J . Phys. Chem. 1989, 93, 3363-3368 the magnitude of AGO9 was calculated to be 143.0 kJ/mol. At this pH, a small ionic strength correction was also made. This was the only case where such a correction was necessary.
Discussion Kinetics. The value of k, = 6.4 X lo8 M-' 6' for 'COT radicals abstracting hydrogen atoms from L(SH)z at pH 4 may be compared with 8.3 X lo8 M-I s-l reported by Akhlaq, Schuchmann, and von SonntagZZfor the same radicals abstracting hydrogen atoms from dithiothreitol, which is another disulfhydryl molecule. The existence of a back-reaction, that is, the abstraction of hydrogen from organic molecules by S-centered radicals, has been propqsed in earlier s t ~ d i e s . ~For ~ *the ~ ~specific case of the reaction of LS2H with formate, concrete evidence came from the observation of a chain reaction16 in the reduction of oxidized lipoamide, which is a cyclic disulfide, in irradiated formate solutions. Chain reactions were also reported by Elliot, Simsons, and SopchyshynZ5 for the reduction of other disulfides in formate solutions. The propagating steps in these chain reactions consist of the reverse of reaction 1 plus reaction 18. Relatively few reports of rate PSSP
+ T O 2 - + H+ = 'PS + PSH + C 0 2
(18)
constants for the abstractions of hydrogen atoms by PS' radicals in aqueous solutions have been given, but similar reactions have been investigated in organic systemsz6 Free Energy Changes and Reduction Potentials. The values of Eo1 for penicillamine and P-mercaptoethanol were measured relative to chlorpromazine in ref 5 , with a relative uncertainty of f0.02 V. The value of Eo8on the same basis is known to within k0.04 V. The values of AGO9 for T O 2 - , in Table 11, from the three sulfhydryl systems lead to an average of -143.6 f 3 kJ/mol or an Eo9value of 1.49 f 0.03 V, which is within the error range for the three PSH standard compounds. The accuracy of the chlorpromazine potential in the pH used here is probably about f 0 . 0 3 V, for a total uncertainty of f0.06 V. (22) Akhlaq, M. S.;Schuchmann, H.-P.; von Sonntag, C. Inf. J . Radiat. Biol. Relat. Stud. Phys., Chem. Med. 1987, 51, 91. (23) Elliot, A. J.; Sopchyshyn, F. C. Radiat. Phys. Chem. 1982,19,417. (24) For earlier work see the review by: von Sonntag, C.; Schuchmann, H.-P. In Chem. Ethers, Crown Ethers, Hydroxyl Groups Their Sulphur Analogues; Patai, S . , Ed. Wiley: Chichester, England, 1980; Vol. 2. (25) Elliot, A. J.; Simsons, A. S.; Sopchyshyn, F. C. Radiat. Phys. Chem. 1984, 23, 377. (26) See for example: Pryor, W. A. Free Radicals Biol. 1976, 1, 1.
3363
Recently, Schwarz and Dodson2' obtained a value of -1.90 0.05 V for EO191
f
C 0 2 + e- = TOz-
(19) The sum of this and Eo9 yields a value of -0.41 V for the twoelectron reduction of C 0 2 , Le., reaction 20. C 0 2 + 2e-
+ H+ = HC0,-
(20) This agrees with -0.36 V calculated from standard free energy datal9 within the uncertainties of the Eo measurements. Combination of the present Eo9with EozOgives EO19 = -1.85 f 0.06 V. The free energy change of reaction 21 can be calculated from AGO9 and the known free energy of protonation of 'C0,- l 8 and HC02-.19 Since the present AGO values are all relative to the 'C0,H
+ e- + H+ = HCOZH
'C02H
+ 1/2H2(g) = HCOzH
(21) standard hydrogen half-reaction, the result, 157.2 kJ mol-', is also the AGO value of (22)
Assuming identical energies of solution of T 0 2 H and H C 0 2 H and taking the known free energy of dissociation of Hz(g)28and the entropies of the 'C02H(g), 'H(g), and HC02H(g) species from published data,2sq2gone can estimate the H-C02H bond dissociation energy. The result is 396.7 kJ mol-' or 94.8 kcal mol-', which agrees within -2 kcal mol-' with 92.7 kcal mol-' given in ref 1.
