I

i

CENTRAL POTENTIALS FOR POLYATOMIC MOLECULES: I A SURVEY OF MGRSE

POTENTIALS DETERMINED FROM VISCOSITY AND THE SECOND

VIRIAL COEFFICIENT *

by

D a n i e l D. Kenowalow

The Morse p o t e n t i a l f u n c t i o n i s used t o approximzte t h e p a i r

i n t e r a c t i o n p o t e n t i a l f o r a wide v a r i e t y of hydrocarbons. The

e f f e c t s of molecular asymmetry (narrcwing and deepening of t h e

p o t e n t i a l w e l l ) are most c l e a r l y e v i d e n t i n p o t e n t i a l s determined

from e x p e r i m e n t a l second v i r i a l c b e f f i c i e n t s . Such p o t e n t i a l s a r e

shown t o be more r e a l i s t i c t h a n those p o t e n t i a l s determined from

e x p e r i m e n t a l v i s c o s i t i e s which a r e much less s e n s i t i v e t o asymmetry.

* T h i s r e s e a r c h w a s suppor ted by N a t i o n a l Aeronaut ics and Space A d m i n i s t r a t i o n Grant NsG-275-62 and t h e U. S . A i r Force C o n t r a c t AF 33(657) - 7 311.

+ NATO Fel low 1963. Permanent Address: I s t i t u t o d i Chimica F i s i c a , U n i v e r s i t a D e g l i S t u d i D i Milano, Milano, I t a l y .

https://ntrs.nasa.gov/search.jsp?R=19660003793 2019-08-12T17:31:05+00:00Z

t ' ! .' The s t u d y of t h e gas-phase thermal p r o p e r t i e s of polyatomic

molecules has b o t h p r a c t i c a l and t h e o r e t i c a l i n t e r e s t . The

d e s c r i p t i o n of t h e p a i r i n t e r a c t i c o s of po lys tomic molecules has

proved t o be such a f o r n i d a b l e ttisk t h a t t h e p r e s e n t s t a t e of t h e

a r t i s q u i t e rudimentary , Thus, ir_ i s custoinary t o a t t e m p t a rough

d e s c r i p t i o n of t h e p r o p e r t i e s of d i l u t e g a s e s i n terms of model

p o t e n t i a l f u n c t i o n s c o n t a i n i n g a few a d j u s t a b l e c o n s c a n t s which are

f i x e d e m p i r i c a l l y .

Of t h e w i d e v a r i e t y of semi-empir ica l p o t e n t i a l f u n c t i o n s which

have been proposed (5,6,17-19) t o p r e d i c t t h e p r o p e r t i e s of d i l u t e

gases , t h e most wide ly s t u d i e d has been t h e two-parameter Lennard-

J o n e s (12-6) model:

T h i s p o t e n t i a l p r o v i d e s a f a i r compromise between s i m p l i c i t y and

a c c u r a c y i n i t s a p p l i c a t i o n t o s p h e r i c a l molecules ,

e s t a b l i s h e d , however, t h a t the (12-6) model i s i n s u f f i c i e n t l y

f l e x i b l e t o r e p r e s e n t t h e i n t e r a c t i o n o f asymmetrical molecules e

A s shown by P i t z e r (26) and Balescu ( 2 ) t h e p o t e n t i a l w e l l narrows

and deepens as t h e corresponding molecule d e v i a t e s from s p h e r i c a l

symmetry a

It i s now w e l l

These shape e f f e c t s have been demonstrated q u i t e c l e a r l y i n

Some r e c e n t i n v e s t i g a t i o n s of model p o t e n t i a l s deduced l a r g e l y

from e x p e r i m e n t a l second v i r i a l c o e f f i c i e n t d a t a . A t l e a s t i n

1

2

i p a r t , they a r e b u i l t i n t o t h e Kihara p o t e n t i a l (18) when i t s hard

c o r e i s f ixed on t h e b a s i s of molecular s t r u c t u r e a lone (7,8,12) .

