Interaction of molecular vibration and rotation in the diffusion theory of the thermal decomposition of diatomic molecules

THEORETICAL AND EXPERIMENTAL CHEMISTRY 115

INTERACTION OF MOLECULAR VIBRATION AND ROTATION IN THE

DIFFUSION THEORY OF THE THERMAL DECOMPOSITION OF DIATOMIC MOLEC ULES

M. N. Safaryan, E. V. Stupochenko, and N. M.

Teoreticheskaya i Eksperimental'naya Khimiya,

UDC 539.196.6+541.124.7

P r u c h k i n a

Vol . 5, No. 2, pp. 173--182, 1969

The thermal decomposition of diatomic molecules in a light inert gas was examined within the scope of the diffusion theory with allowance for interaction between vibration and rotation. It was shown that molecular rotation changes the value of the dissociation constant within an order of magnitude and appreciably affects its temperature dependence. The results, obtained by computer, can be applied directly to the dissociation of Iz, Brg, and C12.

I n v e s t i g a t i o n of t h e r m a l d i s s o c i a t i o n in d i a t o m i c m o l e c u l e s i s u s u a l l y l i m i t e d to e x a m i n a t i o n of the e n e r g y d i s t r i b u t i o n func t ion a m o n g the v i b r a t i o n a l d e - g r e e s of f r e e d o m of the m o l e c u l e . It i s known, h o w - e v e r , tha t r o t a t i o n of the m o l e c u l e l e a d s to the a p p e a r - a n c e of an " e f f e c t i v e " d i s s o c i a t i o n e n e r g y , which d e - p e n d s on the a n g u l a r m o m e n t u m and c o n s i d e r a b l y c h a n g e s the d y n a m i c b e h a v i o r of the m o l e c u l e in the r e g i o n of h igh e n e r g i e s ( n e a r the d i s s o c i a t i o n energ~r w h e r e d i v i s i o n o f the i n t e r n a l m o t i o n of the m o l e c u l e in to r o t a t i o n and v i b r a t i o n i s s t r i c t l y s p e a k i n g not va l id . Owing to the c o m p l e x i t y of the r e s p e c t i v e c a l c u l a t i o n s , r i g i d a l l o w a n c e fo r m o l e c u l a r r o t a t i o n h a s not b e e n m a d e wi th in the l i m i t s of any s p e c i f i c m o d e l ; the v a r - i ous c o r r e c t i o n s to the e x p r e s s i o n fo r the d i s s o c i a t i o n r a t e wh ich d e t e r m i n e the c o n t r i b u t i o n of r o t a t i o n h a v e b e e n m a d e on the s u p p o s i t i o n that t h e r e is no d y n a m i c r e l a t i o n s h i p b e t w e e n m o l e c u l a r v i b r a t i o n and r o t a t i o n [1].

Wi th in the s c o p e of d i f fus ion t heo ry , the p r e s e n t p a p e r e x a m i n e s the t h e r m a l d i s s o c i a t i o n of d i a t o m i c m o l e c u l e s wi th a l l o w a n c e fo r m o l e c u l a r r o t a t i o n . S p e - c i f i c r e s u l t s a r e ob ta ined fo r a s y s t e m of d i a t o m i c m o l e c u l e s in a l igh t i n e r t gas , a c t i ng as a t h e r m o s t a t wi th a f a i r l y h igh t e m p e r a t u r e (I 2 and Br2 in He).

STATEMENT OF THE PROBLEM

We define the internal state of a diatomie molecule

with known interatomic interaction potential V(r)

(where r is the interatomic separation in the molecule) by the values of the total energy E and the angular mo-

mentum L. We introduce the following notation:

E L r V (z) -- ~-~ "~ - - -D- ' b' - - l / ~ i T b , z = r ~ - '

where D is the dissociation energy of the molecule, r e is the equilibrium interatomic separation, I = = (1/2)Mr e is the equilibrium moment of inertia, and M is the mass of the atom in the molecule (a homo- nuclear molecule).

