Isotope Effects in Chemical Processes - ACS Publications

6

Isotope and Ortho-Para Separations

of the Molecular Hydrogens by

Adsorption at Low Temperatures

1

Downloaded by CORNELL UNIV on August 22, 2016 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0089.ch006

W. J. HAUBACH, DAVID WHITE

2

P.

RADHAKRISHNA,

A. KATORSKI,

3

R.

WANG,

and

University of Pennsylvania, Philadelphia, Pa.

The

theory of isotope and ortho-para

molecules,

by preferential

is reviewed,

and a comparison

plane

surface

approximation

for

action

coupling

the angular

motions

between

the

mental

geneity

and

factors)

of interaction. on the

and

and

experi-

hydrogens.

In

a molecule-surface

the

inter-

with the center of mass

good

agreement

is

found

separation

factors.

(solely on the basis of the to establish

experi-

the exact form of the

influence

is discussed

reasonable

is obtained

diatomic

temperatures,

theory

calculated

The

separations

isotopic)

and experiment ous

surface,

not possible

separation

potential

(spin

to the

measured

It is, however,

A

between molecular

normal

of

at low

ment is made for the isotopic

vibrations

'T'he

separation

adsorption

agreement

for monolayers

of surface and

hetero-

in both

between on a

cases theory

heterogene-

surface.

s e p a r a t i o n of t h e o r t h o - p a r a n u c l e a r s p i n species

nuclear hydrogens

of t h e h o m o -

( 4 , 5, 18, 2 1 ) b y a d s o r p t i o n at l o w t e m p e r a t u r e s

has g e n e r a t e d a c o n s i d e r a b l e interest i n t h e a n g u l a r d e p e n d e n t s u r f a c e i n t e r a c t i o n s (7, 22).

molecule-

These anisotropic interactions, responsible

f o r t h e o r t h o - p a r a separations p l a y a n i m p o r t a n t r o l e i n t h e t o t a l b i n d i n g e n e r g y of a d i a t o m i c o r p o l y a t o m i c m o l e c u l e t o t h e s u r f a c e

(13)

and

t h e r e f o r e a r e i m p o r t a n t i n i s o t o p i c separations b y p r e f e r e n t i a l a d s o r p t i o n . W h e r e a s , i n t h e case o f a d s o r b e d i s o t o p i c a t o m i c species t h e s u r f a c e

field

c o n s t r a i n s o n l y t h e mass d e p e n d e n t t r a n s l a t i o n a l m o t i o n s , i n t h e case of 'Present address: Mound Laboratory, Monsanto Research Corp., Miamisburg, Ohio. Present address: Atomic Energy Commission, Roskilde, Denmark. Deceased. 2

3

73

Spindel; Isotope Effects in Chemical Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

74

ISOTOPE E F F E C T S IN C H E M I C A L

PROCESSES

d i a t o m i c m o l e c u l e s the same surface field also influences the a n g u l a r motions. T h i s latter effect w h o s e m a g n i t u d e is i n p a r t d e p e n d e n t o n the r o t a t i o n a l constant

(16,

has also b e e n s h o w n to be i m p o r t a n t i n

22)

a c c o u n t i n g for differences i n v a p o r pressures of isotopic h e t e r o n u c l e a r d i a t o m i c s . T h e m o d e l (1,8)

u s e d to a c c o u n t for these differences is v e r y

s i m i l a r , i n m a n y respects, to that u s e d i n the d e s c r i p t i o n of p h e n o m e n a (16,

surface

22).

I n this p a p e r a b r i e f r e v i e w of the t h e o r y of o r t h o - p a r a a n d isotopic separations of

diatomic molecules, b y

a d s o r p t i o n o n surfaces at

temperatures is p r e s e n t e d together w i t h a c o m p a r i s o n b e t w e e n Downloaded by CORNELL UNIV on August 22, 2016 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0089.ch006

a n d experiment.

