Low-Temperature Calorimetric Study of Methane in Linde 5A Sieve

49 Low-Temperature Calorimetric Study of Methane in Linde 5A Sieve Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on May 27, 2018 | https://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0102.ch049

H. J. F. STROUD and N. G. PARSONAGE Imperial College, London, S.W. 7, England

Heat

capacities

measured taining

have

been

for empty 5A sieve and for the same sample

over the range

con

2 different

are more reliable

amounts

20°

of CH . 4

than previous

to 300°Κ These

calorimetric

measurements results for

tion systems

in that the calorimeter

is not connected

metal

tube

at room

filling

Sorption

to the

isotherms

on the same sample 300°K. model, tions.

A Monte assuming

apparatus

and isosteric

have been determined

Carlo evaluation Lennard-Jones

The isotherm

temperature.

heats of CH , C H , 4

sorp by a

2

6

from

and Kr 194°

has been made of a potentials

and qst predictions

for all for CH

4

interac are

but the Cs v s . Τ curve fails to show the large "hump" 150°Κ

which is the main feature

of the experimental

T i J " e a s u r e m e n t of Cp of the s o r b e d phase (C ) s

A

to

"cell" good, near

results.

p r o v i d e s a v e r y sensitive

test of theories b o t h because of t h e w i d e r a n g e of temperature

w h i c h c a n b e c o v e r e d a n d because of t h e nature of the q u a n t i t y itself. C a l o r i m e t r i c Cp measurements of s o r p t i o n systems a l w a y s h a v e b e e n c a r r i e d o u t w i t h a tube c o n n e c t i n g t h e s a m p l e to t h e r o o m - t e m p e r a t u r e parts of the apparatus ( 7 ). T h i s leads to large heat leaks a l o n g the t u b e a n d to u n c e r t a i n t y a b o u t t h e t e m p e r a t u r e w h i c h is to b e assumed f o r the gas w h i c h is d e s o r b e d f r o m the s a m p l e d u r i n g the experiment.

Studies

h a v e b e e n m a d e of the latter p o i n t ( 7 ), b u t i n the present w o r k w e h a v e a v o i d e d b o t h t h e leak a n d t h e a m b i g u i t i e s b y m a k i n g the s a m p l e c o n tainer a c l o s e d system, a l t h o u g h this leads to v e r y c o n s i d e r a b l e e x p e r i m e n t a l difficulties. C o r r e c t i o n f o r t h e heat associated w i t h d e s o r p t i o n i n t o t h e s m a l l a m o u n t of dead-space i n t h e s a m p l e container w a s m a d e u s i n g s o r p t i o n isotherms d e t e r m i n e d f o r the same s a m p l e of zeolite. F o r the t h e o r e t i c a l treatment of the system, w e h a v e u s e d M o n t e C a r l o ( M C ) m e t h o d s , w h i c h h a v e b e e n v e r y successful f o r t h e m a n y -

138

Flanigen and Sand; Molecular Sieve Zeolites-II Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

49.

Methane in Linde 5A Sieve

STROUD A N D PARSONAGE

139

b o d y p r o b l e m s of fluids ( 9 ) . T h e present p a p e r contains, as f a r as w e are a w a r e , t h e first a p p l i c a t i o n o f s u c h a t e c h n i q u e to a s o r b e d phase. Low—Temperature

Calorimetry

M e a s u r e m e n t s w e r e m a d e f r o m 2 0 ° to 3 0 0 ° Κ u s i n g a n a d i a b a t i c

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on May 27, 2018 | https://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0102.ch049

calorimeter.

Before each

filling

of t h e container, t h e zeolite w a s b a k e d

out f o r at least 1 w e e k w h i l e p u m p i n g o n t h e system.

