefficient according to the rigorous theory of Chapman and Enskog is given by
[Dlz]x =
( .M 1 ! M2,~ 0.002628
\ 2M1M2 ]
T 3/2 (T*)'
/'/"""'212~aaO(1'1)'12
(7)
578
A.S.M. WAHBY, A.J.H. BOERBOOM AND J. LOS TABLE I V
The potential parameters and the absolute values of the diffusioncoefficients of NO in noble gases at 299.14 K and 760 torr Gas mixture
Absolute values D~./(cm2/s)
Potentialparameters
(e/k)~j (K)
thj (,~)
4He-NO Ne-NO Ar-NO Kr-NO
0.690 0.307 0.181 0.148a
39.0 62.0 113.5 126.5
3.088 3.260 3.611 3.687
(e/k)t1 a6t (K A6) 3.38 x 10+4 7.44 x 10+4 2.52 x 10+5 3.18 × 10+5
a Obtained by extrapolating the data of Singh et al) 1) using our least square fit. As a consequence the ratio of the diffusion coefficients at two temperatures To and T at the same pressure is given by [D12(To)]x
[D12(T)h
=
,r,a/2c~o,1)* ~o .~-~2 (T*) ,r3/20-.1)* :T*~" .L ,~a12 I lO)
(8)
In these expressions p, M1, Mz and T denote pressure, molecular weights of species 1 and 2, and temperature, respectively. 0 (1,1)* is the reduced diffusion collision integral which is a function of e12. So the potential parameters (e/k)12 can be obtained from the measured ratio D12(To)/D12(T) by varying (e/k)12 in the diffusion collision integral in conjunction with formula (8). Now using the value of (e/k)~2 which gives the best agreement between the calculated ratio with the experimental one in formula (7), one can evaluate a12 from the absolute value of the diffusion coefficient. All the calculations described here have been carried out using the Lennard-Jones (12-6) potential.
3. Discussion. 3.1. C o m p a r i s o n w i t h o t h e r d i f f u s i o n e x p e r i m e n t s . To make a comparison between our data with other diffusion measurements is rather difficult due to the lack of suitable measurements which coincide with our range of temperature. Nevertheless, relevant for this study are the data on the system K r - N O as determined by Singh, Saran and Srivastava11). They reported measurements on the binary diffusion coefficients for the mixtures K r - N O and K r - C O by the twobulb technique of Ney and Armistead lz) in the temperature range from 237.3 K to 318.2 K. In figs. 1 and 2 we give a plot of log10 (T) against the deviation in l~ercent of our experimentally determined ratios with respect to the calculated ratios by using the
DIFFUSION OF NITROUS OXIDE IN He, N¢, Ar AND Kr
DEXp OTHEOR. ,J
579
in p e r c e n t
+0.3; +0.2¢
• ~He - N O
m Ne-NO
-0.1,
•
Ar - NO
+ 0.0~
0 -0.08 -0.16 -0.24 -0.32 -0.40 I
2.36
2.40
l
I
2.45
I
2.50
2.55
I
l
2.60
2.65
#
2.70
LOglo(T),T= Degrees Kelvin
Fig. 1. Plot of loglo (T) versus the deviation in percent of the experimental points with respect tO the calculated ratios using our potential parameters for the systems 4He, Ne and Ar in NO.
• 1.
\D---~'-~-e~.-
] in p e r c e n t Kr-NO (this work) • Kr- NO (reference 11)
•
"0"! .0. *0.
-o2o
-
-O.C~ -0 I -0.8 -1.
z~
2'4o
2J4s
2'so
21ss
2:60
2'.65
~170
Log ,o (T), T =Degrees Kelvin
Fig. 2. Plot of loglo (T) versus the deviation in percent of the experimental points with respect to our potential parameters and those of Singh e t al. t l ) for the system Kr-NO.