Acknowledgment. We are grateful for the use of the linear accelerator facilities at the Radiation Laboratory at the University of Notre Dame and to the Director, Professor R. H. Schuler, and his staff for their help and support in this work. We also acknowledge very helpful discussions with H. A. Schwarz and thank the authors of ref 27 for making a preprint of their paper available to us. The financial support of the Natural Sciences and Engineering Research Council of Canada was under Grant No. A357 1. Registry No. PenSH, 52-67-5; P-RSH, 60-24-2; L(SH)*, 3884-47-7; HCO,-, 71-47-6; 'CO*-, 14485-07-5. (27) Schwarz, H . A,; Dodson, R. W. J . Phys. Chem. 1989, 93, 409. (28) JANAF Thermochemical Tables, 2nd ed.Natl. Stand. Re$ Data Ser. (US., Natl. Bur. Stand.) 1971, NSRDS-NBS 31. (29) Golden, D . M.; Benson, S. W. Chem. Rev. 1969, 69, 125. ONeil, H. E.; Benson, S. W. Int. J . Chem. Kinet. 1969, 1, 221.
Effect of Mass Transfer Coefficient on the Elution Profile in Nonlinear Chromatography Bingchang Lin, Sadroddin Golshan-Shirazi, and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996- 1600, and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 (Received: June 20, 1988; In Final Form: September 28, 1988)
The kinetic model of chromatography includes a mass balance equation and a mass transfer rate equation. A numerical analysis of the system of partial differential equations obtained is presented, and the characteristics of the finite difference method used to solve it are described. The results obtained are discussed, and the influence of the value of the mass transfer coefficient on the chromatographic elution profile is analyzed. In the limit case of linear chromatography (linear equilibrium isotherm), the result is very similar to the result obtained from the solution of the ideal nonequilibrium equation. In the other limit case, at large values of the rate constant, the result is very similar to the one obtained by solving numerically the equations of nonlinear equilibrium chromatography.
Introduction The influence of the different sources of mass transfer resistance and of their kinetics on the shape of the elution profile of 'Author to whom correspondence should be addressed at the University of Tennessee.
0022-3654/89/2093-3363$01.50/0
large-concentration bands is an important problem in preparative ChromatograPhY. Although, in most cases involving small molecules, the chromatographic process takes place under experimental conditions that are near equilibrium, this is not true for large molecules, such as those found in biochemical "des where the rate of mass transfer is much slower, nor in the case of affinity 0 1989 American Chemical Society
3364
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
chromatography where the kinetics of adsorption-desorption may be very slow. In the former case (fast mass transfer kinetics) the solutions of the ideal, equilibrium models of chromatography give excellent results. The finite character of the rate of mass transfer is properly accounted for, in semiideal models,l by the selection of suitable values for the space and time increments of the numerical integration. In the latter case (slow mass transfer kinetics), we should consider the kinetics of mass transfer. It must be emphasized at this stage, however, that only the kinetic model of chromatography (see below) is general and can be rigorous. If we know how to write properly the kinetics of mass transfer between phases, including the kinetics of sorption/desorption, or in other modes of chromatography, the kinetics of association/dissociation of the solute(s) with the stationary phase, we should be able to describe correctly the profile of the elution band(s), or for that matter, whatever concentration signal is obtained at column exit for a change of the inlet conditions (Le., frontal analysis, displacement, large volume injection, etc.). Only in the case when the kinetics is fast is it possible to use the equilibrium model. Since it is much easier to determine thermodynamic data (e.g., equilibrium isotherms) than kinetic data (e.g., rate constants), the equilibrium models are preferred to the nonequilibrium ones (kinetic) whenever possible. The effect of the mass transfer coefficients on the elution profile has been discussed in the linear case (Le,, linear equilibrium isotherm between stationary and mobile phases) by using analytical and numerical method^.^.^ The mathematics of nonideal (Le., nonequilibrium) linear chromatography have been discussed by Lapidus and Amundson.