T h e amount of

sorbate sealed into t h e p l a t i n u m - i r i d i u m s a m p l e container w a s deter m i n e d gasometricalry, a n d a s m a l l k n o w n a m o u n t of H e also w a s a d d e d to f a c i l i t a t e t h e r m a l e q u i l i b r i u m as is n o r m a l p r a c t i c e i n l o w - t e m p e r a t u r e c a l o r i m e t r y . Since t h e s a m p l e c o n t a i n e r was b a k e d at Τ > 300 ° C , i t w a s necessary to use glass-coated w i r e s a n d p y r o m e l l i t i m i d e v a r n i s h f o r t h e heater o f t h e s a m p l e container.

T h e zeolite w a s a s a m p l e o f 5 A sieve

g i v e n b y U n i o n C a r b i d e C o r p . a n d h a d t h e c o m p o s i t i o n : 0.23 N a 0 , 2

0.77 C a O , A 1 0 , 1.89 S i 0 . 2

3

2

Results R u n A : d e h y d r a t e d zeolite = He = +

23.4338 grams, C H

4

=

0.02597 m o l e ,

0.00026 m o l e . C o n t r i b u t i o n of sorbate t o heat c a p a c i t y o f sorbent

sorbate w a s 7 . 7 % at 5 0 ° K a n d 4 . 7 % at 2 5 0 ° K .

zeolite =

23.4338 grams, C H = 4

0.01283 m o l e , H e =

R u n B : dehydrated 0.00026 m o l e . T h e

average n u m b e r o f C H m o l e c u l e s p e r large c a v i t y ( < n > ) w a s 1.808 i n 4

Figure 1. Molar heat capacity of the sorbed CH ( C ) vs. temperature. Experimental results for = 1.808: Exp A (+). Experimental results for = 0.8933: Exp Β (Ο). Corresponding calculated results from the spherical cell model: IA and IB (for 6-12 potential), HA and IIB (for 6-18 potential). h

s


140

M O L E C U L A R

SIEVE

ZEOLITES

II

R u n A a n d 0.8933 i n R u n B . A r u n w i t h t h e same a m o u n t of zeolite a n d He

also w a s c a r r i e d o u t . O n s u b t r a c t i n g t h e latter results f r o m those

f r o m R u n s A a n d Β a n d d i v i d i n g b y t h e n u m b e r of moles of m e t h a n e , C , t h e m o l a r heat c a p a c i t y o f s o r b e d m e t h a n e f o r t h e 2 samples, w a s s

f o u n d ( F i g u r e 1, E x p . A , E x p . B ) . T h e above C values are d e r i v e d f r o m measurements f o r w h i c h t h e s


i n i t i a l c o o l i n g w a s s l o w ( f r o m r o o m t e m p e r a t u r e to 8 0 ° K i n — 4 0 h o u r s ) . For

l o w e r temperatures the f u r t h e r c o o l i n g w a s faster, e.g., 8 0 ° to 5 0 °

or 2 0 ° K over 100 m i n u t e s . I n one set of experiments of R u n A ( n o t i n c l u d e d i n F i g u r e 1 ), the s a m p l e was " q u e n c h e d " d e l i b e r a t e l y f r o m , ~ 3 0 0 ° to — 8 0 ° K i n ^ 3 0 minutes.

Cp measurements

were then n o r m a l until

~ 9 8 ° K , a b o v e w h i c h large, w a r m i n g drifts w e r e o b s e r v e d ( ^ - 0 . 0 0 1 5 d e g minute" ).

A t 1 1 2 . 6 ° K , t h e d r i f t r e m a i n e d a p p a r e n t l y constant over 2

1

hours. T h i s b e h a v i o r is s i m i l a r to that f o u n d w h e n a system is " f r o z e n " first

i n t o a metastable

state, f r o m w h i c h i t escapes w i t h e v o l u t i o n of

energy w h e n t h e r m a l excitations b e c o m e sufficiently large to p e r m i t i t . A n estimate of the a m o u n t of f r o z e n - i n energy w a s m a d e b y h e a t i n g a q u e n c h e d s a m p l e f r o m 9 6 . 6 ° K , b e l o w t h e t e m p e r a t u r e at w h i c h a n n e a l i n g occurs, to 1 2 3 . 9 ° K , at w h i c h r a p i d e q u i l i b r a t i o n occurs. T h e a m o u n t of heat r e q u i r e d w a s c o m p a r e d w i t h the a m o u n t c a l c u l a t e d f r o m t h e true Cp c u r v e . T h i s gave ^ 5 . 4 0 K J ( m o l e C H ) 4

f o r the f r o z e n - i n heat.