580
A.S.M. WAHBY, A.J.H. BOERBOOM AND J. LOS
potential parameters (e/k)~j, obtained from our data. In fig. 2 the data of Singh et aL 11) are plotted in the same way. A good agreement has been obtained between our data and those of Singh et al. Regarding the system helium 4He-NO, it can be concluded from fig. 1, that there actually is no value for (e/k)u, which gives a best fit with the experimental points. The deviation of the experimental points with respect to the calculated ones shows a systematic trend, negative at low temperature and positive at high temperature. Therefore, it m a y be concluded that the Lennard-Jones (12-6) potential is not the proper potential to be used for the system H e - N O in contradiction to the systems N e - N O , A r - N O and K r - N O . A comparison was also made between calculated values using the potential parameters obtained from viscosity and virial coefficient data la,s) and our values. The discrepancies are of the same order of magnitude as those shown in figs. 1 and 2. 3.2. T h e c o m b i n a t i o n r u l e s . In this part of the discussion we confine ourselves to the validity of formula (3) for the mixtures under consideration. In order to carry out the test we have calculated the quantity eutrrj for all mixtures. F r o m these values, which are given in table IV, it is possible to find the ratio e.ijtr~j/e,ktr6k for any pair of noble gases. As an example, we calculate the ratio E220"22//3330"33, 6 6 where 2 and 3 refer to Ar and He, respectively. This ratio can be obtained by dividing the quantity e120"62 of the system N O - A r by the corresponding one for the system N O - H e . Here the suffix 1 denotes NO. Similarly all the other ratios can be obtained. TABLEV The potential parameters of the pure components together with other literature values
a u (A)
(elk), (K)
(e/k)u trrl (K A6)
Gas 4He Ne Ar Kr
This work
Ref. 7
This work
Ref. 7
This work
2.37 2.72 3.42a 3.57
2.57 2.77 3.42 3.63
14.3 36.0 121.0 149.0
10.5 36.0 121.0a 173.0
2.53 × 10 3 1.46 x 104 1.9 x 105~ 3.1 x 10s
Van Heijningen 4.24 × 10a 1.56 × 10 4 1.9 x 105 4.1 × 105
a Values from ref. 7. b Value from Van Heijningen. In general the ratio of any pair of gases can be evaluated from the expression = ~j~j~/ek~r~.
(8)
This procedure yields the quantity e,tr6~ for the noble gases relative to the value for argon. Table V gives these values for the different noble gases where normali-
DIFFUSION OF NITROUS OXIDE IN He, Ne, Ar AND Kr
581
zation has been done by equating the value for Ar to the value obtained by Van HeijningenT). Comparing the results for the other noble gases with those of Van Heijningen, we observe a good agreement for Ne; large discrepancies, however, for He and Kr. We now infer that this is due to the fact that Z,j, rot, that is Zrot in collisions between NO and a noble gas atom, for Ar and Ne are approximately equal, while those for He and Kr are larger. Indeed it has been found by Kistemaker et aL 14) who measured Z~j,~ot in mixtures of N2 and the noble gases He, Ne, Ar and Xe, that Z~j.~otfor Ne and Ar are approximately equal (Zrot = 3), for He and Xe, however, larger. Because of the similarity of N2 and NO we feel justified to extend their conclusions to NO. If, as is suggested by the loaded-sphere calculations [viz. formula (2)], there is a close relationship between Ztj.~ot and ( 6 ~ j - 5), the temperature dependence of the diffusion coefficient in the mixtures Ne-NO and Ar-NO is affected in approximately the same way by rotational inelastic collisions. This also will apply to the value of the diffusion coefficient itself. In other words the relative behaviour of these two mixtures will be affected to the same extent by the rotational inelastic collisions, which means that apparently the combining rules are still valid. We return now to test the usual combining rules: e~j = (e,ejj) ~, ~r,j = ½ ( a , +
(9)
~j)
(10)
which follow from formula (3). In formula (3) it is assumed that the molecules can be considered as hard spheres of diameter a and the size of the molecules do not differ too much from the other ones. For the present purpose the potential parameters for the noble gases relative to those of argon can be obtained by the following relations: 2 (~,j - a~,) = ~ j j - ~**,
(11)
(~,/~,k) 2 = ~jj/ek~.
