J . Phys. Chem. 1990, 94, 6141-6144
6141
Decomposition of Pernitric Acid in Aqueous Solution Cerhard Lammel, Dieter Perner, and Peter Warneck* Max- Planck- Institut fur Chemie (Otto-Hahn-Institut), Mainz, FRG (Received: June 20, 1989: In Final Form: January 8, 1990)
The unimolecular decay of pernitric acid (PNA), H 0 2 N 0 2 , in aqueous solutions is strongly pH dependent and yields predominantly nitrite above pH = 5. This observation is of particular interest for atmospheric processes involving aerosol and cloud water chemistry. The detailed kinetic analysis of the underlying processes shows good agreement with the kinetics of a related phenomenon, the oxygen production from PNA solutions as observed in 1981 by Kenley, Trevor, and Lan. Both sets of data are analyzed on the basis of a common mechanism, and a number of important and hitherto unknown rate constants M. The rate-limiting are evaluated. For reaction 1 H 0 2 N 0 2e H+ + 02N02-,the equilibrium constant is (1-2.5) X step at large pH appears to be reaction 2, O2N0T O2+ NO2-, which must be rather temperature sensitive. Values found for k2 were 0.014 and 0.045 s-I at 284 and 295 K, respectively. Below a pH of 3-4 the unimolecular decomposition into s-l. neutral products (3), H 0 2 N 0 2 H 0 2 + NO2, is the main reaction path with k , = 4.6 X
-
-
Introduction The first unambiguous evidence for the existence of pernitric acid (PNA), H 0 2 N 0 2 ,was obtained in 1977 by Niki et al.,' who recorded its infrared spectrum in the gas phase. Previous effortsL3 to synthesize PNA from H 2 0 2and N 2 0 s , or H N 0 3 , had shown the formation of a compound with strong oxidizing power but no definite proof for the existence of hydrogen peroxynitrate could be obtained. In the gas phase, PNA is in equilibrium with the precursors H 0 2 and N O 2 NO2 + H 0 2 s HO2N02 but since H 0 2 radicals undergo disproportionation to H 2 0 2and oxygen HO2
+ H02
-
H202
+0 2
PNA is unstable and decomposes. The equilibrium constants, the rate coefficients associated with the forward and reverse reactions, and their temperature dependence have been determined in several studiesee6 The high- and low-pressure limiting rate coefficients are known as well.' In aqueous solution, PNA is expected to decompose in a similar manner, but now some additional reactions are involved. First, N2O4 which is in equilibrium with NO2, reacts with to give H N 0 2 and HNO,. Next, the H 0 2 radical dissociateslo into and the 02-anion may react with N O 2 by the ions H+ and 02-, charge transfer" to form NOT. Finally, PNA may also dissociate into ions,I2 H+ + 0 2 N 0 2 - . The PNA anion is again unstable and decomposes, but the nature of the products is not entirely clear. 0 have been proposed as prodBoth NO2- O2 and NO,u ~ t s . l ~ - Nevertheless, '~ it is evident that two decomposition pathways exist for PNA in aqueous solution, one for the undissociated acid, the other for the anion. This aspect was first
+
+
( I ) Niki, H.; Maker, P. D.; Savage, C. M.; Breitenbach, L. P. Chem. Phys. Left. 1977, 45, 564-566.
(2) D'Ans, J.; Friedrich, W. Z . Anorg. Chem. 1911, 73, 325-359. (3) Schwarz, R. Z. Anorg. Chem. 1948, 256, 3-9. (41 Graham. R. A.; Winer. A. M.: Pitts. J. N. J . Chem. Phvs. 1978. 68. 450514510. (5) Howard, C. J. J. Chem. Phys. 1977, 67, 5258. (6) Sander, S. P.; Peterson, M. E. J . Phys. Chem. 1984,88, 1566-1571. (7) Baulch, D. L.: Cox, R. A.; Hampson, R. F.; Kerr, J. A.; Troe, J.; Watson, R. T. J. Phys. Chem. Ref.Dara 1984, 13, 1259-1380 and references
therein. (8) Gratzel, M.; Henglein, A.; Lilie, F.; Beck, G. Eer. Bunsen-Ges. Phys. Chem. 1969, 73. 646-653. (9) Park, J.-Y.; Lee, Y.-N. J . Phys. Chem. 1988, 92, 6294-6302. (IO) Bielski, B. H. J.; Cabelli, D. E.; Arudi, R. L.; Ross,A. B. J. Phys. Chem. Ref.Dara 1985, 14, 1041-1 100. ( I I ) Warneck, P.; Wurzinger, C. J . Phys. Chem. 1988, 92,6278-6283. ( I 2) Kenley, R. A.; Trevor, P. L.; Lan, B. Y. J . Am. Chem. SOC.1981, 103, 2203-2206. (13) Wagner, F.:Strehlow, H.; Busse. G. 2.Phys. Chem. (Fronkfurr) 1980, /23, 1-33.