The region S of existence of the molecule in the (x, y)-plane is bounded by the lines y = 0, x = #I(Y),

and x =/~2(Y) (F ig- 1), w h e r e

2 g2 (y)=V(zo)+ "--~-, ~(y)=V(z,)~--T, (i)

ZO ,

and Zo and z, are, respectively, the positions of the minimum and maximum "effective" potentials V(z) + + (y2/z2), i.e. , roots of the transcendental equation

d-Z-Y - 2 Y - L = o. ( 2 )

dz z ~

We a s s u m e that the m o l e c u l e d i s s o c i a t e s w h e n it r e a c h e s the u p p e r l i m i t o f the r e g i o n x = ~2(Y).

Le t f ( x , y, t) be the d i s t r i b u t i o n func t ion fo r the m o l e c u l e s in the (x, y ) - p l a n e . We w i l l s u p p o s e that the m e a n s q u a r e i n c r e m e n t A x (and Ay) as a r e s u l t of c o l l i s i o n of the m o l e c u l e wi th an a t o m of the t h e r m o - s t a t is s m a l l c o m p a r e d wi th the ~( '~) r a n g e , on e x t e n - s ion of wh ich the f - f u n c t i o n c h a n g e s a p p r e c i a b l y in v a l u e . Then , as in [3 ,4] , as k i n e t i c e q u a t i o n f o r f , i t is p o s s i b l e to u s e the F o k k e r - P l a n e k equa t ion , w h i c h h a s the fo l l owing f o r m :

at _ _ divj , (3) at

f ~ ) +

A- B~y ( ~--~-q Olnf~ (3a)

]y = - - ]B~I ~['Of __Oy ~o ~d~_)q_Olnf ~ ,

+B,~(~ f ~a'nf~ \ox (3b)

B.x -- ((Ax)~)2, ' B~ v _ (Ax@)2, . B~ -- ((@)2)2~ (4)

Here Ax and Ay are the increments of the x and y values resulting from collision with the inert gas atom, (~>

denotes averaging over all the molecular collisions in unit time, T is the mean free time of the molecule, f0 is the equilibrium distribution function corresponding to temperature T of the thermostat and normalized to the density of the number of molecules in the dis = sociation equilibrium state, no.

The boundary conditions follow from the equality to zero of the particle flux through the boundaries of the region y = 0 and x = p~(y)"

116 TEORETICHESKAYA I EKSPERIMENTAL~NAYA KHIMIYA

] u ] v = o = O, (in) 'x=.,<~,) = 0 (5)

(n is the no rma l to the cu rve x = ~l(Y)) and f r o m the equa l i ty of the "absorp t ion" of the p a r t i c l e s at the l ine x = #~.(y) [2] to zero (we a r e cons ide r ing d i s soc ia t ion in the absence of recombina t ion) ,

q = 0

We r e s t r i c t o u r s e l v e s to examina t ion of the r eg ion

of t e m p e r a t u r e s in which r d >> T r (~'d and 7r are , r e s p e c t i v e l y , the c h a r a c t e r i s t i c t imes for d i s soc ia t ion and r e l axa t ion in the sys tem) , and he re the d i s s o c i a - t ion k ine t ics a re d e t e r m i n e d by the slow change of the q u a s i - s t a t i o n a r y m o l e c u l a r d i s t r ibu t ion at t > 7r. With cons ide ra t ion of the r e s u l t s f r o m [2], it is poss ib l e app rox ima te ly (D/kT ~> 5) to wr i t e , for q u a s i - s t a t i o n -

a r y dis t r ibut ion,

div j = 0. (7)

