1366
S. P. PAVLOU AND B. S. RABINOVITCH
Energy Transfer in Thermal Isocyanide Isomerization. Noble Gases in the Ethyl Isocyanide System' by S. P. Pavlou and B. S. Rabinovitch* Department of Chemistry, University of Washington, Seattle, Washington 98106
(Received November 18, 1970)
Publication costs assisted by the National Science Foundation
Relative collisional activation-deactivation efficienciesof added inert bath gases for the thermal unimolecular isomerization of ethyl isocyanide have been measured in the lower region of falloff just above the second-order regime at 231'. The effect of He, Ne, Ar, Kr, and Xe on the rate of isomeriaation was studied at 231' and at total pressures between 7 X lo-* and 1 X 10-1 Torr; Po( a) increases monotonically from 0.27 to 0.44 with increase of atomic weight. The behavior of the experimental efficiencies as a function of the degree of falloff (reaction order) and of the dilution of substrate by bath gas is illustrated. Their relation to the predicted behavior given by calculations based on a stochastic model is discussed. The average energy ( A E ) removed from the substrate by the bath molecules per down-transition was evaluated. The effect of structural and internal parameter changes of the substrate molecule on the process of intermolecular energy exchange is examined in the light of the data obtained in earlier work with the simpler CHaNC homolog.
Introduction
for a homologous substrate series in which Eo is also nearly invariant, as is the case for isocyanides.8 Thus the structure of the substrate molecule is significantly changed, but the low-pressure reaction parameters are scarcely altered (Table I); an internally consistent reaction series is obtained for a given deactivator, and the effects of the internal parameter changes, ie., vibration frequency pattern, internal rotation, low-frequency bending modes, and total number of vibrational modes, on the transition probabilities can be evaluated. Our initial study in this system is of the effect of noble gases on the isomerization of CzH5NC. This paper presents a detailed description of the experimental relative collisional efficiencies and their behavior as a function of dilution and region of falloff.
Gas phase thermal unimolecular reactions, especially a t low pressures, provide a simple and convenient technique for the study of some aspects of intermolecular energy transfer by molecules a t high levels of vibrational excitation. I n the second-order region the rate of reaction is the rate of activation by collision, and studies in this nonequilibrium region are uniquely advantageous. 2-4 For reasons discussed p r e v i ~ u s l y ,the ~ ~ ~detailed documentation of a specific reaction system and the systematic investigation of the parameters affecting the energy-transfer process is a desideratum for successful exploitation of this technique. The thermal unimolecular isomerization of isocyanides to nitriles was chosen to provide such a system. Experimental Section To this time, the methyl isocyanide system has been Materials. Ethyl isocyanide was prepared by Mastudied in some detail. The behavior of a large number of bath gases has been i n ~ e s t i g a t e d . ~The , ~ temperature dependence of the relative collisional efficiency of (1) (a) This work was supported by the National Science Foundation: (b) abstracted in part from the Ph.D. thesis of S. P. Pavlou, inert bath gas is also under in~estigation.~Dilution University of Washington, Seattle, Wash., 1970. effectsa and relative cross sections of inert gases' have (2) M. Volpe and H. S. Johnston, J. Amer. Chem. Soc., 78, 3903 also been examined. I n order to further our under(1956). (3) F. J. Fletcher, B. S. Rabinovitch, K. W. Watkins, and D. J. standing of the nature of collisional transition probabilLocker, J. Phys. Chem., 7 0 , 2823 (1966). ities, the experimental work on bath gas behavior is here (4) 5. C. Chan, B. S. Rabinovitch, J. T. Bryant, L. D. Spicer, T. extended to the next homolog, the thermal isomerizaFujimoto, Y. N. Lin, and S.P. Pavlou, ibid., 74, 3160 (1970). tion of C2H6NCto CzH5CK. This substrate has been (5) S. C. Chan, J. T. Bryant, and B. 8. Rabinovitch, ibid., 74, 2055 (1970). investigated previously in some detail by RSaloney and (6) Y . N. Lin and B. S. Rabinovitch, ibid., 7 2 , 1726 (1968). Rabinovitch.8 It is one of the advantages of thermal (7) Y . N. Lin, S. C. Chan, and B. S. Rabinovitch, ibid., 7 2 , 1932 unimolecular reaction systems that the magnitude of (1968); S. C. Chan, J. T. Bryant, L. D. Spicer, and B. S. Rabinovitch, ibid., 74, 2058 (1970). (E+)otends to follow classical statistical behavior and is only weakly dependent on substrate c ~ m p l e x i t y . ~ (8) K. M. Maloney and B. S. Rabinovitch, ibid., 73, 1652 (1969) (MR). Thus the pragmatic operational test of bath molecule (9) D. W. Placzek, B. S. Rabinovitch, G. 2. Whitten, and E. Tschuiactivation-deactivation efficiency is virtually invariant kow-Roux, J. Chem. Phys., 43,4071 (1965). The Journal of Physical Chemistry, Vol. 76, No. IO, 1971
1367
ENERGY TRANSFER IN THERMAL ISOCYANIDE ISOMERIZATION Table I : Reaction Parameters for the CHsNC and CzHsNC Systemss CHaNC
CzHaNC
A,, see-'
1013'6
Eo,kcal mol-'
37.9 1.22 2.49 4.6 3.3
10'"8 37.6 1.37 4.50 6.1 4.5
(E+),,o, kcal mol-' kcal mol-' no 8)
' Observed Slater n value.