* Van Deemter et aL4 were able to prove that, when the heights of the mixing stage, 2D/u, and mass transfer stage, 2 u K / ( 1 Q 2 k f(where u is the linear velocity of the mobile phase, D is the axial diffusion coefficient, K is the column capacity factor, and kf is the mass transfer coefficient) are much smaller than the length of the column, the profile equation derived by Lapidus and Amundson may be simplified to a Gaussian function. But this equation still predicts that, when the mass transfer coefficient is very small, the elution profile is no longer a Gaussian curve. In the general case of nonlinear chromatography, there is no analytical solution. One approach would be to assume that the rate-determining step is the adsorption-desorption kinetics itself, i.e., that the adsorption-desorption processes are slow compared to all diffusion processes. Recently, Wade et aL5 have used this approach. They have derived an analytical solution for the nonlinear system of partial differential equations that describe the problem, in the case when the axial diffusion can be neglected. Although this assumption is realistic in the case of ion-exchange and affinity chromatography, it is not realistic for physical adsorption, since physical adsorption is usually fast, but can be slowed down considerably, due to mass transfer resistances arising in the transfer of the solutes from the bulk liquid to the adsorption sites. Another limitation of this approach is that it cannot be extended to the study of the separation of multicomponent mixtures. In this work we discuss a new numerical solution of the kinetic model of chromatography. We have solved the system of the mass balance and the kinetic equations for one single component by using a numerical method, which can be easily extended to the case of complex, multicomponent mixtures.6
+
Numerical Solution I . Mathematical Model. The mass balance equation of a single compound in a chromatographic column can be written
Lin et al.
ac
-4-
at
ac az
a2c azl
u-+F-=D-
at
where t and z are the time and the abscissa along the column, respectively, C and are the concentrations of the studied solute in the mobile and the stationary phase, respectively, F = (1 - c)/e is the phase ratio, u is the velocity of the mobile phase, and D is the axial dispersion. The kinetics of mass transfer is assumed to be given by the solid film linear driving force model. The time differential of the concentration in the stationary phase is proportional to the difference between the actual concentration and its equilibrium value:
aqiat
= -kf(q - 4 )
(2)
where q is the concentration of the solute in the stationary phase in equilibrium with the concentration C in the mobile phase and kf is the mass transfer coefficient. The boundary and initial conditions of the problem are those corresponding to the injection of a narrow plug of solution of the compound studied at a concentration Co, during a time t,, in an empty column: C(0,t) =
co
t, x
>0
The set of the mass balance equation (1) and the kinetic equation ( 2 ) constitutes a system of partial differential equations, which we have solved numerically by using a finite difference method. 2. Difference Type. Since in liquid chromatography, the molecular diffusion coefficient is very small, we have ignored the axial diffusion term in our model. In the finite difference method, however, the artificial dispersion term of numerical origin can be made to compensate exactly for the axial diffusion term by choosing the proper time and space increments.' So eq 1 and 2 can be written in forward finite difference form (implicit form) as follows:
and (4)
while the boundary and initial conditions become O
Cd = Co
O
q is related to C by the adsorption isotherm 9
=Ac,
(5)
3. Courant Condition. The analysis of the stability of the numerical solution in the linear approximation gives the following conditions:
r u , / h > 0 or m Z / h< -1 where ru,/h = a is the Courant number and
( 1 ) Guiochon, G.; Golshan-Shirazi, S.; Jaulmes, A. Anal. Chem. 1988, 60, 1856.
(2) Lapidus, L.; Amundson, N. R. J . Phys. Chem. 1952, 56, 984 ( 3 ) Villermaux, J. NATO ASI Ser., Ser. E 1978, No. 83. (4) Van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 27 1. ( 5 ) Wade, J . L:; Bergold, A. F.; Carr, P. W. Anal. Chem. 1987, 59, 1286. (6) Golshan-Shirazi,S.; Lin, 8. C.; Guiochon, G., submitted for publication in J . Phys. Chem
u, =
.,( + 1
F Z )
4 . Step of Length and Time Increments and Arti3cial Diffusion Term. If we replace the partial differential equations ( 1 ) and (2) ( 7 ) Lin, B. C.; Guiochon, G . Sep. Sci. 1988, 24, 31.