0022-3654/90/2094-6141$02.50/0
recognized by Kenley et al.,12who had measured the rate of oxygen evolution during PNA decomposition and found it to increase with rising pH. In the present paper we report product yields for the decomposition of PNA in the pH range 2-7. Nitrite was found to be the dominant product at pH 1 5. We have also analyzed our results and those of Kenley et al.I2 in terms of a generalized reaction mechanism that enabled us to evaluate the rate coefficients associated with P N A decomposition as well as the dissociation constant for the equilibrium H 0 2 N 0 22 H +
+02N0y
Pernitric acid formation and decomposition is deemed important in all chemical systems in which H 0 2 and NO2 are simultaneously present. An example of particular interest to atmospheric chemistry is the behavior of PNA in clouds. Here, PNA is formed in the gas phase due to photochemical reactions generating H 0 2 in the presence of sunlight. If the gas-liquid partition coefficient of PNA is similar to that of either H202or HNO,, pernitric acid will be rapidly scavenged into the aqueous phase, where it decomposes to form products different from those in the gas phase.
Experimental Techniques Pernitric acid, PNA, was prepared from nitronium tetrafluoroborate, N02BF4,as described by Kenley et al.:I2 0.2-0.5 g of the compound (Fluka) was dissolved without further purification in I O mL of concentrated (>90%) hydrogen peroxide in a glass impinger. The mixture was kept at 5 "C. H 2 0 2 was concentrated from a commercial 85% solution by reducing its liquid volume at room temperature to one-fifth of the initial by continuously pumping its vapor phase. Pure, dry nitrogen, when bubbled through the solution at flow rates of 5-10 cm3/s, provided a continuous source of PNA for several days with slowly decreasing activity. In constructing the experimental setup, only glass, Teflon, and stainless steel were used as materials. The presence of PNA in the gas stream was confirmed by means of its infrared spectrum in the matrix after the cocondensationwith C 0 2 at liquid nitrogen temperature. After correction for the expected wavelength shifts, the spectrum compared well with that reported by Graham et al.4 for PNA in the gas phase. For a quantitative determination of gas-phase PNA concentrations, the gas stream was bubbled for 1-2 min through 10 mL of a solution of 1% KBr in 0.1 M HCI, and the amount of Br3- formed due to the liberation of bromine was measured spectrophotometrically at 266-nm wavelength. In using this method it was assumed that each mole of PNA reacted with Brto produce 1 mol of Br2 (HOONO, + 2Br- + H+ Br2 + NO3+ H 2 0 ) . Tests demonstrated that H 2 0 2 ,which is present in the gas stream as an impurity, does not liberate bromine under our experimental conditions. Calibration of Br3- was achieved by means of the oxidation of Br- by bromate. The reaction is
-
0 1990 American Chemical Society
6142
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990
well-known to produce 3 mol of Br, for each mole of Br0,consumed.14 The reaction was comparatively slow for the Br03concentrations used here, 3 X to 2 X IO4 M (corresponding to 10” to 6 X 1O4 M dissolved PNA). Maximum absorption was reached 50 min after mixing the reagents and at that time the Br3- concentration had already started to decay due to reactions of its own. Accordingly, in taking the maximum absorption as a measure for the Br3- concentration, an upward correction was required that amounted to 25% as estimated from the biexponential growth curve. Good linearity was nevertheless obtained for the optical density at 266 nm as a function of Br03- concentration. In this manner, the range of gas-phase concentrations of PNA was determined to be (1-5) X I O l 5 molecules/cm3. In order to study the production of nitrite and nitrate in aqueous solution, the PNA-containing gas was conducted for 60 s through an impinger filled with I O mL of Milli-Q water adjusted to the desired pH. In the pH range 4-7, phosphate buffers were used, and lower pH values were established by means of dilute acids (HCI or H2S04). The nitrite produced was determined by the diazotization procedure of Rider and M e l l ~ n ’followed ~ by