A solut ion to Eq. (7) can only be obtained by n u m e r i - c a l means , and i t is t h e r e f o r e convenien t to r e d u c e the r eg ion of va lues (x, y) inwhich the solut ion is sought. Cons ide r ing that the re is p r a c t i c a l l y a Bol tzmann d i s - t r ibut ion for t > r r at va lues x ~ (kT/D)(E ~ kT), it is poss ib le ins tead of S to cons ide r the r eg ion St - {y = = 0, x = Pl(Y), x = #2(Y), x = x0} with boundary condi- t ion at the l ine x = x0(1 - (kT/D) > x0 ~ kT/D) :

g 0 I (Xo, v) = ~ f . (8)

The p r o b l e m under cons ide ra t ion is then d e s c r i b e d by Eq. (7) with condit ions (5), (6), and (8).

The d i s soc ia t ion constant is d e t e r m i n e d f r o m the e x p r e s s i o n fo r the flow of p a r t i c l e s through the upper boundary of the r eg ion x = ~z(Y):

1 dN 1 K = N dt N d ]vdx-- Lay (9)

x=p, dy)

where N = f f ] ( x , y ) dxdy is the concen t r a t ion of m o l e - s

cules . We not ice that, as in [2], it is poss ib le to con- s ide r the d i s soc ia t ion p r o c e s s in the p r e s e n c e of r e - combina t ion (nonzero value for condit ion (6)) and to take account of the subsequent phys ica l ly m o r e r e a l approx imat ion when f inding the q u a s i - s t a t i o n a r y d i s t r i -

bution (Eq. (7) inhomogeneous) . To obtain spec i f ic r e su l t s , i t is n e c e s s a r y to c a l -

cu la te the d f f fus ioncoef f ic ien t s Bxx, Bxy, and Byy. We cons ide r a s y s t e m in which the d ia tomie m o l e c u l e s c o m p r i s e a s m a l l addi t ion to a light i n e r t gas : N << << n a, m << M, na is the concent ra t ion , and m is the m a s s of the i n e r t gas atom. As be fo re [2,3], we a s - sume that: a) the mo lecu l e s and a toms i n t e r a c t a c c o r d - ing to the laws of c l a s s i c a l mechan ic s ; b) co l l i s ions be tween the m o l e c u l e s and i n e r t gas a toms a re l a r g e l y nonadiabat ie; e) during col l i s ion , the light a tom only r e a c t s with one of the a toms in the molecu le .

If p i s the m o m e n t u m for r e l a t i v e mot ion of the a toms of the mo lecu l e in the c e n t e r - o f - m a s s s y s t e m of the molecu le , P is the momen tum of the m a s s cen-

t e r of the molecu le , and Pt and Pz a r e the m omen ta of the individual a toms of the molecu le , the fol lowing can be wr i t t en :

1 1 ? l = - 2 - P + p , p ~ . = ~ P - - p

and, with the conc lus ions r e a c h e d about the na tu re of co l l i s ion , the change in the va lue of p a f te r co l l i s ion is equal to

1 Ap = - - ~- Apo, (10)

where Apa is the change in m o m e n t u m of the light gas

a tom. We in t roduce the o r i en ta t ion of the co l l i s ion . Let

a 1 and fit denote the o rb i t a l and az imutha l angles of a c e r t a i n vec to r a, a = {oq,/?l). Then, r = {0,0}, p =

={cr L ={Tr/2,Tr/2}, AL{Tr /2 ,0 r /2 )+90} , Ap = = {~, A}, and in the approx imat ion under c o n s i d e r a -

tion, with a l lowance fo r Eq. (10),

AE pAp,~ Ax = -D- = ~ M - cos

(cos y = sin @ cos q0 sin cz + cos @ cos a), (11)

A ILI AL cos (p zApa Ay V2-iD ]/ 2ID 2 V-M--D sin ~cos % (12)

Averag ing o v e r all the va lues of the angles r and and taking account of Eqs. (4), (11), and (12), we ob-

tain

1 p~ 2b, p~ B~ = 2M-~ ( ~ - ~ (ap~)~cos~v) -- - d - i N - g ) ' (13)

where

1 , (AG) ~ D b~ = T3- k 4-4M~)' a ~ hT ; (14)

