RESONANCE ABSORPTION OF AN ULTRASONIC WAVE IN A WEAKLY ASSOCIATED LIQUID

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RESONANCE ABSORPTION OF AN ULTRASONIC WAVE IN A WEAKLY ASSOCIATED LIQUID

A. Narasimham

To cite this version:

A. Narasimham. RESONANCE ABSORPTION OF AN ULTRASONIC WAVE IN A WEAKLY ASSOCIATED LIQUID. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-223-C8-230.

�10.1051/jphyscol:1979838�. �jpa-00219543�

RESONANCE ABSORPTION OF AN ULTRASONIC WAVE IN A WEAKLY ASSOCIATED LIQUID A.V. Narasimham

department of Physios, Indian Institute of Technology, Madvas-600 036, INDIA.

Abstract.- A theory has been developed for absorption and dispersion of an ultrasonic wave in a weakly associated liquid, taking into account resonance absorption of the wave. The state of the liquid has been described by a distribution function. The effect of the wave on the distribution function has been considered by introducing a term Fx/kT, where F is the force exerted by the wave on a molecule, x is the displacement of the molecule under the action of the force and F = op, where a is the area of cross-section of the molecule and p is the pressure due to the ultrasonic wave. The distribution function, in the presence of the wave, is evaluated in terms of the relaxa- tion time, the constants of the wave, the mass, area of cross-section and natural frequency of the molecules and the distribution in the absence of the wave. It is found that the molecular density

in the presence of the wave can be represented by a power series in p (pressure due to the wave).

This results in a non-linearity in the compressibility of the liquid. This non-linearity exists even after neglecting terms of higher order. Hence it turns out that the compressibility and hence the absorption and dispersion of an ultrasonic wave in a liquid are non-linear, with dependence on the amplitude of the pressure wave. This dependence is used to evaluate the relaxation time and na- tural vibrational frequency of the molecules about their equilibrium positions. It is found that the relaxation time depends both upon velocity and absorption and their pressure variation while the natural frequency depends upon only velocity and its pressure variation. Agreement of calcu- lated values with available values is found to be fairly good.

1. INTRODUCTION.- Absorption of an ultrasonic wave in a liquid can take place in two ways : (1) Ab- sorption due to displacement of the molecules, bound elastically to their equilibrium positions.

Due to such displacement, the molecules execute oscillations, with their natural frequency of vi- bration ( fQ) . As the frequency of the applied ul- trasonic wave approaches the frequency of natural vibrations, the absorption of the wave becomes ve- ry high. This absorption is termed as "resonance absorption" and has its maximum value at a fre- quency of the wave, equal to the resonant frequen- cy (or natural frequency of vibration) of the mole- cules of the medium.(2) The second type of absorp- tion is due to transitions of molecules between two equilibrium positions separated by a potential

barrier. These transitions are usually described b y a certain transition probability and a relaxa- tion time ( T ) . This is known as "relaxation ab- sorption" and has its maximum value at a frequen- cy of the wave, equal to 1/2 TTT, or when W T = W x = l Resonance absorption occurs at frequencies much higher than w„, i.e. f « f = ( = w /2ir ) . Usual-ly r r o o ly w x » 1. The relaxational type of absorption is due to time lag between the application of the wave and the response of the medium. Resonant ab- sorption takes place, when the vibrations of the molecules are in phase with the pressure varia- tions of the wave.