A l t h o u g h the m o d e l u s e d to represent the

low

theory surface-

m o l e c u l e interactions is a n o v e r - s i m p l i f i c a t i o n of the p h y s i c a l s i t u a t i o n , it w i l l nevertheless be seen that a l l of the p r e d i c t i o n s of the t h e o r y h a v e , i n d e e d , b e e n v e r i f i e d b y e x p e r i m e n t . T h e g o o d agreement b e t w e e n theory a n d the a v a i l a b l e e x p e r i m e n t a l results, the latter b e i n g e n t i r e l y t h e r m o d y n a m i c i n n a t u r e , is m o r e a result of the i n s e n s i t i v i t y of the d a t a to a p p r e c i a b l e changes i n the parameters d e s c r i b i n g the

surface-molecule

interactions, t h a n the v e r i f i c a t i o n of the p a r t i c u l a r a n a l y t i c a l f o r m u s e d i n the a p p r o x i m a t i o n . It is, h o w e v e r , clear that regardless of the exact f o r m of the p o t e n t i a l of i n t e r a c t i o n , b o t h the a n g u l a r a n i s o t r o p y of the surface field a n d the strong c o u p l i n g of the c o n s t r a i n e d r o t a t i o n a l a n d v i b r a t i o n a l m o t i o n of a n a d s o r b e d m o l e c u l e are the i m p o r t a n t factors i n a c c o u n t i n g for the observations. Model for

Adsorbed Diatomic

Molecules

T h e m o d e l d e s c r i b e d b e l o w is that p r e v i o u s l y g i v e n b y W h i t e a n d Lassettre (22).

T h e adsorbent is r e g a r d e d as a p l a n e - s u r f a c e d s e m i -

infinite s o l i d . T h e forces b e t w e e n the s o l i d adsorbent a n d the a d s o r b e d d i a t o m i c m o l e c u l e s are a s s u m e d to b e c e n t e r e d at the positions of the component

atoms of the m o l e c u l e .

T h e total interaction between

the

m o l e c u l e a n d the surface is s i m p l y the s u m of the atom-surface i n t e r actions (16, 22).

T h e i n t e r a c t i o n p o t e n t i a l for e a c h a t o m of the a d s o r b e d

m o l e c u l e is g i v e n b y f(z- )

w h e r e Z\ is the distance of the i - t h a t o m

x

m e a s u r e d n o r m a l to the surface. L e t the distance f r o m the center of mass to the atoms of mass mi a n d m , r e s p e c t i v e l y b e bi a n d b , as s h o w n i n 2

2

F i g u r e 1. T h e p o t e n t i a l energy, V , of the a d s o r b e d d i a t o m i c m o l e c u l e is then V(z,0)

= / ( « i ) + f(z ) 2

= /(2 + b osO) + f(z - b cosO) lC

2

(1)

w h e r e z is the distance of the center of mass of the m o l e c u l e f r o m the surface a n d 0 the a n g l e the axis of the m o l e c u l e makes w i t h the z axis. A n i m p o r t a n t feature of E q u a t i o n 1, regardless of the f o r m of f

(z ) {

is t h a t the response of h e t e r o n u c l e a r d i a t o m i c m o l e c u l e s to this surface


6.

Isotope and Ortho-Para

HAUBACH E T A L .

75

Separations

field c a n n o t be the same as for h o m o n u c l e a r s . T h i s c a n r e a d i l y b e seen b y e x p a n d i n g (1) i n a p o w e r series i n rj = V(z,0)=2f(z)

cos0,

+ (b -b )f(z) + ' 1

2

v

b

2 l

l

b 2 2

f W

+ -

() 2

D e f i n i n g the i n t e r n u c l e a r distance b = foi -\- b a n d A the d i s t a n c e ( a l o n g 2

the b o n d j o i n i n g the t w o a t o m s )

b e t w e e n t h e center of mass of

the

m o l e c u l e a n d the m i d p o i n t b e t w e e n the t w o atoms (see F i g u r e 1 ) , + 2Af'(z) +^-f"(z)


V(z,6)=2f(z)

v

F o r h o m o n u c l e a r d i a t o m i c s , bi =

+ W

2

v

b, A = 2

W

+ • • • •

() 3

0, the p o t e n t i a l is r e p r e s e n t e d

b y even p o w e r s i n 77 a n d a l l terms c o n t a i n i n g A v a n i s h . O n the other h a n d , for heteronuclears the p o t e n t i a l consists of b o t h e v e n a n d o d d p o w e r s of 77 m u l t i p l i e d b y coefficients c o n t a i n i n g A . T h i s A has b e e n r e f e r r e d to as the d i s p l a c e m e n t of the center of i n t e r a c t i o n ( I , 8 ) of the m o l e c u l e f r o m the center of g r a v i t y .

//>////S// Figure 1.