1 / p (AP~) 2 Bxv = -2- ~ / r ~ 2MD cos y sin ~ cos q)) =

b 1 ,pzsincr bl ,

By v 1 ! (A p,~)z z 2 sin 2 xb cos 2 q~) : bl : 4 \ 2MD - ~ (zz)"

(15)

(16)

The bl va lue is the r e s u l t f r o m a v e r a g i n g o v e r the Max- wel l d i s t r ibu t ion of the l ight gas; f rom [3] we have

//eT ",1/~ = 3 ~ '*~ ,2~ m / (r*), (17)

where T* = k T / e l 2, a12, and crl2 a r e p a r a m e t e r s of the i n t e r ac t ion potent ia l for the a tom of the molecu le and the i ne r t gas atom, r e p r e s e n t e d as V(1R) = el2~((h2/1R); ~*(tA~(T*) is the co l l i s ion in t eg ra l used in the kinet ic theory of gases . The (pZ/MD) and (z 2) va lues c o r - r e spond to the mean va lues at g iven x a n d y values ; h e r e ( . . . ) denotes ave rag ing o v e r all the points on the su r f ace for E = c o n s t , L = c o n s t ( o r x = eonst , y = = const) in the phase space (p, r) of the m o lecu l e s . This ave rag ing is p e r f o r m e d along the l aye r between the s u r f a c e s (E and E + 6E) and (L and L + 5L) at


dE ~ 0 and 6L ~ O, a s s u m i n g cons tant p robab i l i ty dens i ty within the layer . In the v a r i a b l e s

r p, p z . . . . {~, '1, ;}, -- -- {p{, p;, p~}

~ VTN-6

the des i r ed average of a given quant i ty k(p ' , z) is w r i t - ten in the following form:

(L)=- l im t { X d o ( i d~

y-~u l y , y+Sy ky ,y+Sy / )

where the phase volume element is e~o = ~ n < , ~ p ; ~ .

This exp res s ion can be reduced to the fo rm

z~

OxOy ~do j ~z 0 ~ dz

O~ ) = x.,~ = ~, zz

O= ,t ~ dO (" Z ~ O~W .1 ez

x,(] z l

where zl and z2 a r e the roots of the equat ion

(18)

y2 x - - ~ - - V(z) = 0, (19)

p~2 + p;2 Jt" p 2 ~<X -- V (Z) p~2-I-p;2 ~' �9 "<7~" ( 2 0 )

Integration in Eq. (20) gives

v V'q j (21)

Taking Eq. (21) into account , f rom Eq. (18) we obtain

-= 2 ~_. dZ dz

s ) v - s t Z, Zt

(22)

Subst i tut ing for X in Eq. (22) by p2/MD = x - V and z 2, r e spec t ive ly , we obtain

/ p2 Aa A2 ~ ) - & , (z~)= & , (23)

where the integrals

Z2

A1 = d z y2

x - - V - ,] z 2 zl z2

A 2= i" V z2dz y2 '

. x - - V Z2 zl z~

= C * - v (24)

3 V y2

X - - Y Z2

z t

can be obtained by numerical integration for a given intramolecular interaction potential V(z).

For the equ i l i b r ium d i s t r i bu t ion funct ion f~ y), we obtain

A 2 fo (x, y) dxdy ~" e-a~do = e-'~dxdy O ~ x

Z2

S " . . . o - a . . , [" 2 0~W �9 x am e axay I z ~ az. (25) .! Oxuy zt

Hence, by taking Eq. (21) and the n o r m a l i z a t i o n condi t ion into account, we obtain

where

no -ax p (x, y) = -&o e vA,, (26)

Ao = l l e-a* A - y ~dxcty. (27)