2. THEORETICAL CONSIDERATIONS.- Let us consider a non (or weakly) - associated liquid, so that we Résumé.- On présente une théorie sur l'absorption et la dispersion d'une onde ultrasonore dans un liquide faiblement associé en prenant en compte l'absorption résonante de l'onde. L'effet de l'onde sur la fonction de distribution est pris en compte par l'introduction d'un terme Fx/kT, où F est la force excercée par l'onde sur une molécule, x le déplacement de la molécule sous l'action de la for- ce et F = op, où a est T a i r e de la section de la molécule et p la pression due à Tonde ultrasono- re. La fonction de distribution en présence de Tonde, est évaluée en fonction du temps de relaxa- tion, de la fréquence naturelle des molécules et de la distribution en l'absence de Tonde. On trou- ve que la densité moléculaire en présence de Tonde peut être décrite par un développemement en puissance de p. Il en résulte une non linéarité dans la compressibilité du liquide, cette non linéa- r i t é subsiste si Ton néglige les termes d'ordre les plus élevés. En conséquence, la compressibili- t é , l'absorption et la dispersion d'une onde ultrasonore dans un liquide sont non linéaires (elles dépendent de l'amplitude de la pression de Tonde). Cette dépendance est utilisée pour évaluer le temps de relaxation et la fréquence de vibration naturelle des molécules autour de leur position d'équilibre. On trouve que le temps de relaxation dépend à la fois de la vitesse et de l'absorption et de leur variation avec la pression tandis que la fréquence naturelle ne dépend que de la vitesse et de sa variation avec la pression. L'accord des valeurs calculées avec les résultats connus est satisfaisant.

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979838

~ 8 - 2 2 4 JOURNAL DE PHYSIQUE

can n e g l e c t t h e mutual i n t e r a c t i o n s between t h e various molecules o f t h e l i q u i d . This means t h a t each molecule i s a f f e c t e d o n l y by the a p p l i e d pres- sure and n o t by t h e neighbouring molecules. I n the absence o f any e x t e r n a l force, a molecule executes n a t u r a l v i b r a t i o n s about i t s e q u i l i b r i u m p o s i t i o n and i t s equation o f motion i s given by

The s o l u t i o n o f equation (1) can be w r i t t e n as

where x i s t h e displacement o f t h e molecule from i t s e q u i l i b r i u m p o s i t i o n , a t any time t, and (1/2) mA 2 wo2 i s the energy o f t h e molecule. I f und.istur- bed, t h e molecule continues w i t h t h i s energy f o r any l e n g t h o f time. However, the molecule has d i f - f e r e n t e q u i l i b r i u m p o s i t i o n s corresponding t o d i f - f e r e n t energy s t a t e s and i t s energy changes c o n t i - nuously. Thus, i t can be assumed t h a t t h e molecu- l e s s u f f e r c o l l i s i o n s w i t h molecules o f t h e sur- rounding medium, r e s u l t i n g i n a change i n t h e ener- gy o f a molecule and the establishment o f thermal e q u i l i b r i u m , due t o exchange o f energy between the molecules.

I n t h e presence o f an e x t e r n a l f o r c e (F), t h e equation o f motion o f t h e molecule can be w r i t - t e n as,

where "mu i s t h e mass o f t h e molecule and "a" i s i t s a c c e l e r a t i o n . A t any i n s t a n t t, l e t x and v ( = d x / d t ) represent t h e p o s i t i o n and v e l o c i t y o f a molecule. L e t

0

(x, v ) represent a d i s t r i b u t i o n f u n c t i o n such t h a t , a t the i n s t a n t t, 0 (x, v) dx dv represents t h e number o f molecules p e r u n i t vo- lume, w i t h t h e i r displacements between x and x + d x , and v e l o c i t i e s between v and v

+

dv. Hence, i f N i s t h e t o t a l number o f molecules p e r u n i t volume ( i .e. molecular d e n s i t y ) i n t e g r a t i n g over a l l va- l u e s o f x and v, we get,

I +- I ^fl

( x , v) dx dv = N

-CO

since a1 1 t h e molecules should have t h e i r p o s i t i o n and v e l o c i t y coordinates between

-

^and

+

a t any time t. The d i s t r i b u t i o n f u n c t i o n and the mole- c u l a r d e n s i t y N depend upon t h e c o n d i t i o n s o f t h e medium ( f o r example : whether any e x t e r n a l f o r c e i s a c t i n g upon the medium o r n o t e t c ) . The energy o f a molecule, executing harmonic o s c i l l a t i o n s w i t h a frequency fo about i t s e q u i l i b r i u m p o s i t i o n , i s given, i n t h e absence o f any e x t e r n a l force, by Eo, where

I n the presence o f an e x t e r n a l f o r c e (F), an amount o f work, equal t o Fx, i s done by t h e f o r c e on the molecule, and t h i s energy i s s t o r e d as po- t e n t i a l energy i n the molecule. Hence t h e t o t a l energy o f t h e molecule, i n the presence o f the ex- t e r n a l f o r c e F, i s given by E where E = Eo