Mass Transfer Coefficient in Nonlinear Chromatography
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3365
Results and Discussion 1 . Results. First, we have carried out numerical simulations
€
5 4
3
io0
500
300
700
k
900
-
1100
1300
1
3
Time ( s e d
Figure 1. Simulated chromatograms, using the kinetic model and a linear equilibrium isotherm: isotherm, q = 16C; sample size, 0.083 mmol; column, 25 cm long, 4.6-mm id.; flow rate, 1 mL/min. Values of kf: 1, 0.004; 2, 0.02; 3, 0.1; 4, 0.5; 5, 1; 6 , 10.
by the difference equations (3) and (4), as we have done above when D = 0, we introduce an error that can be estimated in a first approximation by using a Taylor expansion of the concentration terms:
C,$+’ = C:
+
[
7
($1 (211+
-
-h
[ $ (5):.5
($)-7h
in the linear case (see Figure 1). The results agree very well with the conclusion of previous authors.24 Except for very small values of kr, the resulting profiles are Gaussian. Their retention times depend only on the slope of the equilibrium isotherm, while their bandwidths depend only on the mass transfer kinetics. For very small values of kf, on the contrary, the elution profile becomes unsymmetrical (see Figure l ) , the retention times decrease with decreasing values of kr, and a very elongated tail appears (see Figure 1 ) . Eventually, when the rate constant becomes zero, the band profile is one of a nonretained compound. The results of the numerical solution in the nonlinear case for a Langmuir isotherm are shown in Figure 2. We make the foliowing observations: ( 1 ) When kr is very large, the profile exhibits a strong selfshsrpening effect. In this case, the profile obtained is very similar to :hose which we have generated by using the numerical solution of the semiideal model,’ when the column has a high efficiency. ( 2 ) When kf decreases, starting from a large value, the retention time of the band (Le., the time when the maximum concentration of the band is eluted) increases at first, a behavior that is very different from the one observed in the linear case, and the band profile becomes broader and broader. When kr becomes very small such that the height of the mass transfer unit, 2uK/( 1 + K)’kr, is not very small with respect to L, the retention time stops increasing and then begins to decrease. (3) When kf 0, the shape of the elution band profile is nearly the same as it is in the linear case. (4) The effect of the axial dispersion coefficient, D, on the band profile becomes very small when the column is strongly overloaded, such as in the case illustrated in Figure 2. This observation is in agreement with the results obtained by Guiochon et al.’ in the study of the properties of the equilibrium model of chromatography. 2. Analysis. Because in liquid chromatography the axial dispersion coefficient, D, is very small, the following analysis is based on the simplifying assumption that D = 0. We shall distinguish three cases, whether the rate constant is very large, moderate, or very small (in fact the essential parameter that controls the shape of profile is the ratio of the height of the mass transfer stage, Le., 2uK/(1 + K)*kr, to the column length, L, but in our simulation u, K , and L are kept constant): (1) In the first case, kf m , for all practical purposes, and in practice, kf is larger than 10 with the parameters we used (see Figure 2 caption). In this case, from eq 2 we have
I:)”(
(7)
at az
Inserting this result into eq 1 when D = 0 yields
Inserting eq 6 and 7 into eq 3, we obtain:
at
az
or (1
It can be proven similarly:’
+ Ff’) -ac at+
ac
uaz = 0
or
ac -+--The left-hand side of eq 9 is an artificial dispersion term, and the artificial dispersion coefficient is given by the relationship
D, = ( h u / 2 ) ( a
+ 1)
(10)
If we choose for the length increment a value equal to h = ( 2 D ) / ( 3 u )and if we take the Courant number such that a = 2, the artificial dissipation term will become equal to the axial dispersion term (D, = D). The axial dispersion term is equal to the sum of the molecular diffusion and the eddy diffusion terms. Because D is very small, we can achieve a good simulation of the self-sharpening effects of the profile.