We wr i te the f inal fo rm of the o r ig ina l equat ion (7) with al lowance for Eqs. (13), (15), (16), (23), and (26). Using the nota t ion j0 = (2Ao/Nbt ) (aea j ) , q) = (no/ / N ) ( f / f o ) , we have

divj ~ = 0, (28)

�9 o . a(,-x)- O(p O(p (28a) Ix = aye f~a ~ - -i- 2y~e a('-X)A 1 Oy "

.o ,e.(Z_.)A O~ Oq~ Iv = v 2 ~y + 2y2e~~ " (28b)

The d i ssoc ia t ion cons tan t (9) is equal to

K = blagl (a) e -a,

where

(29)

1 ~ ] ~ i~ (30) g~ (a) =

x=~(y)

The gl(a) value is de t e rmined by n u m e r i c a l solut ion of Eq. (28) with the boundary condi t ions (5), (6), and (8).

In addit ion to obtaining va lues for the d i ssoc ia t ion cons tan t d i rec t ly , it is i n t e r e s t i ng to d e t e r m i n e the ro le of dynamic i n t e r ac t i on be tween m o l e c u l a r v i b r a - tion and ro ta t ion while cons ide r ing the d i s soc ia t ion p r oc e s s and thereby to d e t e r m i n e the poss ib i l i t y of va r ious approx imat ions in the computa t ion of ro ta t ion . We the re fo re desc r ibe this p r o c e s s within the scope of p r ob l e m (18), f o r m a l l y d i s r e g a r d i n g the v i b r a t i o n a l - ro ta t iona l dynamic (but not s ta t i s t i ca l ) i n t e rac t ion , i. e . , a s s u m i n g that the momen t of i n e r t i a of the mo le - cule does not change as a funct ions of x and y. F o r this purpose, it is n e c e s s a r y to a s s i g n appropr i a t e va lues to the coeff ic ients (Ai in tegra l s ) in Eq. (28). We w r i t e ~ = x - y2, V = y2 (~ and V h e r e a r c not r e - lated to z), and in this case ~ and rl a re c l e a r l y ( r e - duced) v ib ra t iona l and ro ta t iona l ene rg i e s of the m o l e - cule, r e spec t ive ly . Changing f rom the v a r i a b l e s x / rod y to ~ and V in Eq. (28) and reduc ing by ,71/2, we obtain

divjo=O,

1~ -= (4An - - 8A~l + 4A2~) e -(~+~)~ Oq~

118 TEORETICHESKAYA I EKSPERIMENTAL'NAYA KHIMIYA

+ (4A:] - - 4A~q) e -~(~+n) ~ ,

&p ]'~ = 4A2~]e-a(~+~) --d~l + (4A(~ - - 4A2~) e -~'(~+n) -0~-'0q) (31)

We fur ther a s s u m e that the coeff ic ients of the Fokker - P lanck type equat ion (3) in the va r i ab l e s ~ and V, which

i '/ )l- / / !

J - / /

/,0" tq ,I I / /

J f ~ ~:J:~ ._ o" L~ t.o D y

Fig. 1

a r e not difficult to obtain by analogy with Eqs. (13)- (16) supposing that at co l l i s ion z = i (see also [2, 4]), for the Morse potent ia l have the following form in the g iven approximat ion :

B~ = ~ V l -~(l--Vl-~),

261 Bnn := - - ~ ~l, Br ~ O. (32)

By taking aecotmt of Eq. (32) and also f0(~, ~?) ~ e-a(~+V)/( 1 _ ~)~/2, we find f rom Eq. (31) that the o r ig ina l equation (28) de sc r ibes the p rob lem in the absence of dynamic in t e rac t ion between v ib ra t i on and r o -

(E,

20

t5

10

L2O

o oA 1.0 g"

Fig. 2

tat ion with the following values of the coeff ic ients (for a Morse osc i l l a to r ) :

AO= ~ 1 . c l / f ~ x + v ~

A~176 A ~ 1 7 6 (33)

and the ~I(Y) boundary he re becomes #~(y) = y2.