Observed Kassel s value.
loney8 and was purified by gas chromatography. Analysis revealed no detectable impurities. Helium, neon, argon, krypton, and xenon were assayed reagent grade gases from Air Reduction Co. These gases were used without further purification. Apparatus and Procedure. Rate determinations were made in a static system. The reaction vessel was a 229-1. Pyrex sphere heated in a stirred air furnace. Pumping of the reaction vessel was by an 8-1. sec-l Hughes VP-8R ion pump and a cryogenic molecular sieve sorption pump. After baking above 400°, a pumped-down pressure before each run a t 231" of mm was obtained, as monitored by a cold cathode ion gauge and by a thermopile gauge which read down to mm. The furnace temperature was controlled by a proportional controller and was measured by using eight calibrated chromel-alumel thermocouples placed in good thermal contact with various points of the reaction vessel. During a run the temperature was constant to f0.3" ; the agreement between all thermocouple readings was *0.2". A standard temperature of 231" was used; occasionally, over the history of the work, the vessel temperature differed a little from that value. Runs with seasoned and unseasoned vessels were performed and no difference in rate determinations was observed; seasoning between runs was therefore unnecessary except in cases where air had been admitted into the reactor. No detectable outgassing of the reaction vessel was observed on the thermopile gauge when the background pressure was monitored over time intervals corresponding to actual reaction times. A conventional glass vacuum apparatus was used for gas handling, storage, and pressure measurements. Pumping was by an oil diffusion pump and mecha.nica1 forepump. Mercury vapor was excluded from the gas handling system and reaction vessel. Amounts of substrate and inert gas were measured in standard volumes with a glass null-point Bourdon gauge. A reaction mixture of CzH5NC containing -6y0 C H 8 N as an internal analytical standard was used for kinetic runs. Because of the small initial pressure mm) of CzH5NCused (2.0 X mol), (-3 X
the whole sample was removed for analysis. Isomerization was carried to between 10 and 40% reaction. Kinetic runs were ended by pumping the mixture through a high-conductance trap maintained at - 196". Analysis. Unreacted CzH5NCwas quantitatively removed by passage of the sample through a AgCN column. All gas chromatographic analyses were performed on a Hewlett-Packard F & 14 Model 700 chromatograph equipped with a dual flame ionization detector, a Model 5771A electrometer, and a Honeywell Electronik 19 recorder. A 12-ft, 5% tricresyl phosphate on 6080 mesh, acid-washed Chromosorb G column was used for analyses. Calibrations of the column were made with standard mixtures of C2H5CN-CHsCN before and after each series of analyses. Reproducibility was achieved to within 1%.
Results Rate constants for the isomerization were calculated by using both an "absolute" and an "internal standard" basis. Since in most cases reasonable reproducibility between internal and absolute values was achieved, an average value was taken. When isocyanide interference with the acetonitrile peak occurred due to inadvertent saturation of the AgCN column with isocyanide, only absolute values were used.
Corrections to the Data. a. Temperature Correction. All rate data were brought to a standard temperature of 231.0" with use of the energy of activation for pure C2H5NC8a t a reaction order equal to that for the inert gas kinetic run. b. Dead Space. This amounted to and no correction was necessary, especially since relative inert gas efficiency is desired. c. Pump-out Time. At the conclusion of a run the pressure drop in the reaction vessel was observed as a function of time. Time correction for removal of the sample varied from 1 to loyo,depending on the total pressure, the inert gas used and the run time; it was determined quite accurately from plots of pressure os. pumping time measured for each gas. d. Heterogeneity. Increase of the rate due to wall effects occurs a t pressures below 4 X mm (w = 6 X lo4 sec-') in the CzH5KC system.* It was found
Table 11: Some Calculated and Pragmatic Heterogeneity Functions p, mm
2 5
x x
u, sec-'
10-4 10-4
1.5 x 10-3
x 6.4 x 2.5
10-2
3.2
x
8.0 X
(H(E)')'
103
loa
2.4 x 104 4 . 0 x 104 1 . 0 x 106
0.21 0.13
0.04 0.02
0.001
(H(E)')
+x
0.74 0.58 0.33 0.20 0.05
From ref 8. The Journal of Physical Chemistry, Vol. 76, N o . 10, 1971
1368
S. P. PAVLOU AND B. S. RABINOVITCH
by MRBthat the correction for simple heterogeneity, i.e., simple wall activation collisions, was not sufficient to account for the observed rate a t lower pressures. Some surface catalysis appears to occur. Table I1 shows values for the simple wall activation correction function (H(E)’)a t a givenh omogeneous collision rate w , as given by MR, and the pragmatic deviation a t various collision rates, defined similarly as
+
[(H(E)’) X ]
= Ak/ktot;
Ak =
ktot
- khom