+

Fx o r

Assuming Boltzman's d i s t r i b u t i o n f o r t h e d i s t r i b u - t i o n f u n c t i o n 0 (x, v), i n equation (5), we have

0

( x , V ) = B exp ( - E / k t ) (8) where B i s the constant (independent o f x and v) which can be determined by s u b s t i t u t i n g equation ( 8 ) i n equation (5), k i s t h e Boltzman's constant, T i s the absolute temperature and bF i s t h e d i s - t r i b u t i o n f u n c t i o n , under t h e a t i o n o f e x t e r n a l force, F. Equation ( 8 ) a p p l i e s t o e q u i l i b r i u m con- d i t i o n s , s i n c e Boltzman's d i s t r i b u t i o n a p p l i e s t o equi 1 i brium c o n d i t i o n s . Substi t u t i n q equation (7) i n equation ( 8 ) , we have

B exp ( - Eo/kT) exp ( - Fx/kT) (9)

gF = Bo exp ( - Fx/kT) (10)

where go i s t h e value o f BF f o r F = 0 and i s given by

Expanding equation ( 9 ) , we have

considering t h e case o f a weak e x t e r n a l force, i .e.

i f Fx << kT, we have, n e g l e c t i n g t h e cube and h i - gher powers o f (Fx/kT), i n equation (12),

I n t e g r a t i n g over a l l p o s s i b l e values o f x and v, we have

The i n t e g r a l on t h e l e f t hand s i d e i s t h e molecular d e n s i t y i n t h e presence o f e x t e r n a l f o r - ce F and l e t i t be designated by NF. Likewise, the f i r s t i n t e g r a l on t h e r i g h t hand s i d e i s designa- t e d by No, t h e molecular d e n s i t y i n t h e absence o f t h e e x t e r n a l f o r c e F. The second i n t e g r a l on the r i g h t hand s i d e vanishes s i n c e go i s an even func- t i o n o f x and x takes a l l values from

-

^{m t o}

+

For e v a l u a t i n g the l a s t i n t e g r a l on t h e r i g h t hand side, o f equation (14) we use equation ( l l ) , from which we g e t t h a t

Hence we have

B u t

Since go vahishes b o t h f o r x = +m and x = ac- cording t o equation (11). Thus we have from equa- t i o n s (14) t o (17) and ( 5 )

Hence, u s i n g equation ( 5 ) we have

where NF

-

No i s t h e change i n t h e e q u i l i b r i u m molecular d e n s i t y , due t o t h e a c t i o n o f an e x t e r - n a l f o r c e F. I t should be remembered t h a t equations ( 8 ) t o (19) represent equi 1 i brium c o n d i t i o n s

.

So f a r , F has been considered t o be constant, i .e. independent o f t i m e and p o s i t i o n . We can now consider t h e a c t i o n o f an u l t r a s o n i c ( o r apressure) wave, represented by

p = po exp ( i w t ) exp ( - i K x ) (20)

L e t o be t h e e f f e c t i v e area o f c r o s s - s e c t i o n o f a molecule, exposed t o t h e pressure wave, assuming t h e molecule t o be s p h e r i c a l

.

^Then,

F = o po exp ( i w t ) exp ( - i K x ) (21)

where K i s the wave v e c t o r = 2a/X = w/V and V i s t h e v e l o c i t y o f t h e u l t r a s o n i c wave. p, po, i

,

w, t and x have t h e usual meaning. The e x t e r n a l f o r c e F i s now a l t e r n a t i n g i n time t (and a l s o i n x ) . Hence t h e e q u i l i b r i u m d i s t r i b u t i o n bF (equations 8 t o 14) a l s o a l t e r n a t e s a t t h e frequency o f t h e wave. Thus, a t any i n s t a n t , t h e instantaneous ex-

t e r n a l f o r c e i s given by equation (21) and an equi- 1 i brium d i s t r i b u t i o n corresponding t o t h i s value o f F, i s s e t up w i t h a time lag, which i s indepen- dent o f x,v o r t. Thus, t h e e q u i l i b r i u m d i s t r i bu- t i o n a l s o i s a f u n c t i o n o f time, due t o t h e impres- sed u l t r a s o n i c wave and has t h e same frequency as t h e frequency o f t h e wave, w i t h s i m i l a r v a r i a t i o n s . It i s now r e q u i r e d t o determine t h e time-dependent d i s t r i b u t i o n f u n c t i o n 0, due t o t h e a c t i o n o f t h e wave, so t h a t