If we now in t roduce 0~p/0y = 0 (0q0/0~ = 0) into Eq. (28), with the coeff ic ients given in (33), the value obtained for the d i s soc ia t ion cons tan t wil l co r r e spond (at a >> 1) to the r e su l t in [2], which to a f i r s t approximat ion takes account of the con t r ibu t ion of molecu la r rota t ion; the subsequen t approx imat ion is de sc r ibed by Eq. (28) with coeff ic ients (33).

RESULTS

The n u m e r i c a l in tegra t ion of the p rob lem was p e r - formed on an M-20 computer . As a ru l e the i n t eg ra - t ion step amounted to 0.08 with r e spec t to y and f rom 0.07 with x = x0 -~ 0.4 to 0.02 with x ~- 1 with r e spec t to x; the Pl(Y) and P2(Y) l imi t s we re accu ra t e ly r e p r o -

0q~ t ~nTl~0;. r y) = 1 ; ~p(~2(y),y)= duced, and ~ v=0 =

= r y) = 0 were used as bounda]ry condi t ions . The n u m e r i c a l in t eg ra t ion method was based on [5, 6].

/ /

0.8 ~= /

a7 /,/ :/./

a5 t I /

/ 0.5 2 / /

i / / I 0.# / / I ~ V

/~ ~7 6 0 0,3 ~ =.

82

o.7 . . . . . . , . . . . . ~ . ' o J o,J Fig. 3

The i n t r a m o l e c u l a r i n t e rac t ion potent ia l was ap- proximated by the Morse o sc i l l a to r :

V (z) = (1 - - e -c 1~-11)2. (34)

The ma in r e s u l t s were obtained with the 12 molecule . The reg ion S for the T2 molecule (c = 4.94) is shown in Fig. i (sol id l ines) . The r e s u l t s f rom the computer ca lcu la t ion a r e g iven in Figs . 2 - 6 . F igures 2 and 3 show the < z2> and <pZ/MD) va lues , r e spec t ive ly , as func- tions of y at va r ious x values ; the sol id l ines r e p r e s e n t the r e a l va lues of Eq. (23), and the dashed l ines r e p - r e s e n t the approximat ion in Eq. (33), i. e . , on the a s - sumpt ion that there is no dynamic i n t e r ac t i on between v ib ra t ion and ro ta t ion of the molecule . The t rue va lues for the momen t of i n e r t i a (z2) and the mean kinet ic energy <p2/MD> differ apprec iab ly f rom the approxi - mated values ; as would be expected, the i r dependence on y is sl ight at s m a l l x va lues and i n c r e a s e s nea r the d i s soc ia t ion boundary x ~ ~2(Y).

The va lues for the coeff ic ient gl(a) in our exp res - s ion for the t he r ma l d i s soc ia t ion constant ,

K -- n~ (~2~* tl.1) (T*) ~-~ ~ - ] •


( D \ 1/2 s) • ~ - ) g, (a)e --ff (a-~-DtkT) (35)

a r e g iven in Fig. 4 (solid l ine); gl(a) changes app rox i - m a t e l y f r o m 2 to 10, with the t e m p e r a t u r e dependence gl(a) ~ a~, 1 ~> ~ ~> 1/2 at va lues in the 5 ~ a-< 30 range.