where horn is the homogeneous rate constant that would prevail in the absence of any wall effects; ktot is the observed rate constant having homogeneous and wall activation contributions, plus some additional wall-catalysis component x. The experimental data for the noble gas series show pragmatically (Figure 1) that deviation from the theoretical falloff curves is comparable with that of the pure substrate in the same range of falloff, i.e., between k / k , values of 0.032 and 0.011 Relative collisional efficiencies were calculated from rate data obtained over the homogeneous region of falloff and extending into a region, k / k , 0.02, where only nominal deviation due to heterogeneity occurred. The data below this region were not used for determinations of efficienciesbut solely for comparative and diagnostic purposes. Interrelation of Relative Collisional Eficiencies, p. A detailed discussion of several alternative definitions of relative collisional efficiency quantities in thermal unimolecular systems has been given previously. lo Since these ideas are central t o our work and t o the discussion that follows, we will briefly review and summarize some aspects of this subject. The notation p,(D) employed in this paper is the same as that given by TR.1° The subscript w indicates in terms of the collision rate-whether a ,0, or intermediate falloff, w-the region of behavior of k / k , ; the parenthetic quantity D refers t o the degree of dilution of the substrate by heat bath inert gas, w ( M ) / 4 ) . Collisional efficiencies may be classified as “differential” or “integral” quantities. The need for recognizing and defining these arises from both experimental custom and necessity. a. Digerential. At finite dilution, the relative efficiency equals the ratio of the increments of total specific collision rate of the parent, due either to addition of more substrate, A ~ ( A )or , of inert gas, Aw(M), required to produce a given increment in the specific reaction rate, Ak; the meaning of this quantity is clearest when these increments are small so that the dilution is low. l1 Thus
>
p , ( ~ ) = A ~ ( A ) / A ~ ( M )A; ~ ( A = ) A ~ ( M ) (1) A related differential quantity may also be defined as
p’,(D)
=
Ak(M)/Ak(A); Au(A)
=
Au(M)
The Journal of Physical Chemistry, Vol. 76,No. 10, 1971
(1’)
7
Figure 1. Plots of k / k , us. w for CtH6NC at 504’K: CzH&C, 0; He, 0 ; Ne, n; Ar, A ; Kr, X ; Xe, A; (1)RRKM 300(400) strong collider model;8 (2) 710 em-’ P model; (3) 635 cm-1 P model; (4) 545 cm-l EXP; and (5) 495 cm-1 EXP. The x-dashed curve represents the predicted dilution path for the 495 cm-1 E X P model; the dash-dotted line shows the predicted behavior after the correction for heterogeneity is applied.
The designations A and 14 which represent parent and inert gas signify strong collider and weak collider bath gas, respectively; of course, not all bath gases are weak colliders, but all noble gases are such, here. b. Integral. At high dilution of parent in inert gas mixtures, in other than the low-pressure second-order region, it is more suitable t o use an efficiency ratio which refers to total specific rates, i.e., to an “integral” quantity expressed as
B,(D) = w(A)/w(Mix); k(A)
= k(Mix)
(2)
k(Mix)/k(A); w(A) = w(Mix)
(2’)
or as
B’,(D)
r:
where Mix refers to the inert gas mixture of specified dilution. The relative collisional efficiencies are functions of the order of reaction, 4 [ = 1 d(ln k)/d(ln w ) ] , of the dilution of substrate, D , and of the strength of collision which was specified by T R in terms of the reduced parameter E’, characteristic of the bath molecule for a given model of collisional transition probabilities; E’ = (AE)/(E+), where ( A E ) is the average energy removed per deactivating collision by the bath molecule and
+
(10) D. C. Tardy and B. S. Rabinovitch, J. Chem. Phys., 48, 1282 (1968) (TR). (11) This definition of differential efficiencies reflects the conventional experimental practice of adding a finite and variable amount of bath gas to the pure substrate to give a final mixture of dilution D. This corresponds to averaging over the whole dilution range covered by the increment. A more exact definition would involve infinitesimal increments of substrate or of bath gas to an initial mixture of dilution D ,but this quantity cannot be measured in practice with any precision. The fundamental significance of differential quantities becomes more obscure in studies above the second-order region.
1369
ENERGY TRANSFER IN THERMAL ISOCYANIDE ISOMERIZATION 1.0,
1
0.6
k 0A
0.2
I
0.0
k/k-
+
I
I
IIIII 10
D
100
-, and B,(D),---, Figure 4. Calculated plots of B,’(D), 504’K, 385 us. D for various values of (k/km),nit. For C~HSNC, em-1 SL; solid lines refer to the primed quantities and dashed lines correspond to unprimed quantities; sets 1, 2 and 3 correspond to the calculated behavior for (k/km)in it values of 0.0025, 0.025, and 0.5, respectively. Set 1 corresponds to the low pressure limit and the behavior for &(I)),, . , and PO’@), X X X, is also displayed.