(12)
The values thus obtained are summarized in table V, together with the values obtained from literature as selected by Van Heijningen. Also in this case we notice a reasonable consistency between the potential parameters found for pure argon and for pure neon, and large discrepancies with respect to the other noble gases. This notwithstanding that Van Heijningen from his extensive study on mixtures of the noble gases had to conclude that the normal combination rules are not appropriate. Concluding we might state that the potential parameters obtained from diffusion measurements in mixtures of molecular gases and noble gases are apparent
582
A.S.M. WAHBY, A.J.H. BOERBOOM AND J. LOS
or effective parameters, which only give a best fit of the data obtained to the calculated values applying the monatomic Chapman-Cowling theory. It has been shown, that rotational inelastic collisions do affect these apparent potential parameters. More work, however, has to be performed in order to estimate quantitatively the effects of the rotational degree of freedom on the transport properties. A c k n o w l e d g e m e n t s . We gratefully acknowledge the continuous interest of Professor J. Kistemaker and of Dr. J. van de Ree. We also express our thanks to Mr. F.L. Monterie for the precision mass spectrometer analyses. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappetijk Onderzoek (Netherlands Organization for the Advancement of Pure Research). Note added in proof. The cos 2 0 dependence of the dipole-induction part of the potential leads to a small transition probability on averaging over the orientation angle 0. The next term in the series, containing the quadrupole moment of lhe polar molecule is much larger (1 x 10 -2 with respect to unity). This term depends on cos 30/r 7 and does not average on integration over the orientation angle 0. For the lighter collision partners, He and Ne, this term can become prohibitive for the use of combining rules based on the Lennard-Jones potential (ref. 8, p. 1027).
REFERENCES I) Monchick, L., Sandier, S.I. and Mason, E.A., J. chem. Phys. 49 (1968) 1179. 2) Van de Ree, J. and Scholtes, T., J. chem. Phys. 57 (1972) 122. Van de Ree, J., Physica 37 (1967) 584. 3) Wang Chang, C.S., Uhlenbeck, G.E. and De Boer, J., Studies in Statistical Mechanics 11, North-Holland Publ. Comp. (Amsterdam, 1964). 4) Wahby, A.S.M., Boerboom, A.J.H. and Los, J., Physica 74 (1974) 85. 5) Wahby, A.S.M., Boerboom, A.J.H. and Los, J., Physica 75 (1974) 560. 6) Van de Ree, J. and Los, J., Physica 75 (1974) 548. 7) Van Heijningen, R.J.J., Harpe, J.P. and Beenakker, J.J.M., Physica 38 (1968) 1. 8) Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., Molecular Theory of Gases and Liquids, John Wiley and Sons Inc. (New York, 1954). 9) Vugts, H.F., Boerboom, A.J.H. and Los, J., Physica 44 (1969) 219. 10) Chapman, S. and Cowling, T.G., The Mathematical Theory of Non-uniform Gases (Cambridge, 1939). 11) Singh, Y., Saran, A. and Srivastava, B.N., J. Phys. Soc. Japan 23 (1967) 1110. 12) Ney, E.P. and Armistead, F.C., Phys. Rev. 71 (1947) 14. 13) Krieger, F.J., The Viscosity of Polar Gases, Proj.-RAND Report RM-646 (1961). 14) Kistemaker, P.G. and De Vries, A.E., Physica, to be published.
D I F F U S I O N OF N I T R O U S O X I D E I N He, Ne, Ar A N D Kr A.S.M. WAHBY, A.J.H. BOERBOOM and J. LOS FOM-lnstituut voor Atoom- en Molecuulfysica, Amsterdam, Nederland
Received 13 May 1974
Synopsis The relative diffusion coefficients of NO in noble gases have been measured as a function of temperature. The mixtures investigated were 4He-NO, Ne-NO, Ar-NO and Kr-NO. The temperature range is from 237 K through 420 K. The accuracy of the measurements is about one or two per mill. It is found that the combining rules eua6j = (etla,ejjajj) 6 6 are not valid for these mixtures. The deviation is attributed to rotational inelastic collisions which influence the temperature dependence of the diffusion coefficient. Seemingly, good consistency could be obtained by applying these combining rules on the systems Ne-NO and Ar-NO only as in these mixtures Zu, rot is approximately equal. Effective potential parameters for the various mixtures have been obtained from the measui'ed temperature behaviour of the diffusion coefficients. 1. Introduction. The transport properties of molecular gases can be described