at
u
ac - 0
+ iy az = u / ( l + Ff’j is a function of C, at
1
This means that (dzldt), least as long a s f i s not constant (Le., the isotherm is not linear). Accordingly, the direction of the characteristics are different, and some of them will intersect. The corresponding parts of the profile will exhibit self-sharpening, while the other parts of the profile will be nonsharpening, Le., continuous. These features of the profile are characteristic of the nonlinear process.’ If the injection is a pulse, the intersection of the characteristics corresponding to the concentrations of one side of the pulse will take place immediately. In this case, the shock is stable. For
3366 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 convex isotherms such as the Langmuir isotherm, the front of the elution band is a shock, while the rear part is tailing (see Figure 2 when k f = IO). The profile of this continuous tailing part may be derived exactly m and the isotherm is any convex function. In this when kf case the general corresponding equation becomes
-
where Q(C) is any convex function. Lax* has given an asymptotic solution of this class equation. It has the form A - B ( t / z ) , where A and B are constant. This result is consistent with the results obtained by Houghtong and by Yaroshenkova et al.IO These authors have used the Burgers equation and the Cole-Hopf transform. Thus, they have obtained an expression for the continuous part of the profile which is proportional to t / z . (2) When k f decreases, but still is large (i.e., between 0.1 and 5 with the parameters we used; see Figure 2 caption), we derive from eq 2 that 1 a4 4 = - - kf - +at
f(C)
As a first approximation, we may write
-
4=---
kf
+f(C)
at
Inserting this result into eq 1 and assuming that D = 0, we obtain (1
F azf ac + u ac = + Ff') at az kf a t 2
This equation shows that along the characteristics of the first-order part of eq 16, we have the following relationship: d2 -
i=
u,2
at2
a2 -
az2
So we have (17) From the preceding discussion, it is clear that the reverse of the rate constant, I/kf acts like an axial dispersion coefficient. Its smoothing effect is the same as that of the diffusion. Due to this smoothing effect, the peak broadens and its height decreases. So, with a Langmuir isotherm, the velocity of the shock decreases with decreasing peak height, while the retention time increases with decreasing kf (see Figure 2 when k f = 1). When kf becomes smaller, the smoothing becomes more intense; the effect of the kinetics of mass transfer on the smoothing of a sharp front and on the increase of the retention time becomes stronger (see Figure 2 when k f = 0.1). Since the directions of the characteristics of the first-order part of eq 17, are still different, one part of the profile remains self-sharpening while the other one stays nonsharpening and the profile still cannot be Gaussian. (3) When k f 0 (Le., kf is smaller than 0.02 with the parameters we used), from eq 2 we now derive
-
a4/at = -kf[qkr=O -f(C)l so aq/at = o Combining with eq 1, we obtain ~
~
(8) Lax, P. D.Commun. Pure Appl. Math. 1957, 10. 537. (9) Houghton, G. J . Pfiys. Cfiem. 1963.67, 84. (IO) Yaroshenkova, G V ;Volkov, S. A,; Sakodinski, K. I. J . Chromatogr 1980, 198, 377
Lin et al.