_ 1

T h e t r i v i a l e x p l a n a t i o n that t h e " q u e n c h i n g " p h e n o m e n a

can be

a s c r i b e d to s p a t i a l i n h o m o g e n e i t y of t h e C H , i n d u c e d b y the t e m p e r a 4

ture g r a d i e n t d u r i n g c o o l i n g , a n d subsequent r e l a x a t i o n c a n b e d i s m i s s e d o n t h e f o l l o w i n g g r o u n d s . F o r this e x p l a n a t i o n to b e possible, i t w o u l d b e necessary f o r t h e g r a p h of q

st

vs. < n >

to b e s h a r p l y concave to t h e

axis. I n fact, o u r measurements s h o w e d that q

st

was independent

of c o m p o s i t i o n over t h e range of o u r i s o t h e r m measurements. even if q

st

vs. < n >

Indeed,

w a s as c u r v e d as i t is f o r C H , i t w o u l d b e v i r t u a l l y 2

6

i m p o s s i b l e to e x p l a i n the m a g n i t u d e of o u r observations i n this w a y . In

n o r m a l experiments, u s i n g the s l o w - c o o l i n g m e t h o d , 2 regions

w e r e f o u n d f o r w h i c h t h e rate of e q u i l i b r a t i o n after heat i n p u t w a s u n u s u a l l y l o n g . T h e s e w e r e near 3 0 ° K (^~50 m i n u t e s ; n o r m a l , ~ 1 0 m i n u t e s ) a n d at 1 5 5 ° ~ 2 0 0 ° K ( ^ 5 0 m i n u t e s ; n o r m a l , — 1 5 m i n u t e s ) .

I n the

latter r e g i o n , this caused a scatter i n the total Cp of u p to 0 . 4 % . Sorption Isotherm

Results

F o r < n > b e t w e e n 0.2 a n d 3, q w a s 17.76 a n d 22.36 K J m o l e " f o r K r a n d C H , r e s p e c t i v e l y , these results b e i n g i n d e p e n d e n t of c o m p o s i t i o n a n d t e m p e r a t u r e . F o r C H , o n the other h a n d , t h e b e h a v i o r w a s m o r e c o m p l e x . T h u s , b e t w e e n 2 3 0 ° a n d 2 8 0 ° K , i t w a s 25.19, 29.37, 32.26, a n d 34.10 K J m o l e " at < n > = 1, 2, 3, a n d 4, r e s p e c t i v e l y . st

4

2

6

1


1

49.


STROUD AND PARSONAGE

141

Discussion B a k a e v ( I ) has s h o w n h o w t h e G r a n d P a r t i t i o n F u n c t i o n f o r t h e s o r b e d phase m a y b e expressed i n terms o f the C a n o n i c a l P a r t i t i o n F u n c tions f o r single cavities c o n t a i n i n g 1,2,3 . . . , n m o l e c u l e s .

T h e o n l y as

s u m p t i o n necessary is that there is n o i n t e r a c t i o n b e t w e e n particles i n


different cavities.