Figure 2. Calculated plots of us. k / k , for (1)n = 4.6 and (2) n = 6.1, based on the Slater function In(0).
0.6
0.6
>i‘ 0.4
----------_ -------------
02
I
I 4
I
III
I ,
0.01
0.001
, ! & I 0.1
I
I
,
0.5
(k/km) init Figure 3. Calculated plots of B’,(D), and p,(D),- - -, us. (k/km)init for CzHsNC, 504’K, 385 cm-’ SL; curves 1, 2, 3, 4, and 5 correspond t o D = 200, 50, 10, 3, and 1, respectively.
(E+) is the thermodynamic equilibrium value of the average active energy possessed by the reacting molecules above the critical threshold, Eo. A correlation of order with k / k , for methyl and ethyl isocyanide (for which Slater’s n is 4.6 and 6.1, respectively) is given in Figure 2. The limiting values of all the fl quantities and their general trend as a function of falloff region and dilution is given in Table 111. The numerical magnitudes of the various p quantities are illustrated in terms of a typical weak collider for which E’ = 0.75 ((AE) = 1.03, (E+) = 1.37 kcal mol-’) in Figures 3-5. A maximum spread and 8, and between p‘, and p, occurs a t between high values of 4. All quantities converge to the same value in the second-order region a t D = a . P‘, passes through a minimum value at intermediate dilution for all values of 4, while B, increases continuously as D 0; p’, passes through a minimum value a t intermediate dilution if k / k , is less than 0.05 for D = 0; 0, behaves similarly t o 8;, however, its magnitude at low dilution is less than fl,. Three-dimensional plots of fl’, and 8, us. both D and the reference value of k / k , for the pure substrate (Figure 5 ) show these qualitative relationships more clearly : two surfaces
B’,
-
Figure 5 . Calculated three-dimensional plots for B’,(D), vs. (k/k,)i,it and D,for C~HSNC; 504’K, 385 cm-1 SL; the dashed and solid surfaces depict the behavior of the B’ and B quantities, respectively. The horizontal curves lie on various dilution planes slicing the two surfaces along all values of k/k,, while the curves on vertical planes span values of the efficiencies for all dilutions on various intersecting k lk , planes.
---, and &,(D), -,
span all regions of D, from 0 to 0 0 , and all regions of falloff;both surfaces converge a t low values of k / k , for all values of D, converge on unity for D = 0, a t all values of k/k,, and diverge at high values of k / L , for all D > 0. For a stronger collider, similar behavior is predicted. However, the surfaces are shallower and closer to a plane a t a value of unity. Noble Gas Fallo$ Data. The measured rate conThe Journal of Physical Chemistry, Vol. 76, No. 10, 1971
S. P. PAVLOU AND B. S. RABINOVITCH
1370
~
Table I11 : Relationships of Relative Collisional Efficiencies a t Various Regions of Falloff and Dilution Second order
Dilution
0
D m
Falloff
B’O(0) = Bo(0) = 1 8’0(0) = PO(0)”
First orderb>O
P ’ O ( 0 ) = Po(D) < Po@)
P’u(0) = Pu(0) < Po(0) B’o(D) > sw(D)d P ’ w ( D ) > P,(D)
D’o(m) = B o ( m ) = P ’ o ( m ) = Po( PdO)
P’,(m)
B’o(D)
=
p m ( 0 ) = Pm(O) = 1 {P’m(O) = P m ( 0 ) < Pu(0) P’m(D) = p’m(D) > Bm(D) = P m ( 0 ) < Pw(D)i Pm(0) m) = Ofm(-) = 1 > P m ( m ) < P m ( D ) , &(
B‘,(O) = B,(O) = 1
Bo(D) < 1
B’w( m )
1
Bo( ) > Pw(m)
a Magnitude less than unity, in general, and may be obtained from quasiuniversal curves given by T R and depend only on E’ for a particular collisional transition probability model. w + m ; k / k m -+ 1. ‘ This region is not experimentally useful. P’,(D;) > p’,(Dj) > Q < p’,(Dk) . . . , < p’,(Dm) and p,(Di)> p,(Dj) . , . . > p,(D,); where i < j < k , and Q is a minimum value. The same trend is observed for B‘ and B quantities; see Figure 2.