ac/at
+ (ac/az)= o
-
(18)
This means that, when kf 0, the velocity of the peak is nearly constant (see Figure 2, the curves for kf = 0.004 and 0.02) and the retention time tends toward to. Influence of CurGature of the Isotherm. The effect of the curvature of the isotherm on the band profile is very different, depending on the value of the rate constant. Figure 3 shows simulated chromatograms obtained for various values of the isotherm curvature, at constant sample size, for a fast kinetics of mass transfer, Le., a large value of the rate constant. The results are identical with those obtained previously by numerical simulation, using the semiideal model, and published separately.' They illustrate the influence of the thermodynamics on the band profiles. Figure 4 shows the results of the same calculation carried out under the same set of experimental conditions (same isotherms, same sample size, same column), but with a much lower value of the rate constant, 250 times smaller. The influence of the isotherm curvature on the shape of the band profile is much less important than in the case of a fast kinetics. For the same isotherm, the chromatogram obtained with a small rate constant has a larger retention time, has a smaller height, and is wider than the chromatogram obtained with a large rate constant. At the difference of what is observed with a large value of the rate constant (see Figure 3), the chromatograms obtained for the whole set of isotherms have almost the same peak height. Only a very small increase is noted with increasing isotherm curvature, while the bandwidth hardly changes. This contrast is explained by the fact that when the curvature is small or moderate, the nonlinear effect is small and the band profile is mostly determined by the kinetic effect. It broadens the band and smoothes its sharp front. However, when the isotherm curvature becomes more important, its influence on the shape of the band profile increases and the influeqce of the kinetic effect decreases in relative terms. The essential conclusion of our work, however, is that in nonlinear chromatography, the influence of the kinetics and the thermodynamics on the band profiles cannot be separated. They interact, and both retention times and bandwidths are determined by a combination of the effects of the mass transfer kinetics and the equilibrium isotherm. Accordingly, the influence of the parameters that affect simultaneously the rate and the equilibrium constants (e.g. temperature, mobile phase composition) may be complex and difficult to account for. The investigation of some of these phenomena will be reported later.6 Comparison with Experimental Results. We have not acquired any experimental results that could be compared to the numerical solutions of the system of partial differential equations, (1) and (2). However, we have calculated numerical solutions in the case previously studied by Wade et al.s These authors have reported experimental elution profiles for bands of p-nitrophenyl-a-Dmannopyranoside eluted on concanavalin A. They have fitted these data on their kinetic model by using a Simplex procedure and have calculated the best values of the four parameters, assuming a slow kinetic of adsorption-desorption. We have derived the two parameters of the Langmuir isotherm from their parameters k( and b. We have no correct value for the overall mass transfer coefficient (kf, eq 2), however, so we tried to adjust the numerical solution of eq 1 and 2 on the profile shown in Figure 7 of ref 5 by varying kf. We found that the optimum value of k f is equal to 2 s-l. The kinetic equations used by Wade et al. (Langmuir kinetics) and us are slightly different. There is no visible difference between the profile derived from their analytical solution of the kinetic model of chromatography and the numerical solution we have calculated with the same set of parameters by using our algorithm and program. Since the agreement between the experimental and theoretical results of Wade et al.s was excellent, our numerical results must be satisfactory too. The advantage of this numerical approach is that it is possible to calculate readily the band profile corresponding to any sample size, on any chromatographic system, as long as the kinetic and equilibrium isotherm equati'ons as well as
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3367
Mass Transfer Coefficient in Nonlinear Chromatography (0
‘1
~l *
x
h
6,
‘1-
&$
8
-
x
x
0
100
360
560
760
9AO
llb0
1300
1
Time ( s e d
Figure 2. Simulated chromatograms, using the kinetic model and a Langmuir equilibrium isotherm: isotherm, q = 16C/(1 8C); same sample size and column a for Figure 1. Values of kf: 1, 0.004; 2, 0.02; 3, 0.1; 4, 0.5; 5, I ; 6 , IO.
+
50
100
300
150
TIME.(S~C i50
Figure 3. Band profiles simulated by using the kinetic model and Langmuir isotherms, showing influence of the curvature of the isotherm: isotherm, q = 25C/(1 + bC); same sample size for all profiles, 4.15 mmol; flow rate, 5 mL/min; column length, 25 cm, id., 4.6 mm; phase ratio (Vs/V,,,),0.25; rate constant, 50. Values of the second coefficient,
b: 1, 0; 2, 0.1; 3, 0.2; 4, 0.5; 5, 1; 6 , 2.
the numerical values of the constants involved are known. Thus, it is possible to study the influence of the parameters of a chromatographic experiment by simulation. The drawback of the approach is that a large number of numerical solutions must be calculated to investigate the interactive behavior of many parameters.