T h e p r o b l e m reduces itself to that o f e v a l u a t i n g t h e

c o n f i g u r a t i o n integrals, Z Spherically

Symmetric

M

= f

e~ / d.T . u kT

n

Model

B e c a u s e o f the h i g h s y m m e t r y o f the m a i n cavities o f sieve A , i t m i g h t seem reasonable, i n t h e s p i r i t o f the c e l l t h e o r y of l i q u i d s ( 5 ) , to assume that t h e v a r i o u s c h a r g e d species o f t h e zeolite a r e d i s t r i b u t e d w i t h c o m plete s p h e r i c a l s y m m e t r y a b o u t the center o f e a c h c a v i t y . T h i s , of course, leads t o zero electric field t h r o u g h o u t t h e c a v i t y a n d h e n c e t o zero c o n t r i b u t i o n f r o m i o n - i n d u c e d d i p o l e a n d s i m i l a r forces.

T h e remaining

terms, d i s p e r s i o n a n d r e p u l s i o n , are a s s u m e d to b e o f t h e L e n n a r d - J o n e s f o r m a n d a r e t a k e n to b e a d d i t i v e . T h i s t h e o r y t h e n becomes essentially the same as t h e c e l l t h e o r y o f l i q u i d s , except that w e p e r m i t several m o l e c u l e s p e r c e l l . T h e t o t a l P . E . is g i v e n as U

= Χφ(η) +

LJ

i

2 ti(r ), i ;

i>j

w h e r e φ(τι) is t h e P . E . f o r t h e i t h m o l e c u l e - w a l l i n t e r a c t i o n w h e n t h e m o l e c u l e is r f r o m t h e center a n d u(rij) is t h e P . E . f o r t h e i n t e r a c t i o n {

b e t w e e n t h e i t h a n d ; t h particles. T h e d i f f i c u l t process n o w is to evaluate the Z . T h e I m p o r t a n c e S a m p l i n g M o n t e C a r l o M e t h o d (6,9) is r e l e v a n t n

to this p r o b l e m , b u t i t c a n y i e l d values o n l y f o r average quantities s u c h as < X > = fx

exp ( - U j / k T ) d r / f L

n

exp ( —U j/kT)dr ; L

n

i t does n o t

give, i n p a r t i c u l a r , a v a l u e f o r t h e d e n o m i n a t o r itself. T o o v e r c o m e this, w e h a v e u s e d a c o u p l i n g p a r a m e t e r m e t h o d ( 3 ) . W e i n t r o d u c e cut-offs i n t o the p o t e n t i a l s u c h that i f either a n y 2 particles a r e less t h a n r apart 0

or a n y p a r t i c l e is less t h a n r ' f r o m the w a l l , t h e n P . E . = 0

P . E . as C7(£) — UHS + éULJ, w h e r e UH8 = 0 i f a l l r ^ ro b u t U

H8

Zn(()

=f

fu

L J

4i

^ r0 a n d ( a - n )

oo o t h e r w i s e , w e are interested i n U(l); o r , p u t t i n g

exp (-UU)/kT)dr ,

that l n Z ( l ) w

=

oo. W r i t i n g t h e

n

= lnZ (0) H

exp {-U{è)/kT)drJ/exp

- f*

w e require Z ( l ) . n

LJ

It can be shown

d£/kT w h e r e

è

LJ

(-U(é)/kT)dr

n

è

—

· Ζ „ ( 0 ) , w h i c h re

fers t o t h e f r a c t i o n o f a l l configurations o f t h e h a r d spheres w h i c h are allowed, was evaluated numerically.

ç, LJ

was evaluated b y the

a b o v e M o n t e C a r l o m e t h o d f o r a b o u t 10 values of ξ b e t w e e n 0 a n d 1, t h e n u m b e r o f t r i a l moves r a n g i n g f r o m 98,000 f o r 5 particles t o 46,000 f o r 8 particles. F o r cells w i t h 4 o r less particles, the f u n c t i o n s w e r e e v a l u a t e d


142

M O L E C U L A R SIEVE

ZEOLITES

d i r e c t l y b y a s i m p l e M C m e t h o d . T h e p o t e n t i a l parameters f o r C H - C H 4

Π

4

w e r e those of Ref. 4. F o r C H - 0 , the attractive p o t e n t i a l w a s c a l c u l a t e d 4

b y the S l a t e r - K i r k w o o d e q u a t i o n ( 8 ) , a n d the r e p u l s i v e coefficient was c h o s e n so as to m a k e the p o t e n t i a l m i n i m u m o c c u r at the v a n d e r W a a l s ' distance.