Even when a l l t h e i r parameters are determined from B(T) d a t a

a lone , t he Kihara a s w e l l as o t h e r th ree-parameter p o t e n t i a l s (27,291

e x h i b i t the narrowing and deepening of t h e p o t e n t i a l w e l l w i t h

i n c r ea s ing mo 1 ecu lar as ymme t r y . It i s t h e purpose of t h i s survey t o assess t h e s e n s i t i v i t y t o

molecular shape of p o t e n t i a l s determined from t h e expe r imen ta l

v i s c o s i t i e s f o r a v a r i e t y of hydrocarbons. It i s no e s s e n t i a l

r e s t r i c t i o n t o conf ine t h e s t u d y t o t h e Morse p o t e n t i a l model,

which has proven modera te ly s u c c e s s f u l when a p p l i e d t o s p h e r i c a l

molecules (19,20). The parameters e and 6 have t h e same

s i g n i f i c a n c e as f o r t h e (12-6) model. The t h i r d parameter , c , i s re la ted t o t h e c u r v a t u r e of t h e p o t e n t i a l a t r through t h e

e x p r e s s i o n

m

Thus, l a r g e va lues of c cor respond t o a narrow p o t e n t i a l w e l l ,

w h i l e s m a l l va lues of c correspond t o a wide one. Apparent ly ,

3

t h i s model h a s t h e r e q u i s i t e r l e x i b i l i t y € o r a p p l i c a t i o n t o n o n - s p h e r i c a l

molecules

To conf i rm t h i s view, the Morse p o t e n t i a l s f o r a few n -a lkanes which

were de termined from B(T) d a t a a lone (21,22), are p r e s e n t e d i n Table I,

There i t i s seen t h a t c / k , c , and Q" i n c r e a s e w i t h t h e i n c r e a s e i n c h a i n

l e n g t h about as expec ted from a c o n s i d e r a t i o n of molecu la r s t r u c t u r e

( 2 , 2 6 ) . Next, Morse p o t e n t i a l s were f i x e d by minimiz ing t h e r m s p e r c e n t a g e

d e v i a t i o n between t h e expe r imen ta l and c a l c u l a t e d

found f r o m t a b l e s of c o l l i s i o n i n t e g r a l s (23) r e c e n t l y developed i n t h i s

l a b o r a t o r y . These "vi scosicy-determined" Morse p o t e n t i a l s are l i s t e d i n

Tab le 11 a long wi th ( 1 2 6 ) p o t e n t i a l s which were a l s o de te rmined from

v i s c o s i t y d a t a ( 1 6 ) , I n view of i t s g r e a t e r f l e x i b i l i t y , i t i s no s u r p r i s e

t h a t t h e Morse por rzn t ia l a f f o r d s apprec i ab ly b e t t e r agreement w i t h t h e

expe r imen ta l ( r > t h a r does t h 2 (12-6) .nodel, However, f o r n e i t h e r model

d o t h e p o t e n t i a l s de te rmined from v i s c o s i t y a f f o r d r easonab le agreement

w i t h t h e expe r imen ta l B ( T ) , The r m s pe rcen tage e r r o r s f o r B(T) are

t y p i c a l l y 40 p e r c e n t o r more, w e l l o u t s i d e t h e range of expe r imen ta l

u n c e r t a i n t y . I n c o n t r a s t , a s shown i n Table I, t h e Morse p o t e n t i a l s

de te rmined from B ( T ) a lone fa re* a p p r e c i a b l y b e t t e r i n r ep roduc ing t h e

expe r imen ta l

under 20 pe r c e n t .

'1 (T), The l a t t e r w e r e

?I

7 ( T ) . I n those c a s e s t h e e r r o r s i n '1 (T) are t y p i c a l l y

For t h e rare gases , which are e s s e n t i a l l y s t r u c t u r e l e s s , s p h e r i c a l

molecules , p o t e n t i a l s deduced f r o m B(T3 a r e u s u a l l y n o t t o o d i f f e r e n t

from those found krom (T) ( 1 7 ) , Fo r t h e n -a lkanes , however, i t i s

e v i d e n t from t h e r e s u l t s t h a t a r e a l dichotomy e x l s t s between t h e Morse

p o t e n t i a l s deduced from each of t h e s e p r o p e r t i e s , An e x p l a n a t i o n

4

may be found by examining t h e s t r u c ture-dependence of t h e " v i r i a l - d e r i v e d "

and "v i scos i ty -de r ived" p o t e n t i a l s .

P i t z e r ( 2 6 ) and Balescu ( Z ) , t h e a t t r a c t i v e p o r t i o n s of t h e " v i r i a l -

der ived" p o t e n t i a l s deepen and narrow f a i r l y smoothly w i t h i n c r e a s i n g

c h a i n l eng th (which we equa te wi.th i n c r e a s i n g asymmetry) : t hose f o r t h e

I n accord w i t h t h e p r e d i c t i o n s of i

"v i scos i ty -de r ived ' ' p o t e n t i a l s do no t , Although t h e same comparison

canno t be made f o r t h e remain ing hydrocarbons w t r e a t , w e a r e a b l e t o

d i s c e r n l i t t l e connec t ion between molecular s t r u c t u r e and t h e pa rame te r s

of t h e " v i s ~ o s ~ t y der ived" p o t e n t i a l s . I t i s w e l l known t h a t f o r a l l

molecules t h e low- tenL*r . r i t u re B(T) y i e l d s in fo rma t ion on t h e dep th as a

f u n c t i o n of t he W L U L ~ of the p o t e n t i a l w e l l , T t appea r s t h a t t h e B(T)