quite well by the monatomic Chapman-Enskog theory. Theoretical considerations indicate that corrections for rotational inelastic collisions and nonsphericity of the intermolecular potentials only amount to a few percent. This concerns viscosity, diffusion as well as thermal conductivity if, in the latter case the Eucken factor for the transport of internal energy is applied. This means that in general transport measurements are not very sensitive to rotational inelastic collisions. There is, however, an exception. The thermal diffusion factor in mixtures where at least one of the constituents is a diatomie molecule is very sensitive to isotope substitution. The formal theoretical explanation for this phenomenon has been given by Monchick, Sandier and Mason 1) and independently by Van de Ree2), who applied the theory of thermodynamics of irreversible processes. These theoretical considerations, based on the Wang Chang, Uhlenbeck and De Boer 3) approach, in which spin polarization is neglected, show the appearance of new terms in the expression for the thermal diffusion factor. These terms, which are linked with the partial internal heat conductivities of the components can explain the anomalies observed in thermal diffusion of molecular gas mixtures. 573
574
A.S.M. WAHBY, A.J.H. BOERBOOM A N D J. LOS
The coupling coefficients (6E u - 5) are collision integrals depending on the rotational quantum numbers of the colliding molecules. A theoretical evaluation of these collision integrals has been performed for simple models, like for example the loaded-sphere model. As, however, (6C~j - 5) also appears in the temperature derivative of the diffusion coefficient, measurements on diffusion as a function of temperature for different isotopic molecules in noble gases should be consistent with thermal diffusion measurements on the same mixtures. The theoretical relation between the thermal diffusion factor and the thermal dependence of the diffusion coefficient indeed was verified for helium-hydrogen and isotopic hydrogen mixtures4). For lorentzian mixtures in which the heavy abundant component is monatomic this relation reduces to the simple expression aL=2_
(tglnD'] . \ O In T i p
(1)
Very good agreement was obtained between thermal diffusion factors measured for the almost lorentzian mixtures A r - H 2 , HD, D2 and K r - H 2 , HD, D2 5) and the values for ~ obtained from diffusion-coefficient measurements as ,a function of temperature for these same mixtures6). On the other hand calculation of (6(7u - 5) for the simple case of loaded sphere - degenerate loaded sphere of equal mass and size gives an approximate relation with Z, ot (6(7,~ - 5) "~ 0.75/Zu, rot.
(2)
Although for more realistic models this relation won't apply, it is still expected that (6(7 - 5) will depend on the rotational collision numbers. As in a moleculargas-noble-gas mixture Zu. rot will depend on the mass of the noble gas atom it is expected that the thermal dependence of the diffusion coefficient in these mixtures also will be a function of the mass of the noble-gas atom. The most appropriate method seems to be the determination of the potential parameters e12 and al z for these mixtures in which the molecular component is the same and to test if the combining rules are still valid. Van Heijningen et al. 7) have measured the diffusion coefficients for ten binary mixtures of the noble gases 4He, Ne, Ar, Kr and Xe as a function of temperature and concentration. The temperature range was from 65 K through 400 K. From these measurements they concluded the validity of the combining rule for the dispersion force constant; not, however, the normal combining rules for el z and a~ 2, e.g.
i?tj(T6j =
( ~ . ~ . ~6 j j ~ j j )6
"} ,
(3)
where e is the depth of the potential well and tr is the separation at zero energy. In this formula the subscribts i, j refer to different species of molecules.
DIFFUSION OF NITROUS OXIDE IN He, Ne, Ar AND Kr
575
Our main purpose in the present investigation was to check the validity of formula (3) for a mixture containing one diatomic constituent. We have chosen the diatomic molecule NO together with the noble gases 4He, Ne, Ar and Kr. With these combinations a very suitable range of T* can be reached through which the mixture parameters can be determined accurately. It is relevant here to point out that Hirschfelder, Curtiss and Bird s) have derived that the effective total energy of interaction between a polar and a nonpolar molecule can be expressed in the same form as that between two nonpolar molecules. They have given combining rules similar to those applied for nonpolar molecules, containing a correction factor for the polar molecules. However, the correction factors in these combining rules for e12 , respectively, a12 are very small, due to the small dipole moment of the NO molecule. These corrections are of the order of 1 x 10 -5 (for 4He-NO: 1 × 10 -4) with respect to unity, which is much smaller than the experimental accuracy*. The relative diffusion coefficients of NO against the noble gases 4He, Ne, Ar and Kr have been measured as a function of temperature. The experiments were carried out in the temperature range of 237 K up to 420 K.