Diatomic molecule adsorbed on plane surface

I n o r d e r to o b t a i n the energy levels of the a d s o r b e d d i a t o m i c m o l e cule, it is necessary to solve the S c h r o e d i n g e r e q u a t i o n for the H a m i l tonion

(13). H = ^ 2M

+

2M

Pi + + JPjL 2M

Pi 21

+ V(z,6)

+ 2Zsin 0 2


(4)

76

ISOTOPE E F F E C T S IN C H E M I C A L

PROCESSES

w h e r e M is t h e mass of the m o l e c u l e , 7 t h e m o m e n t of i n e r t i a , a n d

V(z,0)

the p o t e n t i a l energy of i n t e r a c t i o n of t h e m o l e c u l e w i t h t h e surface. Representing the eigenfunction b y the product t = F(x,y)G(4>)R(z,e)

(5)

the S c h r o e d i n g e r e q u a t i o n becomes separable i n the v a r i a b l e s x, y, a n d . F r o m t h e F e q u a t i o n one obtains t h e energy levels of a p a r t i c l e i n a two-dimensional box, from the G equation exp(im),m = 0, ± 1, ± 2

G() = [(2 ) y Downloaded by CORNELL UNIV on August 22, 2016 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0089.ch006

7r

1/2

i

a n d the R e q u a t i o n is

" ( S ^ M ) J? ~ ( 8 ^ / ) { (5b)

fe [

sin0

+ V(z,$)R

!?]

" (l£?)

R

}

( 6 )

= ER.

T o o b t a i n t h e r o t a t i o n - v i b r a t i o n eigenvalues, E , of E q u a t i o n 6 i t is necessary to specify a p a r t i c u l a r f o r m f o r V(z,0).

Substituting a Morse type

function f(z)

(7)

= °{e-*«*-2e-«*}

w h e r e D a n d a are constants, i n E q u a t i o n 1 gives t h e m o l e c u l e - s u r f a c e p o t e n t i a l energy of i n t e r a c t i o n

' M = 7 - [ - ( ^ f )

-(^)]

+

- -"[«"p(fS) «"(^T)] D

where

P

=

b /b 1

=

2

+

m /m 2

1

^ 1 and y =

a n d L a s s e t t r e (22) d e f i n e d y =

(ab/2) ). 2

( T h e definition y =

ab.

u s e d here is that of K a t o r s k i a n d W h i t e (16).

< 8 )

ab

I n a n earlier p a p e r W h i t e

Thus

Thus [/(Reference 16) —2 y/(/(Reference

22)

It s h o u l d b e n o t e d that e v e n t h o u g h t h e constants, a a n d D, of E q u a t i o n 7 c a n , to a g o o d a p p r o x i m a t i o n , b e a s s u m e d t h e same f o r a series of i s o t o p i c molecules, t h e m o l e c u l e p o t e n t i a l energy of i n t e r a c t i o n g i v e n b y (9)

differentiates

between

t h r o u g h t h e constant p.

homonuclear

a n d heteronuclear

(y is t h e same for isotopic m o l e c u l e s ) .

constant is, i n fact, s i m p l y a f u n c t i o n of A


species This

6.

HAUBACH E T A L .


1 +

77

Separations

2A b

(9)

the d i s p l a c e m e n t of the center of i n t e r a c t i o n f r o m the m o l e c u l a r center of gravity. T h e f o r m of V , E q u a t i o n 8, as a f u n c t i o n of z is d e p e n d e n t o n the m o l e c u l a r o r i e n t a t i o n . B o t h the p o t e n t i a l m i n i m a a n d the

corresponding

distance of the center of mass f r o m the surface at the m i n i m u m change


w i t h m o l e c u l a r o r i e n t a t i o n . T h u s , o n a c o m p l e t e r o t a t i o n , if the m i n i m u m energy is m a i n t a i n e d at each angle, the center of mass is d i s p l a c e d f r o m its e q u i l i b r i u m p o s i t i o n w h i c h is z = 0 at TT/2. F u r t h e r m o r e , the m a x i m u m d i s p l a c e m e n t of the center of mass o n a c o m p l e t e r o t a t i o n increases w i t h i n c r e a s i n g p. T h i s is i l l u s t r a t e d i n F i g u r e 2. T h e p o t e n t i a l , E q u a t i o n 8, is therefore not separable i n z a n d 6 (or

the nature of the c o u p l i n g

b e i n g d e t e r m i n e d b y the p a r a m e t e r p. 0.6

27T

37772

e Figure 2. Distance of molecular center of mass from surface at potential minimum as a function of molecular orientation. = 1, homonuclear molecules; = 2, HD; = 3,HT p