spectrophotometry. Within 10 s after termination of the gas flow an aliquot of the solution from the impinger was admixed to 1 mL of an aqueous solution of aminobenzenesulfonic acid (35 mM in HCI, pH = 1.4). Five minutes later 1 mL of the coupling reagent I-aminonaphthalene hydrochloride (34 mM in HCI) was added, and the mixture was buffered to pH = 2.5 with I mL sodium acetate (2 M) and water added to make 50 mL of solution. The optical absorption at 520-nm wavelength was measured 10 min later. Calibration was performed as usual with nitrite solutions of known concentrations. The concentration of nitrate in the PNA solution was determined immediately after termination of the gas flow by means of ion chromatography combined with conductivity detection. It was not feasible to determine nitrite in the same way with comparable sensitivity because of interferences with chloride and/or phosphate ions. I n the few cases when the pH was adjusted with sulfuric acid, the nitrite was determined by ion chromatography as well. Both methods, the diazotization procedure and ion chromatography, gave comparable results. Model Calculations Calculations were performed on the basis of the chemical reactions shown in Table I . For radicals and the PNA anion we applied the steady-state assumption to calculate effective rate product yields resulting from the thermal coefficientsand N02-/02 decomposition of PNA as a function of pH. This leads to the following equations:
Lammel et al. TABLE I: Reaction Mechanism for the Thermal Decomposition of Pernitric Acid (1 ) H 0 2 N 0 2
k-i
H+ + 0 2 N 0 2 -
k
(2) 0 2 N 0 2 -2 0, + NO;
K , = I x 10-50
k2 = 1.4
X
k 3 = 4.6 x 10-30 k
(3) H 0 2 N 0 2
H 0 2 + NO,
0,- + H+
(4) HO,
(5) H02+ H02
( 6 ) 0,-
k..3 = 1.0 X K~ = 1.58 x
H 2 0 2+ 0,
k6 = 9.7 X
$ HOC + H+
(8) 2 N 0 2
k
K7 = 2.5
X
k + H 2 02 H N 0 , + NO3- + H+ k8 = 6.5 X
(9) H N 0 2 (10) 0,-
10-56
k5 = 8.3 X
k + H 0 26 H02- + O2
(7) H 2 0 2
IO7O
b
H+ + NO2-
+ NO2 2NO2- + 0,
K, = 6
X
kIo = 2
X
loBc
Values derived in this paper (T= 284 K, Kenley et al.’,); see text. k in dm’ mol-’ s-], K i n mol dm-). bBielski et a1.I0 CSmith and Marte11,16 Wagman et al.” dGratzel et a1.,8 Park and Lee.g CWarneck and Wur~inger.~
[PNA] is the concentration of undissociated acid. It is related to the total concentration of pernitric acid [PNA], by [PNA], =
Here, K , = k,/k-, and we shall assume that [H+]k-, >> k2. This condition simplifies [H+]/K, + k2/kl to [H+]/K,. In applying these equations to the rates of oxygen evolution reported by Kenley et al.I2 we first derived k3 from the value for kObpat low pH and then estimated and varied KI and k2 at higher pH until a best fit to the experimental data was obtained. In treating the yield of HNO, NO2- from our own experiments, we first calculated the loss of [PNA], during the time of dissolution of PNA in water until the reaction was quenched by the addition of the analytical reagent and then calculated the yield [NO,-])/[PNA], from the above formulas. ([HNO,]
+
+
Results The reaction of PNA with bromide is rapid. We found it essentially complete when the PNA gas flow was stopped and the concentration of Br3- was measured. A second impinger in series with the first, both filled with acidic (pH = 1) KBr solution, showed that about 10% of PNA escaped from the solution with the gas stream. This loss was taken into account when the gasphase concentration of PNA was calculated. Its value fell into the range (1-5) X IOI5 molecules/cm3. The dissolution of PNA in water should have resulted in aqueous concentrations of (50-500) X 10” M if PNA had suffered no losses due to thermal decomposition. In reality, however, such losses did occur as demonstrated by the evolution of nitrite. Accordingly, the PNA concentration after time At must be obtained by integrating the equation (14) Szekeres, L. 2.Anal. Chem. 1958, 160, 198-201. ( 1 5 ) Rider, B. F.; Mellon, M. G. Ind. Eng. Chem. Anal. Ed. 1946. 18,
96-99. (16) Smith, R. M.; Martell, A. E. Critical Sfability Constants Vol. 4 : Inorganic Complexes; Plenum Press: New York, 1976. (17) Wagmann, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, 1.; Bailey, S. M.: Churney. K . L.; Nutall, R. L. J . Phys. Chem. ReJ Data 1982, 1 1 . suppl. 2, 1-392.