2. Experimentalprocedure and results. The details of the experimental method and procedure of the apparatus employed for the measurements of the relative diffusion coefficients have been described fully in an earlier paperg). Again the efficiency of the cycling action of the thermal syphon pump was checked by reversing its direction. Also, the absence of mass discrimination in both the inlet system of the diffusion apparatus and the mass spectrometer has been checked. Our experimental data of the diffusion coefficients for NO against the noble gases 4He, Ne, Ar and Kr are given in tables I and II. We have given in the third column the ratios of the diffusion coefficients at room temperature To and at T, which is varied from 237 K to 420 K, Dlz(To)/D12(T). Each ratio of diffusion coefficients is the mean value of four experimental runs. The room temperature To was kept at 299.14 K during the experimental runs for all cases; the pressure was about 340 torr. The fourth column gives the first-order values of the diffusion coefficients [D]I. These were obtained by applying the second-order Chapman and Cowling1°) correction fD2 on the experimental values. The correction function fD2 deviates only slightly from unity, hence this correction is minimal. In column five the ratio (A In [Dh/A In T)p is recorded as a function of temperature. In the last column of tables I and II we have given the deviation in percent of the ratio of the diffusion coefficients calculated by applying formulae (4) and (6), from the experimental points. In our temperature range, 237 K to 420 K, the corresponding T* interval ranges from 6 to 11 for the system 4He-NO. In this range this function is nearly temperature independent for a Lennard-Jones (12-6) potential as well as for a Bucking* See Note added in proof.
576
A . S . M . W A H B Y , A . J . H . B O E R B O O M A N D J. L O S TABLE I
T h e experimental ratios o f the diffusion coefficients as a f u n c t i o n of t e m p e r a t u r e for N O in 4He and Ne
1
System
exp
1
{AIn[D]I]
[D1-~-SJI
~ A " 7 ] ~ Tn ] v
Deviation in percent f r o m eqs. (4) a n d (6)
'*He-NO
420.36 398.22 373.50 348.32 324.35 271.25 255.50 237.28
0.56848 0.62193 0.69159 0.77596 0.87369 1.17700 1.30070 1.47240
0.56855 0.62200 0.69166 0.77602 0.87373 1.17700 1.30060 1.47220
1.6598 1.6597 1.6606 1.6660 1.6683 1.6651 1.6667 1.6694
+2.10 -2.2 -2.9 +3.3 +1.4 +0.5 + 1.0 --0.6
x x x x x x x x
10 - ¢ 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4
Ne-NO
420.51 398.38 373.33 348.28 324.58 271.55 255.67 237.42
0.56155 0.61527 0.68655 0.77309 0.87088 1.17970 1.30820 1.48580
0.56192 0.61564 0.68689 0.77333 0.87101 1.17940 1.30760 1.48480
1.6850 1.6877 1.6909 1.6943 1.6977 1.7064 1.7093 1.7130
+1.4 +0.3 +2.4 -1.8 -2.1 + 3.2 +0.5 --2.9
x x x x × × × ×
10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4
TABLE II T h e experimental ratios o f the diffusion coefficients as a function o f t e m p e r a t u r e for N O in A r and Kr [D12(To)]
[Da2(To)]
/ d In [Dla]
D e v i a t i o n in percent
(K)
[ D 1 - - - - ~ J e,p
[ DI---'~'~J,
\ A"~n T ] v
f r o m eqs. (4) a n d (6)
Ar-NO
420.49 398.45 373.27 348.62 324.59 271.50 255.47 237.33
0.54722 0.60139 0.67445 0.76123 0.86417 1.19110 1.33040 1.52190
0.54853 0.60262 0.67550 0.76212 0.86473 1.19010 1.32900 1.51940
1.7383 1.7463 1.7560 1.7661 1.7768 1.8033 1.8124 1.8233
+ 1.3 -0.9 +0.3 - 2.1 -0.2 + 1.1 -0.3 -0.7
x x x x x × × x
10 - 4 10 -'~ 10 - 4 10 - 4 10 -'~ 10 - 4 10 - 4 10 - 4
Kr-NO
420.52 398.38 373.56 348.62 324.40 271.68 255.44 237.42
0.54251 0.59712 0.66974 0.75772 0.86293 1.19270 1.33550 1.53010
0.54503 0.59953 0.67184 0.75929 0.86388 1.19130 1.33300 1.52610
1.7528 1.7618 1.7726 1.7841 1.7962 1.8258 1.8361 1.8483
+0.5 -0.2 - 1.9 + 2.2 +0.7 -- 1.7 + 3.2 -- 1.2
x x x x x × x ×
10-* 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4 10 - 4
System
T
•
577
D I F F U S I O N O F N I T R O U S O X I D E I N He, Ne, A r A N D K r
ham exp-6 potential. This allows one to fit the data with a straight line of the form A In [D]~ = A + B A In T.