‘
stants are summarized in the Table IV. Curves of log k / k m vs. log w are given in Figure 1 for the pure substrate and for substrate-noble gas mixtures of increasing dilution. I n practice, it was not convenient to work either a t a constant or near-infinite dilution: t o work at constant dilution would have meant restricting experiments to a value of D significantly lower than infinite, while to work near infinite dilution would have meant restricting rate data to a k / k , range which was higher in falloff. The experiments represent a compromise. The smallest value of substrate pressure was mm) such that accurate analysis chosen (- 3 X could be performed, while the smallest amount of bath gas added was dictated by the heterogeneity artifact described earlier and by the requirement that k(Mix) should be sufficiently increased so that good accuracy was possible. Thus, various amounts of inert gases were added to a constant amount of substrate such that D varied on a collision basis from approximately 20 to 260, depending somewhat on the gas. The lower range of dilution corresponds neither to an infinite value nor to effectively constant dilution. The experimental expediency thus forced on us constituted in part a modest study of the variation of @ quantities with dilution. The collision rates, w , may be related to pressure, and the “integral” efficiencies may be expressed as B,(D) = w(A)/w(Mix) = P(A)/[P(A,D)
+ P(M,D)p,psI; k(A) = k(Mix)
(3)
here k(Mix) is the specific rate constant for the mixture of dilution D; P(A,D) is the pressure of A in the mix, = (SAM/ ture of dilution D,etc.; p,, = ( ~ A A / ~ A M ) ~ ”ps S A A ) ~ , where p is the reduced mass of the collision partners, and SAA and SAM are equivalent hard-sphere diameters; values of intermolecular potential parameters are given in Table V. l 2 Also
S’, (0) = k (Mix)/ k (A) ; P(A) = P(A,D) For the case D = and
, then
+ P(M,D)p&ps (3’)
P(M, ~
>> P(A, w ) ,
) p ~ p ,
The Journal of P h y s h l Chemistry, Vol. 76, No. 10, 1971
6 J w)
= lim BJD) = P(A)/P(M, m )p,ps; D-+m
I ~ ( A= ) k(n4)
p,(w) = limP’,(D) D-
(4)
= k(M)/k(A);
m
P(A) = ~(M,aJ)p,,ps(4’) Expressions 4 and 4’ correspond directly to reIations 2 and 2’, respectively. Relative efficiencies on a pressure-to-pressure basis, Bp(D) and ,Yp(D), and on a reduced mass-corrected basis, B,(D) and B‘,(D), can be obtained from (3) and (3’), or from (4)and (4’),by deleting the factors ps and ppps, respectively. When the dilution, D, is large (D N 100 is for all practical purposes an infinite value), then for a narrow range of dilution for a weak collider (200 > D > 50, in these studies), and a narrow range of falloff (0.03 < k / k m < 0.07, in these studies), the dependence of the @ quantities on D and is small and they can be averaged for all data points in this range.
Discussion Correlation with Stochastic Calculations. The theoretical dependence of S, and S’, on D for He, Ne, Ar, Kr, and Xe is illustrated in Figure 6, as based on various models for the transition probability distribution function. The measured values of P’,(D) for a given bath gas are greater than Pw(D)in agreement with the theoretical prediction for a given collisional model. For He and Ne, the best fit was obtained for values of the average size of down jumps, (AE), of 495 and 545 cm-I (-1.5 kcal mol-’), respectively, based on an exponential (EXP) model, which has been shown to be the correct qualitative form for the transition probabiliFor the heavier memties for low values of ( A E ) . 4 - 7 bers of the noble gas series, the efficiencies are intermediate between strong and weak colliders, i.e., P = 0.4-0.5. Then (AE) is larger and the data are better fitted by a Poisson (P) modello and values of (a) = 635, 710, and 710 cm-I (-2 kcal mol-’), respectively. (12) (a) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N. Y., 1954; (b) R. A. Svehla, NASA Technical Report, R-132, 1962, p 36.
1371
ENERGY TRANSFER IN THERMAL ISOCYANIDE ISOMERIZATION Table IV : Summary of Rate Data Da
10s aec-1
P (Mix), 10-a m m
18 30 31 42 69 72 96 125 138 174 196
1.00 1.47 1.49 2.06 3.00 3.54 4.58 4.77 6.75 8.71 10.17
0.765 1.14 1.16 1.62 2.38 2.81 3.65 3.81 5.40 6.97 8.15
15 24 27 42 65 103 176
0.706 1.11 1.27 1.94 3.17 4.89 6.96
0.915 1.45 1.67 2.57 4.22 6.39 9.31
Usa