C
Conclusion The effect of the mass transfer coefficient (or the rate constant) on the profile of a large-concentration band in the case of a nonlinear isotherm is different from what it is with a linear one. With a linear isotherm, and except for very small values of the rate constant, kf,the elution profile is Gaussian. Its variance depends only on the mass transfer kinetics, not on the slope of the isotherm, while its retention time depends only on the equilibrium isotherm, not on the mass transfer kinetics. On the contrary, our results show that, in the case of a nonlinear isotherm, the effect of the mass transfer kinetics and of the curvature of the isotherm are not independent. Both the retention time and the bandwidth are determined by the combined effect of the mass transfer kinetics and of the nonlinearity of the isotherm. This coupling effect is certainly one of the major features of nonideal, nonlinear chromatography. Numerical simulations of the migration of a large-concentration band along a chromatographic column are an important tool for the study of these nonlinear problems, since the self-sharpening of the bands and the apparition of shock layers is one of the major characteristics of this problem. The successful simulation of correct elution profiles requires both the choice of the proper difference type equation used for writing the program and the selection of the correct value of the space and length increment for its execution. The advantage of a numerical algorithm such as the one described here is the relative ease with which the solution of the single-component problem can be extended to solve
100
150
200
250
300
350
400
TIME. (sec) Figure 4. Same as Figure 3, except for a rate constant of 0.2.
3368
J . Phys. Chem. 1989, 93, 3368-3372
a multicomponent problem and predict the elution profiles of the various components of a mixture. Results in this area will be presented shortly.6 These results will be important and useful for the study of the optimization of the experimental conditions for the separation of
large molecules. In this case, mass transfer resistances play an important role that cannot be ignored. Acknowledgment. This work was supported in part by grant CHE-85 15789 from the National Science Foundation.
Thermal Decomposition of Sodium trans-Hyponitrite J. D. Abata,+ M. P. Dziobak, M. Nachbor, and G . D. Mendenhall* Department of Chemistry and Chemical Engineering, Michigan Technological University, Houghton, Michigan 49931 (Received: May 11, 1987; In Final Form: August 26, 1988)
Samples of amorphous sodium trans-hyponitrite (NazNzO2),heated with a continuous rise in temperature in the absence of oxygen, decompose suddenly between 360 and 390 O C . The decomposition products are Na20 and N20,along with secondary reaction products derived from N20. When Na2N202is heated in an O,-containing atmosphere, the decomposition temperatures are lowered due to the slow oxidation of N20?- to NO, and of NO, to N O , which accumulate and increase the decomposition rate. At P(0,) = 754 Torr, the isothermal induction period for Na2N2O2between 275 and 315 OC is given approximately by log (t(min)) = -14.8 0.3 + (41090 & 600 ca1)/2.3RT. At 298 f 2 ' C , the induction period of Na2N202depends inversely on oxygen pressure (100-745 Torr), with a purified sample showing t(min) = 4.83 + 4902/Po(02,Torr).
Introduction Sodium trans-hyponitrite is the salt that is prepared by most synthetic routes that lead to the hyponitrite ion.' This derivative is rather useful because it is much more stable than the silver salt or the organic esters to storage. Chemical and safety handbooks cite a decomposition temperature of 300 OC for trans-Na2N202,2v3 which appeared to be lower than values we obtained from our samples by preliminary studies with differential scanning calorimetry (DSC) methods. In this paper we report a detailed study of the decomposition of N a 2 N 2 0 2as a function of temperature and atmosphere. Of particular interest is the discovery of the facile cleavage of the N=N bond in Na2NZO2by molecular oxygen to give sodium nitrite. Experimental Section Sodium trans-hyponitrite was prepared by the electrolysis of 3.6 M aqueous sodium nitrite over a pool of Hg in a modification of Polydoropoulos' p r o c e d ~ r e . ~The product was isolated by dilution with absolute ethanol and cooling to 0 OC. The colorless crystals were washed thoroughly with ethanol and dehydrated at oil pump pressures at 25-50 OC for several days. Accurately weighed samples of 0.10 f 0.02 g were analyzed for sodium by acidifying solutions in distilled water with concentrated HCl and concentrating to a dry residue that was weighed. Hyponitrite ion was analyzed for by precipitation of the silver salt, which was washed with water, dried at