(4)
However, according to theory, the logarithm of the diffusion coefficient is strongly temperature dependent when T* is below 6. So the experimental points of the mixtures Ne-NO, Ar-NO and Kr-NO (1.5 < T* < 6), are fitted according to the formula A In [Dh = A + B A In T + C ( A In T ) 2.
(5)
TABLE I I I The temperature dependence of the diffusion coefficient for the different mixtures Gas mixture 4He-NO Ne-NO Ar-NO Kr-NO
A
--2.35 +4.47 +1.67 -1.01
x x x x
10 - 4 10 -4 10 - 4 10 - 4
B
C
1.666 1.702 1.789 1.810
+ 2 . 4 5 × 10 -2 +7.43 x 10 -2 + 8 . 3 5 x 10 -2
Standard deviation
+2.19 +__1.38 +0.30 +0.64
x x x x
10 - 4 10 - 4 10 - 4 10 - 4
The constants A, B and C for the different mixtures obtained by least square analysis are tabulated in table III together with the root mean square deviation of the experimental points with respect to the empirical fit. The function
(~lnT
/p
can be calculated for the mixture 4He-NO from the value of B in formula (4) while for the other mixtures it is calculated according to OIn [Dh = B + 2C In (To~T). dlnT
(6)
Absolute values have been obtained by normalization of our value for the mixture Kr-NO on the value of Singh, Saran and Srivastava 11) for the same system at 288.8 K. Our absolute values for the various mixtures at 299.14 K and at one atmosphere along with the potential parameters are recorded in table IV. In a binary gas mixture the first-order approximation of the diffusion coefficient according to the rigorous theory of Chapman and Enskog is given by
[Dlz]x =
( .M 1 ! M2,~ 0.002628
\ 2M1M2 ]
T 3/2 (T*)'
/'/"""'212~aaO(1'1)'12
(7)
578
A.S.M. WAHBY, A.J.H. BOERBOOM AND J. LOS TABLE I V
The potential parameters and the absolute values of the diffusioncoefficients of NO in noble gases at 299.14 K and 760 torr Gas mixture
Absolute values D~./(cm2/s)
Potentialparameters
(e/k)~j (K)
thj (,~)
4He-NO Ne-NO Ar-NO Kr-NO
0.690 0.307 0.181 0.148a
39.0 62.0 113.5 126.5
3.088 3.260 3.611 3.687
(e/k)t1 a6t (K A6) 3.38 x 10+4 7.44 x 10+4 2.52 x 10+5 3.18 × 10+5
a Obtained by extrapolating the data of Singh et al) 1) using our least square fit. As a consequence the ratio of the diffusion coefficients at two temperatures To and T at the same pressure is given by [D12(To)]x
[D12(T)h
=
,r,a/2c~o,1)* ~o .~-~2 (T*) ,r3/20-.1)* :T*~" .L ,~a12 I lO)
(8)
In these expressions p, M1, Mz and T denote pressure, molecular weights of species 1 and 2, and temperature, respectively. 0 (1,1)* is the reduced diffusion collision integral which is a function of e12. So the potential parameters (e/k)12 can be obtained from the measured ratio D12(To)/D12(T) by varying (e/k)12 in the diffusion collision integral in conjunction with formula (8). Now using the value of (e/k)~2 which gives the best agreement between the calculated ratio with the experimental one in formula (7), one can evaluate a12 from the absolute value of the diffusion coefficient. All the calculations described here have been carried out using the Lennard-Jones (12-6) potential.
3. Discussion. 3.1. C o m p a r i s o n w i t h o t h e r d i f f u s i o n e x p e r i m e n t s . To make a comparison between our data with other diffusion measurements is rather difficult due to the lack of suitable measurements which coincide with our range of temperature. Nevertheless, relevant for this study are the data on the system K r - N O as determined by Singh, Saran and Srivastava11). They reported measurements on the binary diffusion coefficients for the mixtures K r - N O and K r - C O by the twobulb technique of Ney and Armistead lz) in the temperature range from 237.3 K to 318.2 K. In figs. 1 and 2 we give a plot of log10 (T) against the deviation in l~ercent of our experimentally determined ratios with respect to the calculated ratios by using the
DIFFUSION OF NITROUS OXIDE IN He, N¢, Ar AND Kr
DEXp OTHEOR. ,J
579
in p e r c e n t
+0.3; +0.2¢
• ~He - N O
m Ne-NO
-0.1,
•
Ar - NO
+ 0.0~
0 -0.08 -0.16 -0.24 -0.32 -0.40 I
2.36
2.40
l
I
2.45
I
2.50
2.55
I
l
2.60
2.65
#
2.70
LOglo(T),T= Degrees Kelvin
Fig. 1. Plot of loglo (T) versus the deviation in percent of the experimental points with respect tO the calculated ratios using our potential parameters for the systems 4He, Ne and Ar in NO.