k (Mix) ,a S‘dD)
SdD)
S’w ( D )
80 (Dl
Helium 1.73 f (0.02)c 2.61 f (0.02) 2.39 2.82 f (0.01) 3.60 f (0.00) 4.23 5.06 f (0.08) 4.94 f (0.03) 6.46 7.65 f (0.09) 8.08 f (0.08)
0.30 0.35 0.32 0.29 0.28 0.29 0.29 0.27 0.29 0.29 0.27
0.18 0.22 0.19 0.17 0.17 0.18 0.18 0.16 0.17 0.17 0.16
0.36 0.40 0.37 0.34 0.33 0.34 0.34 0.32 0.33 0.33 0.31
0.23 0.27 0.24 0.21 0.20 0.22 0.22 0.20 0.21 0.21 0.19
Neon 1.88 f (0.02) 1.84 f (0.00) 2.16 2.98 f (0.05) 4.32 f (0.11) 5.18 f (0.09) 7.02 f (0.05)
0.29 0.20 0.22 0.22 0.22 0.20 0.22
0.17 0.11 0.12 0.12 0.12 0.10 0.11
0.47 0.34 0.36 0.37 0.38 0.34 0.35
0.34 0.21 0.23 0.24 0.26 0.22 0.23
10-6 aeo-1
48 73 42 92 89 90 101 158 258
1.98 2.78 2.79 3.08 3.56 4.43 4.98 6.57 10.25
2.41 2.88 3.40 3.74 4.33 5.39 6.04 8.01 10.60
Argon 3.38 f (0.05) 4.22 f (0.00) 4.69 f (0.06) 5.26 5.90 f (0.13) 6.32 f (0.09) 6.88 9.23 f (0.27) 10.88 f (0.04)
0.26 0.28 0.28 0.30 0.30 0.28 0.28 0.32 0.31
0.15 0.18 0.17 0.19 0.19 0.17 0.17 0.19 0.18
0.41 0.45 0.45 0.45 0.47 0.43 0.43 0.48 0.47
0.28 0.33 0.32 0.35 0.35 0.31 0.32 0.36 0.34
25 53 45 61 145
1.18 2.44 2.60 3.50 7.53
1.51 3.15 3.34 4.51 9.77
Krypton 3.04 f (0.02) 5.18 f (0.37) 5.69 f (0.34) 6.38 f (0.14) 10.61 f (0.17)
0.33 0.34 0.34 0.32 0.32
0.20 0.21 0.23 0.20 0.19
0.54 0.54 0.57 0.52 0.50
0.41 0.39 0.42 0.40 0.38
18 30 50 54 73 80 93 120 123
0.72 1.72 2.66 2.77 3.52 3.82 4.56 5.83 6.39
0,909 2.08 3.25 3.37 4.31 4.66 5.58 7.15 7.82
Xenon 2.02 f (0.17) 3.82 f (0.05) 5.03 f (0.15) 5.17 f (0.05) 6.74 f (0.57) 5.84 7.51 f (0.04) 8.98 f (0.14) 10.37 f (0.23)
0.32 0.33 0.31 0.31 0.29 0.37 0.33 0.33 0.36
0.19 0.21 0.18 0.19 0.17 0.26 0.21 0.20 0.23
0.49 0.53 0.47 0.49 0.44 0.50 0.50 0.51 0.55
0.35 0.39 0.34 0.37 0.33 0.39 0.39 0.39 0.44
+
+
a Dilution, D,is on a collision basis, where w = w(A,D) w(M,D) = 1.59 X 107P(A,D) 1.97 X 106[s~~2/~~~~’/2]P(M,D); the average value of P(A,D) was 3.2 (f0.2) X 10-4 mm. b k , = 1.56 X sec-1. c Average deviation of absolute and internal standard values from mean.
A summary of the average jump sizes (AE), and related probabilities, ( p ) , for the different models used is given in Table VI. For values of D < 50, the data depart from the infinite dilution curve. The situation is illustrated in particular in Figure 1 for the weakest collider He. The experimental points are in agreement with calculations based
on the stochastic model that fits the data a t D = and lie along the S-shaped dashed curve that connects the pure substrate curve (initial value of k/k, = 1.65 X 10-3) and the infinite dilution curve. Ne is also expected to lie close to this curve. The effect of heterogeneity below D = 100 was introduced into the dilution mixture calculations a t each Q)
The Journal of Physical Chemistry, VoZ. 76,N o . 10,1971
S. P. PAVLOU AND B. S. RABINOVITCH
1372 Table V : Various Parameters for CzHsNC and Noble Gases €/ha Molecule
A.
CzHsNC He Ne Ar Kr Xe
5.00 2.58 2.79 3.42 3.61 4.06
.
u,a
a ~ ~ , c
€AM/kd
OK
PAMb
A
OK
d2~2)*(T*,b,,,)e
400 10.2 36 124 190 229
27.5 3.7 14.8 23.2 33.2 38.8
5.00 3.79 3.90 4.21 4.30 4.53
400 64 120 223 276 303
1.65 0.85 0.96 1.13 1.22 1.26
sAM2
41.25 12.20 14.56 20.03 22,52 25.77
+
p,
PB
1.00 0.30 0.35 0.49 0.55 0.63
1.00 2.72 1.37 1.09 0.91 0.84
+
a All values same as in ref 4 except for CzHsNC from ref 8. * /LAM = M A M M / M A M,. uAM= (uA uM)/2. a,,/k = [(aA,k/k)(~M,k)]1/2, E,,,S jt 0 only for CnHaNC,for which p = 3.93 D (A. L. McClellan, “Tables of Experimental Dipole Moments,” W. H. Freeman, San Francisco, Calif., 1963) and,,,S = 1.12 [L. Monchick and E. A. Mason, J. Chern. Phys., 35, 1676 (196l)l.
effective pressure by weighting the strong-collision probability matrix, PA,by a collision fraction which includes the heterogeneity function, (H(E)’) x ; i.e., the expression of TR for the transition probability matrix corresponding to the mixture of strong and weak colliders, Pmix, is
+
pmix
+ (H(E)’)+ x)/wtIPA-I- [ u ( W / L J ~ I P ~
= [(@(A)
+
20
60
100
140
I80
’+I
260
D Figure 6. Experimental plots of J’,(D), 0, and J , ( D ) , 0, vs. D for: (a) He, 497 cm-1 EXP; (b) Ne, 547 cm-l EXP; (c) Ar, 633 cm-1 P ; (d) Kr, 710 cm-1 P ; and (e) Xe, 710 cm-l P. The solid curves correspond to the theoretical behavior, including a heterogeneity correction. The theoretical curves, without correction, are shown by dashed lines.