F o r the c o m p o s i t i o n s of R u n s A a n d B , q

st

to 20.33 K J m o l e " over the range 1 0 0 ° to 3 0 0 ° K . Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on May 27, 2018 | https://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0102.ch049

AH

are

1.6 a n d ^ 2 . 0

decreases f r o m 21.01 T h e errors i n AG a n d

1

K J m o l e " , r e s p e c t i v e l y , at 3 0 0 ° K . 1

satisfactory, as w e h a v e not u s e d a n y adjustable parameters. c u l a t e d curves of C vs. Τ f o r < n >

=

s

1.808 a n d 0.8933 ( F i g u r e

T h i s is The cal Ι,ΙΑ,ΙΒ)

are, h o w e v e r , c l e a r l y of the w r o n g f o r m . T h e peaks p r e d i c t e d near 6 0 ° Κ m a r k d i s p r o p o r t i o n a t i o n l e a d i n g to a m i x t u r e of e m p t y a n d f u l l cells. E v e n i f this process is t h e r m o d y n a m i c a l l y f a v o r e d , i t w o u l d be u n l i k e l y to o c c u r at s u c h l o w temperatures w i t h i n a reasonable t i m e . A l s o s h o w n are the curves a s s u m i n g a 6 - 1 8 p o t e n t i a l f o r C H - C H 4

( 2 ) , a n d corre

4

s p o n d i n g 6 - 1 5 potentials f o r C H w i t h the w a l l ( F i g u r e

Ι,ΙΙΑ,ΙΙΒ).

4

T h e S p h e r i c a l l y S y m m e t r i c M o d e l is c l e a r l y unsatisfactory f o r p r e dicting C

s

values. C a l c u l a t i o n s are n o w i n progress w i t h the N a , C a , O ,

S i , a n d A l atoms treated as c h a r g e d p o i n t centers of force. T h e s m a l l h u m p s near 6 0 ° Κ i n the e x p e r i m e n t a l C

s

vs. Τ curves m a y

result f r o m escape of H e f r o m s o l u t i o n i n the s o r b e d C H . T h e r u n w i t h 4

e m p t y zeolite s h o w e d a s i m i l a r h u m p near 30 ° K , w h i c h w e a s c r i b e d to d e s o r p t i o n of h e l i u m . Literature (1) (2) (3) (4) (5) (6) (7) (8) (9)

Cited

Bakaev, V. Α., Dokl. Akad. Nauk SSSR 1966, 167, 369. Byrne, Μ. Α., Jones, M. R., Staveley, L. A . K . , Trans. Faraday Soc. 1968, 64, 1747. H i l l , T . L., "Statistical Mechanics," p . 180, M c G r a w - H i l l , N e w York, 1956. Hirschfelder, J. O . , Curtiss, C. F., B y r d , R. B., "Molecular Theory of Gases and L i q u i d s , " p. 1110, W i l e y , N e w York, 1954. Lennard-Jones, J. E., Devonshire, A . F., Proc. Roy. Soc., Ser. A 1937, 163, 53. Metropolis, N., Rosenbluth, A . W . , Rosenbluth, M. N., Teller, A . H., Teller, E., J. Chem. Phys. 1953, 21, 1087. Pace, E . L., "The Solid-Gas Interface," E . A . F l o o d , Ed., V o l . 1, C h . 4, Dekker, N e w York, 1966. Pitzer, K . S., Advan. Chem. Phys. 1959, 2, 59. W o o d , W . W . , "The Physics of Simple L i q u i d s , " Η. Ν. V . Temperley, J. S. Rowlinson, G . S. Rushbrooke, E d s . , C h . 5, N o r t h H o l l a n d , Amster dam, 1968.