d[PNA]/dt = F - k,rf[PNA] which leads to [PNAI = (No/keffAt)ll - exp(-ker4t)l where kerfis the effective first-order rate coefficient for thermal decomposition and F = N o / A t the flowrate of PNA. No is the
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 6143
Decomposition of Pernitric Acid
I
TABLE 11: Yields of Nitrite and Nitrate Resulting from the Absorption of HNO, in Buffered or Acidified Aqueous Solutions
type of buffer phosphate acetate unbuffered phosphate or acetate phosphate phosphate HCI or H2S04 HCI or H,S04
pH
No. pmol
(NOT)/
No, % 7 1.6-4.5 84 14 4.5 82.4 6 4.2 f 0.1 0.7-0.9 34 f 2 4 1.1-3 8 6 f 11 0.6-1.7 22 f 16 3.5 1.1-1.6 I l k 4 3 3.1 4 f 2 3 0.7-3 8.5 f 4.5 2
*
(NO,-)/ n
No, %
I
I
I
I
I
I
I
1
full reaction mechanism
nb
5 1
2 3 5 4 2 3
17f1 17 f I
24f1 27
' N o = amount o f HNOI supplied within 60 s. bNumber of runs. I
1.0
0'
PH
I
1
I
I
3
I
, 5
4 .
, 1
Figure 2. Rate coefficient for oxygen evolution from the thermal de-
9 Y
+ '
N
z 0
Figure 1. Yield of nitrite from the thermal decomposition of pernitric acid. Solid points, pH adjusted with phosphate buffer; open point, pH in pure water after the absorption of H02N02;diamonds, pH adjusted with hydrochloric or sulfuric acid; solid line, calculated with k , = 8.3 X IO-'s-', k2 = 0.1 s-), K , = 2.5 X M. Temperature 295 K.
total amount of PNA absorbed. The fraction of No that was converted to nitrite during these experiments is shown in Table I1 as a function of pH. When a second impinger with 10 mL of buffered water was used in series with the one from which these data were derived, the second solution did not contain nitrite in measurable amounts. Apparently, the absorption of P N A in water was complete regardless of pH. The possibility that at very low pH a small fraction of PNA escaped with the gas stream as observed above cannot be entirely discounted. The lowest pH applied in these experiments was pH = 2. In acidic solutions the yield of nitrite was about 0.08, indicating that 16% of total PNA absorbed probably had decomposed according to reactions 3 and 8. The above equation then provides an estimate for the effective rate coefficient: keff= 5 X IO-, s-l. The value is similar in magnitude to that found by Kenley et al.I2 from the evolution of oxygen from aqueous PNA solutions at low pH. As Figure 1 shows, the NO2- yield rises with increasing pH until it reaches a value in the vicinity of 0.8 at pH 2 5. This observation suggests a change in the decomposition mechanism when the pH is raised. The observation is similar to that of Kenley et a1.I2 who had found a similar increase in the rate of oxygen evolution with increasing pH. Nitrate in aqueous solutions of PNA may be a decomposition product resulting from the reaction of N O 2 with water, reaction 8 (see Table I), or it may arise from the dissolution of HNO,, which is present in the gas stream as an impurity. At pH = 6, the relative concentration of nitrate was [NO3-]/N0 = 0.17 f 0.01. The high yield of nitrite in this pH region is interpreted to arise from the decomposition of the PNA anion. For these conditions nitrate should not be a major product of PNA decomposition. The concentration observed must therefore be due mainly to nitric acid impurity. At a pH of 2-3, the yield of nitrate rose to 0.25 f 0.02, which possibly indicates a contribution of 0.08 f 0.02 from the decomposition of PNA in addition to the background HNO,. Within the margin of error the difference agrees with the yield of nitrite found at pH = 2. In this pH region, according to the reaction mechanism of Table I , the decomposition involves primarily undissociated PNA. The yields of nitrite and nitrate then should be equal because both are formed in equimolar amounts from NO, in reaction 8.