+
+
where ut = w(A) w(M) (H(E)’) x and PMis the particular weak-collision matrix. The calculations on this basis are given by the solid lines in Figure 6 and the agreement with experiment is quite satisfactory. Comparison with the CHINC System. The collisional efficiency of the noble gases in the thermal isomerization of methyl isocyanide was reported for infinite dilution in the second-order region4 where all p quantities have the same value (Table 111). Since the experimental data for CzH5NCrefer to a 4 range of 1.81-1.91, extrapolation of the present high dilution integral quantities, p’,( m ) and pw(a),to their second-order limit, &(a), is necessary for direct comparison of the two systems. The extrapolation was made for each bath gas with use of the theoretical model that fits the data. The Po( 0 3 ) values for all noble gases in the C H 8 C and CzHsNC systems are gathered in Table VII. The Table VII: Summary Values of PO( a ) for the Noble Gases in the CHaNC and CnHsNC Systems a t 504’K
Table VI: Energy Jump Sizes and Related Probabilities in the CzHsNC System
_------ CzHaNC----B’w(m)
(AE)downVb (AE)up,*
Gas
Model
E la
om-’
em-’
(P)down
He Ne
EXP EXP P P P
1.04 1.14 1.32 1.48 1.48
495 545 635 710 710
240 250 440 490 490
0.67 0.68 0.75 0.77 0.77
Ar
Kr Xe
a E‘ = (AE)/(E+), where ( E + ) = 480 cm-’ for CIH.&“ a t 504’K. 6 All values of (AE) were calculated for the infinite dilution curve for each gas.
The Journal of Physical Chemistry, Vole76,N o . 10, 1971
He Ne Ar Kr Xe
0.33 0.36 0.45 0.53 0.50
l%o(m)
0.21 0.24 0.33 0.40 0.38
k(Mix)/kma
0.037 0.035 0.040 0.045 0.046
CHsNC,
BO(=) 0.27 0.30 0.39 0.46 0.44
BO(~)~ 0.27 0.31 0.31 0.27 0.26
a Average value of k(Mix)/k, over the range for which p’,(D) and &J were i’) averaged. * Data of ref 4 at 553.7’K corrected to 504°K according to the measured temperature dependence of PO( ) for He (ref 5 ) (all noble gases have near-equal efficiencies for CHaNC).
ENERGY TRANSFER IN THERMAL ISOCYANIDE ISOMERIZATION latter values are close to the mean of fl'u( m ) and flu(m ) , as is predicted for the present region of 6. The efficiencies of He and Ne are very similar for both systems. Those of Ar, Kr, and Xe are distinctly higher in the CzH5NCsystem; Kr and Xe have equivalent efficiencies and the maximum (or plateau) observed for CH3NC around Ar is shifted t o Kr for the CzHsNC homolog. Several factors influence the observed alterations in efficiencies and suggest some difficulties in making a priori predictions. 1 . Size (Complexity) of the Substrate Molecule. The equal or enhanced efficiencies for CzH5NCrelative to those for CH&C are unexpected if one considers only the complexity of the two substrate molecules; CzH5KC is a larger heat reservoir than C H d " and energy removal by heat bath atoms should be less efficient if a quasistatistical energy accommodation process is involved on collision, as has been proposed.la 2 . Conservation of Angular Momentum Restrictions. As was also pointed out by LR,13& conservation of angular momentum restrictions influence profoundly the efficiency of energy transfer. Qualitative consideration may be made here of the relative magnitudes of the moments of inertia (Table VIII) that enter into the
Table VI11 : Some Values of Momznts of Inertia for the Noble Gas Systems in amu AZ
a
IL = pAnr(b2).
b
CHaNC
CzHaNC
3.2 50.3 50.3 11.6 31.7 40.0 33 155 343 523 690
12.6 97.8 110.4 14.6 110.5 117.0 346 172 395 605 860
1373
is expected (Table VIII) since their related moment of inertia values are more comparable with the I values of CzH5SC. Indeed, excepting Ne, the Po( a ) values in Table VI1 for the heavier bath gases are in accord with this expectation. 3. Efective Number of Transition Modes of the Collision Complex. One further consideration will assist in rationalizing the high Po( a ) value for He in the the molecule figure axis roethyl case. tation was treated as an ineffective heat sink (i-e., there is a large centrifugal effect) for transfer of internal energy from CH3NC. There is a fourfold change between I A and I A + (Table VIII), and this treatment accorded with an earlier successful RRKM treatmenVb of CHINC isomerization, according t o which vibrational energy is not efficiently removed via this degree of freedom; this restriction is removed for C2H&C8 for which I A I A + . Thus LR treated energy transfer from CH3NC into two transition modes only and excluded the methyl torsion of the collision complex (which correlates with the methyl isocyanide figure axis rotation) as inactive; the appropriate number becomes three for C,H5r\lC, with concomitant increase in energy-transfer efficiency as compared with the CH&C case. Thus the observed behavior of inert bath gas efficiencies for CZH5NCis plausibly, although by no means rigorously, explained. A slightly low relative value for Ne may be only apparent inasmuch as the measured Ke value of CH3NC appears t o be a little high and may itself contain some error. Figure 7 summarizes the relative collisional efficiencies of the noble gases for the two isocyanide homologs and for a number of other unimolecular systems. The trend observed for CzHsNC appears similar t o that for IT2O5.l4 The data for the latter system were
-
The values of b used here are those of LR.