R E C E I V E D February 4,

1970.

Discussion K . K l i e r ( L e h i g h University, Bethlehem, Pa. 18015): The

authors

h a v e chosen the c e l l m e t h o d , w h i c h seems to b e q u i t e successful i n ex-


49.

STROUD A N D PARSONAGE

143


p l a i n i n g p r o p e r t i e s of l i q u i d s i n the range of densities w h i c h a d s o r b i n g m o l e c u l e s a t t a i n i n the zeolite cavities. It w o u l d be i n t e r e s t i n g to k n o w w h e t h e r , c o m p l e m e n t a r y to the M o n t e C a r l o m e t h o d u s e d , a n a l y t i c a l solutions c o u l d be o b t a i n e d i n terms of p a r t i t i o n f u n c t i o n b e i n g a f u n c t i o n of average d e n s i t y of the sorbate. v a l u e of o b t a i n i n g t h e i s o t h e r m as ρ =

S u c h solutions w o u l d h a v e t h e —kT(d l o g [ p . f . ] / ô u ) . T h e m o d e l r


suggested here w o u l d b e to assign one c e l l to one m o l e c u l e w i t h i n the c a v i t y , n o t to the c a v i t y itself. N . G . Parsonage: I c a n see n o p o s s i b i l i t y of o b t a i n i n g a n a n a l y t i c a l expression f o r the p a r t i t i o n f u n c t i o n f o r cavities c o n t a i n i n g s e v e r a l m o l e cules.

F o r cells c o n t a i n i n g η m o l e c u l e s , w e w o u l d n e e d to integrate

c o m p l i c a t e d f u n c t i o n over ~ 3 n coordinates.

a

Lennard-Jones and D e v o n

shire c o n s i d e r e d o n l y the case of one m o l e c u l e p e r c e l l , a n d for this n e e d e d o n l y one c o o r d i n a t e , the r a d i a l distance.

E v e n so, t h e y h a d to

evaluate t h e i r integrals n u m e r i c a l l y . If one w e r e to redefine the " c e l l s " so that e a c h c o n t a i n e d o n l y one m o l e c u l e , t h e n one w o u l d find great d i f f i c u l t y since the n e w " c e l l s " w o u l d n o t c o r r e s p o n d w i t h the r e a l cavities. T h e a d v a n t a g e of the present treat ment, that the cells h a v e r e a l p h y s i c a l existence, w o u l d b e lost. H . A . Resing ( N a v a l R e s e a r c h L a b o r a t o r y , W a s h i n g t o n , D . C . 20390) : W h a t d o y o u r experiments s h o w a b o u t m o l e c u l a r rotation? N . G . Parsonage: T h e e x p e r i m e n t does not d i s t i n g u i s h b e t w e e n r o t a t i o n a l a n d other c o n t r i b u t i o n s to the heat c a p a c i t y .

Information on ro

t a t i o n a l f r e e d o m m u s t d e p e n d o n a t h e o r e t i c a l analysis of the d a t a .

In

o u r treatment, w e assume that r o t a t i o n is free over the w h o l e t e m p e r a t u r e range a n d so gives a c o n t r i b u t i o n of 3 / 2 R. T h i s m u s t be w r o n g at v e r y l o w temperatures, b u t the e v i d e n c e f r o m the rather s i m i l a r system m e t h a n e — q u i n o l clathrate, f o r w h i c h a g o o d analysis is a v a i l a b l e , is that the r o t a t i o n remains v i r t u a l l y free d o w n to 50 ° K .

A t this t e m p e r a t u r e , o u r

treatment w i l l a l r e a d y be f a i l i n g w i t h respect to other degrees of f r e e d o m because i t is classical a n d so does not s h o w the q u a n t u m fall-off of the heat c a p a c i t y .


Low-Temperature Calorimetric Study of Methane in Linde 5A Sieve

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