One of the a i m s of the p r e s e n t work was to d e t e r - mine the r o t e of m o l e c u l a r ro t a t ion in the d i s soc i a t i on p r o c e s s and the a c c u r a c y of the app rox ima te methods fo r i ts computat ion. The c o r r e s p o n d i n g r e su l t , in p a r t i c u l a r , is i l l u s t r a t ed in Fig. 5, which g ives the gl(a) va lues obtained in v a r i o u s approx imat ions : 1) c o r - r e spond ing to a c c u r a t e solut ion of the p rob lem; 2) the solut ion of Eq. (28) with (33) and the condi t ion 3m/0y = = 0 (Ocp/O~ = 0); 3) obtained e a r l i e r [2] without a l lowance for m o l e c u l a r ro ta t ion ( co r re spond ing to the coe f - f ic ien t a in [2]). The r e s u l t f rom the ca lcu la t ion of gl(a) using coef f ic ien t s (33) in p l ace of (24) d i f fe r s s l ight ly f r o m the a c c u r a t e va lue; cons ide ra t i on of the dynamic in t e r ac t ion be tween v ib ra t ion and ro ta t ion leads to a s l ight d e c r e a s e of gl(a) at f a i r l y high t e m - p e r a t u r e s (by ---10% at a -~ 5). This is d i r e c t l y r e l a t e d to the b e h a v i o r of the coe f f i c i en t s in (24) and (33), and

;2 I/

9

7 g

4~ 3 Z ~ I

; D 21 Fig. 4

their difference is in fact mutually compensated, since (A2/A1)t(A~/A~) > 1, and (A~/Ai)/(A;/A~) < 1. The ap- p r o x i m a t e ca lcu la t ion on the assumpt ion of no dynamic i n t e r ac t i on be tween v ib ra t ion and ro ta t ion of the m o l e - cule in the g iven t e m p e r a t u r e r ange 5 _< a -< 30 a g r e e s with the a c c u r a t e ca lcu la t ion of the d i s soc i a t i on constant . This r e s u l t makes it pos s ib l e to exami ne the s i m p l e r p rob l em, name ly in the v a r i a b l e s ~ and 7; the co r r e spond ing diffusion equat ion does not conta in a mixed d e r i v a t i v e , the S~ ~) r e g i o n h a s a s i m p l e r f o r m (~ = 0, ~ = 0, ~ = #~(~)), and the p r o b l e m of in t eg ra t ion

of the equat ion (and ca lcu la t ion of the coef f ic ien ts ) is c o n s i d e r a b l y s impl i f i ed .

The s i m p l e s t and mos t convenient for the c a l c u l a - tion is the approx imat ion co r r e spond ing to cu rve 2 in Fig. 5; this approx imat ion takes account of the e f fec t of d e c r e a s e in the d i s soc ia t ion ene rgy of the mo lecu l e p o s s e s s i n g angular momen tum L and can be ca l cu la t ed by subs t i tu t ing the d i s soc i a t i on e n e r g y va lues into the e x p r e s s i o n for the d i s soc i a t i on constant obtained wi th- out a l lowance for ro ta t ion by its " e f f ec t ive" va lue : D* -- = V(z*) + (L2/2Iz 2) - V(z0) - (L2/2Iz20) and subsequen t ly ave rag ing the r e s u l t ove r the Bol tzmann d i s t r ibu t ion of ro t a t iona l e n e r g y [2]. As seen f r o m Fig. 5, this

approximation gives a rather good result; the values corresponding to curves 2 and 1 differ by 20-40~o with

a close temperature dependence. Clearly, the contri-

7 5 5 0

J 2 3 !

I !

Fig . 5

b u t i o n of r o t a t i o n to the v a l u e o f the d i s s o c i a t i o n con - s tan t is strictlycharacterizedby the difference between curves i and 3 in Fig. 5. Rotation changes the value of the dissociation constant within an order of magnitude and appreciably affects its temperature depen-

denc e. The effect of rotation depends on the temperature

of the surrounding gas and on the form and parameters of the intramolecular interaction potential. In Fig. 4 the dotted line shows the gt(a) values obtained for a model of a cutoff harmonic oscillator: V(z) = c2(z - - 1) 2. In this case, when considering the problem in

the variables x, y (or ~, 77) the solution is more sensi-

tive to the form of V(z) than in the one-dimensional (with respect to ~) diffusion problem [2]; according to