For dsjmtletriL n l o l e ~ u l e s i s s e n b i t i v e to rrheir s t r u c t u r e i n a way t h a t

s p e c i f i e s t he well shape even more c l o s e l y S i n c e t h i s s e n s i t i v i t y t o

s t r u c t u r e seems t o be l a c k i n g i n the (T) daca, t h e " v i r i a l - d e r i v e d "

p o t e n t i a l s are expec ted t o be more n e a r l y c o r r e c t , T h i s view i s

suppor ted by the d i s c u s s i o n i n the p reced ing paragraph ,

1

It i s n o t s t r i c t l y v a l i d t o compare p o t e n t i a l s de te rmined from t h e s e

p r o p e r t i e s s i n g l y , s i n c e n e i t h e r one i s s u f f i c i e n t t o s p e c i f y t h e

p o t e n t i a l i n t h e f i r s t p l a c e I n an ensu ing work ( 2 2 ) i t w i l l be shown

t h a t i t i s p o s s i b l e t o f i n d compromise Morse p o t e n t i a l s which a d e q u a t e l y

reproduce both B(T) and (T) f o r a number o f hydrocarbons, '1 Acknowledgements

One of u s ( S . c ' , ) wishes t o thank P r o f e s s o r J. 0. H i r s c h f e l d e r f o r

making a v a i l a b l e t h e f a c i l i t i e s of t h e T h e o r e t i c a l Chemis t ry I n s t i t u t e . We

thank Mrs. Merilene McCloud and Mrs. Linda H u l b e r t f o r t h e i r programming

a s s i s t a n c e , and P r o f e s s o r R. B, B e r n s t e i n f o r h i s c r i t i c i s m of t h i s manu-

s c r i p t .

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9

Not s t ion

C

k

T

Tc

E

= second v i r i a l c o e f f i c i e n t

= Morse pararr,eter governing width s f p o t e n t i a l w e l l .

= Boltznann3s c o n s t a n t

= s e p a r a t i o n between rmleeu la r c e n t e r s

= s e p a r a t i o n a t the energy minimum

I - r m s percentage d e v i a t i o n s

= rms d e v i a t i o n s

= a b s o l u t e tempera ture

= c r i t i c a l t e a p e r s t u r e

= dep th o f p o t e n t l a ;

= f i n i t e separ3 tPon of molecular c e n t e r s a t

ze ro p o t e n t i a l energy

= pcten tLaP energy f u n c t i o n

10

L i t e r a t u r e C i t e d

1- Xnd;;l, 3 * i - L J. D . Cox, P a F , G.. Herr ington , and J. F.

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7 4 (1778 C l P 5 2 \ \ , I

4. RottcrixIev, G. A 5 G R ~ e v e ~ . : and R . Whytlaw-Gray, Proc . Roy.

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6 .

7 , Corrr ic l ly . I F , and K ~ r a d ~ l i c ~ Phys. F l u i d s 3, 463 (1960).

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8 . Cor11-i~; y . 1. F . Fkiy;, FlhJ Ids - 4 , 1494 (1961).

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rhermudyT;;.ol: L -cj Trm. ;par t P r o p e r t i e s " , The American S o c i e t y

of Mec'rcanir%l E n g ~ r > e f i r * ~ New York, (1962), p . 135.

11

L i t e r a t u r e C i t e d ( c o c t I d )

l -

15. E v e r h a r t , W . A . , and E . Mack, J. Am. Chem. SOC., 55, 4894 (1933).

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1 7 . H i r s c h f e l d e r , 3. O. , C . F . C u r t i s s , and R . B. B i r d , "Molecular

Theory of Gases and Liquids" , Wiley, New York, (1954) .

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T h e o r e t i c a l Chemistry I n s t i t u t e Repor t WIS-TCI-74, 7 Dec. 1964.

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I n s t i t u t e Report WIS-TCI-104, 3 June, 1965.

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T h e o r e t i c a l Chemistry I n s t i t u t e Repor ts WIS-AF-19 and

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12 ,

4 L i t e r a t u r e C i t e d ( c o n t ' d )

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Data Book, V o l . I1 :

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b . Table 2009, Dec., 1961.

c . Table 2029, Dec., 1961.

d. Tqble 2006, Dec., 1961.

e. Table 2018, June, 1962.

f . Table 2003, June, 1962.

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