composition of pernitric acid in aqueous solution. Points, experimental data at 284 K from Kenley et a1.;I2curves, calculated rate coefficient either for decomposition of undissociated acid alone (lower curves), or for decomposition of the anion and acid combined (upper curves). Solid curves: k 3 = 4.6 X s-I, k2 = 1.4 X s-I, K I = M. Dashed SKI,K , = 2.5 X curves: k3 = 4.6 X I O - ' S - ~ ; k 2 = 1 X M. The simulation of the experimental data (at 295 K) in terms of the reaction mechanism given in Table I gave k3 = 8.6 X lo-, S-I from our results a t low pH, and k2 = 0.08 S-I together with K l = 2.5 X M from our data at higher pH. The resulting curve for the nitrite yield is shown in Figure 1 together with the experimental data. The uncertainties in the rate coefficients are approximately SO% of the values given. For example, K , cannot be much smaller than lo-' M or the curve would rise too soon, and k2 likewise cannot differ much from that given, or the nitrite yield at high pH would not match the measured value. Since the calculated curve in Figure 1 represents the experimental data reasonably well, we have further simulated the data recorded by Kenley et al.I2 for the evolution of oxygen. In this case, the corresponding effective rate Coefficient kobswas measured as a function of p H at a temperature of 284 K. The results are shown in Figure 2. The best fit to the experimental data, which is shown by the solid curve, was obtained with k3 = 4.6 X lo-, s-I, k2 = 1.4 X IO-2 S-I, and K , = IO-5 M, but the data may still be represented with k2 = 1 X IO-* s-I and K , = 2.5 X loT5M as shown by the dashed curve. Figure 2 includes for these two cases the fraction of P N A decomposition that occurs via the radical formation pathway, reactions 3-10. The initial rise of the rate with increasing pH is due to the formation of 02-in reaction 4 and the faster rate of reaction 6 in comparison with reaction 5, whereas the subsequent decline of kobs as the pH is increased further results from the growing importance of pernitrate anion formation in reaction 1. The higher values of the rate coefficients derived from our data as compared to those of Kenley et al.I2 are partly due to the higher temperatures employed. Kenley et al. reported for T = 293 K a rate coefficient for oxygen evolution of (3.92 f 0.29) X s-', which is about 5 times greater than that at 284 K at the same pH = 4.9. The rate coefficient for NO2- evolution a t T = 293 s-' at pH = 4.7. On the basis K was given as (2.25 f 0.18) X of K l = 1 X k2 = 4.5 X and k, = 5 X IO-, we calculated kobs= 3.9 X at pH = 4.9 and kobs = 2.6 X IO-* s-I at pH = 4.7. These values are in good agreement with those reported, and this shows that k2 increases by a factor of 3 when the temperature is raised from 284 to 293 K, whereas k , increases less strongly with temperature. The difference between our results and those of Kenley et al. thus amounts to less than a factor of 2, which in view of the different experimental approaches must be considered good agreement.