P

P

E v e n t h o u g h the v a r i a b l e s of the differential E q u a t i o n 6 are separable i n the exact sense, a c o n d i t i o n for s e p a r a b i l i t y c a n be

not

imposed

b y a p p l i c a t i o n of the v a r i a t i o n t h e o r e m )uQudr \u d 2

T


(10)

78

ISOTOPE E F F E C T S IN C H E M I C A L PROCESSES

w h e r e Q designates the operator o n the left h a n d side of E q u a t i o n 6 a n d u is a f u n c t i o n o b e y i n g the same b o u n d a r y c o n d i t i o n s as R. H e r m i t i a n , the q u a n t i t y E e i g e n f u n c t i o n of Q. u =

0

S i n c e Q is

takes its extreme v a l u e o n l y w h e n u is a n

If the c o n d i t i o n that u is a p r o d u c t - t y p e f u n c t i o n ,

is i m p o s e d , a p p l i c a t i o n of the c a l c u l u s of v a r i a t i o n s shows

S(z)T(rj),

that E

0

is a n extreme w h e n the f u n c t i o n s S, T satisfy the t w o

dependent

differential equations


-

GOT)

S

^

+

D e

-

2 a z

-

2

?

D

e

^

=

s

e

*

s

(

n

)

and

W h e n the f u n c t i o n s S a n d T are n o r m a l i z e d to u n i t y , t h e n the constants a, p, y , & are g i v e n b y

^ X M f f ^ M ^ ) } ™ - ' h

2

D i +

t

x

I n the limit when

(Xi/X ) ->0 2

g

SL2=

a n d as ( X i / X ) 2

S . (D )/(D )dD

/Jf " m

1

2

1

1

(35)

1

—> oo

g

i Si,

=

2

ro, j D min

S

m

1|2

(36)

j(D )dD . l

(Di)

i

T h e c a l c u l a t i o n of t h e m o n o l a y e r s e p a r a t i o n factor f r o m E q u a t i o n 34 d e p e n d s o n a k n o w l e d g e of the S i ( D ) s , w h i c h are g i v e n i n the p r e v i o u s t 2

i

,

section, a n d t h e d i s t r i b u t i o n f u n c t i o n f(D ) {

c h a r a c t e r i s t i c of t h e adsorbent.

T h e latter c a n b e o b t a i n e d f r o m e x p e r i m e n t a l isosteric heats of a d s o r p tion.

A comparison between

experimental a n d calculated

s e p a r a t i o n factors f o r h y d r o g e n s

adsorbed

monolayer

o n y - a l u m i n a is p r e s e n t e d

below. The

isosteric heats o f a d s o r p t i o n , Q , st

of e q u i l i b r i u m

hydrogen

a d s o r b e d o n y - a l u m i n a as a f u n c t i o n of surface coverage are s h o w n i n F i g u r e 7. T h e s e w e r e d e t e r m i n e d f r o m v a p o r pressure measurements i n the t e m p e r a t u r e r a n g e 5 0 ° to 80 °K. u s i n g a c a l o r i m e t e r d e s c r i b e d b y J o h n s t o n a n d K e r r ( 1 5 ) . T h e y - a l u m i n a u s e d i n these e x p e r i m e n t s w a s t h e 2 0 C r A l s a m p l e d e s c r i b e d i n References 4 a n d 5. It w a s i m p r e g n a t e d w i t h 1.1 X 1 0 " moles of C r 0 4

2

: i

to give r a p i d ortho-para e q u i l i b r a t i o n ;


6.


HAUBACH E T A L .

93

Separations

h o w e v e r , its i s o t h e r m a l b e h a v i o r at 20.4°K. ( v o l u m e p e r g r a m a d s o r b e d as a f u n c t i o n of p r e s s u r e ) w a s w i t h i n e x p e r i m e n t a l error i d e n t i c a l w i t h t h e u n i m p r e g n a t e d m a t e r i a l (4, 5 ) .

T h e difference i n e n e r g y b e t w e e n

the gas a n d the a d s o r b e d phase at the absolute zero (E

— E )

g

f u n c t i o n of the moles a d s o r b e d o n the surface N

s

a

( )

as a

is o b t a i n e d f r o m t h e

isosteric heats u s i n g the expression


( E

' -

= w X "

+

where C

n s

Isotope Effects in Chemical Processes - ACS Publications

Recommend Documents