contributions t o the total angular momentum M from rotational angular momentum J of the isocyanide molecule (IB,I C ) and from orbital angular momentum L due to relative translation of the collision partners (11,).The more disparate these moments, the less effective is their coupling uia the collision complex and the fewer are the possibilities for composing their sum so as to conserve the total, M = J L, after collision and energy transfer. For He and C2H,NC, these restrictions are expected to be more severe than in the CH3NC case since the disparity between moments is larger in the former case, and He would again be expected to be less efficient for CzHsNC. However, for Xe, Ar, Kr, and Xe the reverse
+
0.01
$
20
1
1
40 6 0
I
I
1
1
80
100
120
140
Boiling Paint
I
I
160 180
( O K )
Figure 7. Plots of relative collisional efficiencies of noble gases P-quantities us. boiling point for various unimoledhlar systems: CHsNC, 504"K, 0 ; CzHJSC, 504"K, 0 ; Nz06,223"K, 0 ; NO2C1, 546"K, A; seoCaHB,300"K, X ; cyclobutanone, 296"K, A, (13) (a) Y. I".Lin and B. S. Rabinovitch, J . Phys. Chem., 74, 3161 (1970) (LR) ; (b) F. W. Schneider and B. S. Rabinovitch, J . Amer. Chem. SOC., 84, 4215 (1962).
T h e Journal of Physical Chemistry, Vol. 76,N o . 10,1971
1374 obtained in the second-order region but the dilutions used were between 1 and 40 (on a pressure-to-pressure basis) and the efficiencies correspond to values of &(D) rather than of Po( a ) . I n the case of Ar, D varied from 2.5 to 6 over a klk, range from 0.0025 to 0.004. With use of the quasiuniversal plots of TR for p o ( w ) vs. E', a reduced-energy parameter, together with the curves given by T R for Po( m ) as a function of D a t various values of E' (E' = 0.15 for Ar a t 250), the values reported by Wilson and Johnston may be lowered by -20% when brought to infinite dilution. Similar corrections apply to the other members of the noble gas series; however, the basic trend reported14 would not change significantly. For NO2C1, the p quantities presented in Figure 7 correspond to extrapolated Po( a) values derivedl0 from the original d a h 2 The increase of efficiency with mass is more prominent throughout the series and there is no maximum or plateau between He and Xe. In the sec-butyl radical's system, the behavior of the noble gases is qualitatively similar to that of the CZHSN C system. I n the cyclobutanone system,l8 the efficiency values for He and Ne are very low and their disparity with the values for Ar, Kr, and Xe is excessively large. The heat capacities for cyclobutanone and C2HhNC are similar but the excess energy above the reaction threshold in the former system is much larger ( E + N 45 kcal
The Journal 01Phyekal Chemistry, Vol. 76, No. 10, 1971
S. P. PAVLOU AND B. S. RABINOVITCH mol-l. We believe that interpretation of this system is premature. A fuller interpretation of the behavior in these systems is by no means simple. Comparison with the CzHBNC system involves various counterbalancing factors: (1) one of these is temperature, which governs the magnitude of the thermal criterion energy, E t h , described by LR; collisional efficiency tends to increase at lower temperatures as Eth decreases; (2) a second is the excess energy, E+, which governs the operational definition of collisional deactivation efficiency and whose increase has been demonstrated'O to cause decrease of p; (3) the heat capacity of the substrate molecule whose increase causes decrease of p ; (4) finally, rotational coupling and angular momentum conservation restrictions which vary from case to case.
Acknowledgment. B. S. R. wishes to acknowledge the 'debt of pleasure and profit which he owes to Professor G. B. Kistiakowsky in connection with his tenure as Royal Society of Canada Research Fellow and as Milton Research Fellow a t Harvard University, 1946-1948. This paper reflects that stimulus and the benefits of continuing fellowship with G. B. K. (14) H.8. Johnston, J. Amer. Chem. Soc.., 75, 1667 (1963); D.J. Wilson and H. 5. Johnston, ibid., 75, 6763 (1953). (16) G. H.Kohlmaier and B. S. Rabinovitch, J. Chem. Phgs., 38, 1662 (1963). (16) N.E.Lee and E. K. C. Lee, ibid., 50,2094 (1969).