• 1.
\D---~'-~-e~.-
] in p e r c e n t Kr-NO (this work) • Kr- NO (reference 11)
•
"0"! .0. *0.
-o2o
-
-O.C~ -0 I -0.8 -1.
z~
2'4o
2J4s
2'so
21ss
2:60
2'.65
~170
Log ,o (T), T =Degrees Kelvin
Fig. 2. Plot of loglo (T) versus the deviation in percent of the experimental points with respect to our potential parameters and those of Singh e t al. t l ) for the system Kr-NO.
580
A.S.M. WAHBY, A.J.H. BOERBOOM AND J. LOS
potential parameters (e/k)~j, obtained from our data. In fig. 2 the data of Singh et aL 11) are plotted in the same way. A good agreement has been obtained between our data and those of Singh et al. Regarding the system helium 4He-NO, it can be concluded from fig. 1, that there actually is no value for (e/k)u, which gives a best fit with the experimental points. The deviation of the experimental points with respect to the calculated ones shows a systematic trend, negative at low temperature and positive at high temperature. Therefore, it m a y be concluded that the Lennard-Jones (12-6) potential is not the proper potential to be used for the system H e - N O in contradiction to the systems N e - N O , A r - N O and K r - N O . A comparison was also made between calculated values using the potential parameters obtained from viscosity and virial coefficient data la,s) and our values. The discrepancies are of the same order of magnitude as those shown in figs. 1 and 2. 3.2. T h e c o m b i n a t i o n r u l e s . In this part of the discussion we confine ourselves to the validity of formula (3) for the mixtures under consideration. In order to carry out the test we have calculated the quantity eutrrj for all mixtures. F r o m these values, which are given in table IV, it is possible to find the ratio e.ijtr~j/e,ktr6k for any pair of noble gases. As an example, we calculate the ratio E220"22//3330"33, 6 6 where 2 and 3 refer to Ar and He, respectively. This ratio can be obtained by dividing the quantity e120"62 of the system N O - A r by the corresponding one for the system N O - H e . Here the suffix 1 denotes NO. Similarly all the other ratios can be obtained. TABLEV The potential parameters of the pure components together with other literature values
a u (A)
(elk), (K)
(e/k)u trrl (K A6)
Gas 4He Ne Ar Kr
This work
Ref. 7
This work
Ref. 7
This work
2.37 2.72 3.42a 3.57
2.57 2.77 3.42 3.63
14.3 36.0 121.0 149.0
10.5 36.0 121.0a 173.0
2.53 × 10 3 1.46 x 104 1.9 x 105~ 3.1 x 10s
Van Heijningen 4.24 × 10a 1.56 × 10 4 1.9 x 105 4.1 × 105
a Values from ref. 7. b Value from Van Heijningen. In general the ratio of any pair of gases can be evaluated from the expression = ~j~j~/ek~r~.