Discussion In applying the diazotization procedure we were initially concerned about a conceivable surplus generation of NO,- from the decomposition of PNA during the diazotization reaction. The
6144
Additions and Corrections
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990
lower value of k , for the aqueous phase. The evaluation of k , depends to some extent on the rate constant used for the reverse reaction. From the fitting procedure k-, appears to be 52 X IO7 dm3/(mol s), and we have used k-, = 1 X IO7 dm3/(mol s) in the calculations. This agrees with previous conclusions for the aqueous phase reaction," but the rate constant also is by 2 orders of magnitude smaller than that for the corresponding reaction in the gas phase in the high-pressure limit,7 k-, = 2.5 X IO9 dm3/(mol s). The reduction is to that for the disproportionation reaction 2 H 0 2 H 2 0 2 02.It is possible that these reactions are slowed down in solution due to the hindrance of rotation, which may make reactive encounters less frequent among the total number of all reactant encounters. From k3 and k-, we obtain values for the equilibrium constant of reaction 3 in aqueous solution: K3 = k3/k-, = (4.6-8.2) X mol/dm3. This may be compared with the gas-phase values for the temperature range 284-295 K, which are (1.4-6.2) X IO-" mol/dm3. It appears that in solution more of the PNA is dissociated into H 0 2 and NO2 than in the gas phase. The value for K ,as determined here corresponds to a pK of 4.6-5.0. This pK value is lower than that for pernitrous acid, which falls into the range 6.C-6.6.18-20 Kenley et a1.12 have treated their data at pH > 4 in terms of a simplified mechanism consisting only of reactions 1 and 2. Implicit in their treatment was the assumption that k-,[H+] 5 X IO9 dm3/(mol s). This would make reaction -1 much faster than reaction 2. We calculate k , = K , k - , = 5 X IO4 s-I so that reaction 1 is not rate limiting in the decomposition of the PNA anion. In fact, the last process is slow enough to leave the equilibrium 1 essentially intact, and we have utilized this feature in our calculations.
results show, however, that PNA was efficiently scavenged by sulfonic acid, which was present in large excess, because the rate coefficient that we derived for the formation of NO2- at pH = 2 is very similar to that reported by Kenley et al. for the formation of oxygen in the same pH range. Moreover, our experimental data and those of Kenley et a1.I2can be successfully modeled in terms of a common reaction mechanism with rather similar values for K , , k2, and k3.even though different products were monitored. Thus, we concur with Kenley et al.I2that the major products from the decomposition of the PNA anion are NO2- and 02,and not NO3- and 0 atoms as Wagner et aI.l3 had suggested. Wagner et aI.l3 have studied conductivity changes induced in flash-photolyzed aqueous solutions of nitrate. They observed two slow processes with decay time constants of 40 and 150 s at pH < 4 and assigned them to the decay of H O O N 0 2 and H N 0 2 , respectively. With regard to the first process, its decay rate was found to rise with increasing pH in a manner resembling that observed by Kenley et al.I2 for the evolution of O2 from PNA decomposition. However, the absolute values for the decay constants given by Wagner et al.13 are greater by a factor of IO. Since our own results confirm the data of Kenley et al.lz and the temperature applied by Wagner et al.', were similar to ours, it appears that the signal observed by them must be due to a different species. The second long-lived feature that occurred in their experiments with a time constant of about 150 s would be more compatible with PNA decomposition as studied directly by Kenley et all2 and by us, but this possibility cannot be checked out because detailed data were not given. Wagner et al.13 have suggested K , = 1.2 X but as their signal assignment is uncertain and in addition they used a different reaction mechanism in treating their data, a comparison would be misleading anyway. The preferred value for the high-pressure limit of k3 in the gas phase7 is
k3gas= 0.4
X
I O l 4 exp(-l0420/T)
-
+
Acknowledgment. The present study has, in part, received support from the Deutsche Forschungsgemeinschaft within the program of Sonderforschungsbereich 233 (Dynamics and Chemistry of Hydrometeors). Registry No. H 0 2 N 0 2 ,26404-66-0.
s-I
from which we calculate k3 = 0.035-0.15 s-I for temperatures of 284-295 K . This constrasts with the value for k3 in aqueous solution of k, = 5 X IO-' S-I. Thus, decomposition in solution is slower than that in the gas phase by at least a factor of 10. However, for dissociation to occur in aqueous solution the fragments must receive sufficient energy to enable them to leave the solvent cage before recombination takes place, whereas in the gas phase the fragments are liberated directly following bond cleavage. This difference in mechanism presumably is responsible for the
(18) Barat, F.; Gilles, L.; Hickel, B.; Sutton, J. J . Chem. Soc. 1970, A1970, 1982-1 986.
(19) Shuali, U.; Ottolenghi, M.; Rabani, J.; Yelin, Z. J . Phys. Chem. 1969, 73, 3445-345 I . (20) Keith, W. G . ; Powell, R. E. J . Chem. SOC.1969, A1969, 90.
ADDITIONS AND CORRECTIONS 1989, Volume 93 Marc L. Mansfield: Gambler's Ruin Model of Semicrystalline Polymer Systems with Antiparallel Chain Packing.
Page 6928. A factor of 6 was inadvertently omitted from eq 20. The equation should read x + 2y z =9
+