[2], gi(a)/gl(ct)harm.ose. = c~/~ l = 1 .2 -1 .6 ; he re g t (a ) /

/ g l ( a ) h a r m . o s e . --- 1 .4 -6 .5 , 5 -< a -~ 30. In addit ion, the use of a cutoff ha rmon ic o s c i l l a t o r in the two- d imens iona l p r o b l e m is not so i n t e r e s t i n g in the quant i ta t ive r e s p e c t , s ince the reg ion for this model (shown by the dashed line in Fig. 1) d i f fe rs c o n s i d e r - ably f r o m the r e a l region. C u r v e s 2 and 3 in Fig. 4 w e r e obtained for the M o r s e o s c i l l a t o r with c p a r a m - e te r va lues of c i = 4.5 (Br2 molecu le ) and c2 = 4.03 (C12 molecu le ) , r e s p e c t i v e l y .

F igu re 6 shows the behav io r of the ~(x, y) function as a function of x for va r ious y va lues (with a = 10). The a c c u r a c y of this ca lcu la t ion is d e t e r m i n e d by the poss ib i l i t y of using diffusion app rox ima t ions in de-

a~ ~8

ao

a2

o. ~$ az a9 zl x

Fig. 6

scribing the problem and, in particular, using the con- dition ~ i -~ (((Ax)2))I/2/~ < I. By taking account of Eqs. (4), (13), (17), and (23) and ~ ~> 2/a (Fig. 6), we

120 TEORETICHESKAYA I EKSPERIMENTALrNAYA KHIMIYA

obtain

V - m x f~om 7" D A3

[1 ~ -ffM k T A~ ~ 1. (36)

The r e s u l t s are d i rec t ly appl icable to the I2 + He and Br2 + He s y s t e m s ; i t may be usefu l also for de- t e rmin ing the r ecombina t ion cons tan t of I, Br, and C1 a toms with H2 and H as th i rd bodies . Compar i son of the r e s u l t s obtained with the ava i lab le exper imen ta l data on the d i ssoc ia t ion of I2 in He at 1000 ~ <~ T <~ <~ 3000 ~ K gives good a g r e e m e n t both in t e m p e r a t u r e dependence and in the value of the d i ssoc ia t ion constant; he re (~J2 -~ 2.3 ~. If it is a s sumed that a~2 ~- -~ ( 1 / 2 ) ~ 2 a ((~ma is a p a r a m e t e r of the i n t e r ac t i on po- ten t ia l between the molecule and the light atom), we obtain (rma .< 3.25 /~ and f rom data on t r a n s f e r c h a r a c - t e r i s t i c s at low t e m p e r a t u r e s ( rmag 3.8 /~ [2], i . e . , the ca lcula t ion leads to a f a i r l y r e a s o n a b l e value for the (r12 p a r a m e t e r .

In conclusion, we exp res s our gra t i tude to A. I. Vol 'pe r t and L. N. S te s ik fo r the poss ib i l i ty of c a r r y - ing out the computer ca lcula t ion.

REFERENCES

1. E. E. Nikitin, Modern Theor ies of T h e r m a l Decomposi t ion and I some r i z a t i on of Molecules in hhe Gas Phase [in Russ ian] , Nauka, Moscow, 1965.

2. E. V. Stupochenko and M. N. Safaryan, TEKh [Theore t ica l and E x p e r i m e n t a l Chemis t ry ] , 2, 784, 1965.

3. M. N. Safaryan and E. V Stupochenko, PMTF, no. 4, 1964.

4. M. N. Safaryan and E. V. Stupochenko, PMTF, no . 1, 1965.

5. N. N. Yanenko, DAN SSSR, 125, 1207, 1959. 6. L D. Safronov, Zh. vychis l , matem, i matem.

f iz . , 3, 786, 1963.

23 F e b r u a r y 1968 Moscow State University Branch, Institute of Chemi- cal Physics IkS USSR

Documents

Interaction of molecular vibration and rotation in the diffusion theory of the thermal decomposition of diatomic molecules