(8)
This procedure yields the quantity e,tr6~ for the noble gases relative to the value for argon. Table V gives these values for the different noble gases where normali-
DIFFUSION OF NITROUS OXIDE IN He, Ne, Ar AND Kr
581
zation has been done by equating the value for Ar to the value obtained by Van HeijningenT). Comparing the results for the other noble gases with those of Van Heijningen, we observe a good agreement for Ne; large discrepancies, however, for He and Kr. We now infer that this is due to the fact that Z,j, rot, that is Zrot in collisions between NO and a noble gas atom, for Ar and Ne are approximately equal, while those for He and Kr are larger. Indeed it has been found by Kistemaker et aL 14) who measured Z~j,~ot in mixtures of N2 and the noble gases He, Ne, Ar and Xe, that Z~j.~otfor Ne and Ar are approximately equal (Zrot = 3), for He and Xe, however, larger. Because of the similarity of N2 and NO we feel justified to extend their conclusions to NO. If, as is suggested by the loaded-sphere calculations [viz. formula (2)], there is a close relationship between Ztj.~ot and ( 6 ~ j - 5), the temperature dependence of the diffusion coefficient in the mixtures Ne-NO and Ar-NO is affected in approximately the same way by rotational inelastic collisions. This also will apply to the value of the diffusion coefficient itself. In other words the relative behaviour of these two mixtures will be affected to the same extent by the rotational inelastic collisions, which means that apparently the combining rules are still valid. We return now to test the usual combining rules: e~j = (e,ejj) ~, ~r,j = ½ ( a , +
(9)
~j)
(10)
which follow from formula (3). In formula (3) it is assumed that the molecules can be considered as hard spheres of diameter a and the size of the molecules do not differ too much from the other ones. For the present purpose the potential parameters for the noble gases relative to those of argon can be obtained by the following relations: 2 (~,j - a~,) = ~ j j - ~**,
(11)
(~,/~,k) 2 = ~jj/ek~.
(12)
The values thus obtained are summarized in table V, together with the values obtained from literature as selected by Van Heijningen. Also in this case we notice a reasonable consistency between the potential parameters found for pure argon and for pure neon, and large discrepancies with respect to the other noble gases. This notwithstanding that Van Heijningen from his extensive study on mixtures of the noble gases had to conclude that the normal combination rules are not appropriate. Concluding we might state that the potential parameters obtained from diffusion measurements in mixtures of molecular gases and noble gases are apparent
582
A.S.M. WAHBY, A.J.H. BOERBOOM AND J. LOS
or effective parameters, which only give a best fit of the data obtained to the calculated values applying the monatomic Chapman-Cowling theory. It has been shown, that rotational inelastic collisions do affect these apparent potential parameters. More work, however, has to be performed in order to estimate quantitatively the effects of the rotational degree of freedom on the transport properties. A c k n o w l e d g e m e n t s . We gratefully acknowledge the continuous interest of Professor J. Kistemaker and of Dr. J. van de Ree. We also express our thanks to Mr. F.L. Monterie for the precision mass spectrometer analyses. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappetijk Onderzoek (Netherlands Organization for the Advancement of Pure Research). Note added in proof. The cos 2 0 dependence of the dipole-induction part of the potential leads to a small transition probability on averaging over the orientation angle 0. The next term in the series, containing the quadrupole moment of lhe polar molecule is much larger (1 x 10 -2 with respect to unity). This term depends on cos 30/r 7 and does not average on integration over the orientation angle 0. For the lighter collision partners, He and Ne, this term can become prohibitive for the use of combining rules based on the Lennard-Jones potential (ref. 8, p. 1027).
REFERENCES I) Monchick, L., Sandier, S.I. and Mason, E.A., J. chem. Phys. 49 (1968) 1179. 2) Van de Ree, J. and Scholtes, T., J. chem. Phys. 57 (1972) 122. Van de Ree, J., Physica 37 (1967) 584. 3) Wang Chang, C.S., Uhlenbeck, G.E. and De Boer, J., Studies in Statistical Mechanics 11, North-Holland Publ. Comp. (Amsterdam, 1964). 4) Wahby, A.S.M., Boerboom, A.J.H. and Los, J., Physica 74 (1974) 85. 5) Wahby, A.S.M., Boerboom, A.J.H. and Los, J., Physica 75 (1974) 560. 6) Van de Ree, J. and Los, J., Physica 75 (1974) 548. 7) Van Heijningen, R.J.J., Harpe, J.P. and Beenakker, J.J.M., Physica 38 (1968) 1. 8) Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., Molecular Theory of Gases and Liquids, John Wiley and Sons Inc. (New York, 1954). 9) Vugts, H.F., Boerboom, A.J.H. and Los, J., Physica 44 (1969) 219. 10) Chapman, S. and Cowling, T.G., The Mathematical Theory of Non-uniform Gases (Cambridge, 1939). 11) Singh, Y., Saran, A. and Srivastava, B.N., J. Phys. Soc. Japan 23 (1967) 1110. 12) Ney, E.P. and Armistead, F.C., Phys. Rev. 71 (1947) 14. 13) Krieger, F.J., The Viscosity of Polar Gases, Proj.-RAND Report RM-646 (1961). 14) Kistemaker, P.G. and De Vries, A.E., Physica, to be published.