0

approaches

flF,

a f t e r t h e a p p l i c a - t i o n o f t h e e x t e r n a l f o r c e F. Hence l e t us assume t h a t

~ 8 - 2 2 6 JOURNAL DE PHYSIQUE

The d i s t r i b u t i o n represented by equation (13), i .e.

gF, represents e q u i l i b r i u m d i s t r i b u t i o n . The value o f F, given by equation (21), a t any i n s t a n t t, can be considered t o be t h e value d u r i n g t h e time r e q u i r e d f o r t h e establishment o f e q u i l i b r i u m . Du- r i n g t h i s time 0 + bF, where OF, 0 and F are given by t h e equations (13), (21) and ( 2 2 ) . There i s a time l a g between t h e a p p l i c a t i o n o f t h e f o r c e and t h e establishment o f e q u i l i b r i u m , corresponding t o t h a t f o r c e . During t h i s period, t h e p o s i t i o n s ( x ) and v e l o c i t i e s ( v ) o f the molecules o f the medium change continuously. I n a d d i t i o n t o t h i s , t h e r e i s a l s o a s i n u s o i d a l v a r i a t i o n o f t h e e x t e r n a l f o r c e w i t h time, due , t o t h e a p p l i e d u l t r a s o n i c wave. On removal o f the f o r c e F, both 0 and gF should r e t u r n t o 0,. Hence, n e g l e c t i n g h i g h e r order terms, 0 (x,v,t,) can be expressed as

BF and yF 2 may be considered as small a d d i t i o n s t o t h e e q u i l i b r i u m d i s t r i b u t i o n go ( f o r F = 0). A t time, t = 0, F = 0, and 0 = go. A f t e r t h e a p p l i - c a t i o n o f F, as 0 approaches 0 (BF

+

yF 2 ) approa- ches Po [(- Fx/kT)

+

(Fx/kTli], according t o equations (13) and (23). Comparing the c o e f f i c i e n t s o f F and F2, B approaches ( - g0x/kT) and y ap- proaches -2 1 0, ( x / k ~ ) ' . Eventhough time does n o t e n t e r B and y e x p l i c i t l y . T h e i r time-dependence i s r e f l e c t e d i n t h e time-dependence o f x and v.

A small change i n 0 can be expressed as

Hence, t h e t i m e - v a r i a t i o n o f can be w r i t t e n as

2 2

where v ( = d x / d t ) and a ( = d v / d t = d x / d t ), a r e the v e l o c i t y and a c c e l e r a t i o n o f a molecule. 6 0 / 6 t i s t h e c o n t r i b u t i o n t o t h e t i m e - v a r i a t i o n o f 0, w i t h o u t t a k i n g i n t o account t h e changes i n t h e po- s i t i o n and v e l o c i t y coordinates o f molecules, i .e.

w i t h o u t t a k i n g i n t o account t h e i r motion. Such a c o n t r i b u t i o n can be brought by c o l l i s i o n s , which h e l p t h e establishment o f e q u i l i b r i u m . I t can be assumed t h a t 6 0 / 6 t , due t o c o l l i s i o n s , i s propor- t i o n a l t o ( 0

-

^{gF), i} .e. d i f f e r e n c e between i n s t a n - taneous and e q u i l i b r i u m d i s t r i b u t i o n s . Since t h i s d i f f e r e n c e goes on decreasing, as e q u i l i b r i u m i s

approached, 6 0 / 6 t i s negative. Hence we have,

where T i s a r e l a x a t i o n time, independent o f x,v o r w, and 0 and gF are given by equations (23) and ( 1 3 ) . This means t h a t e q u i l i b r i u m i s approached e x p o n e n t i a l l y w i t h time, w i t h a s i n g l e r e l a x a t i o n time.

The second and t h i r d terms i n t h e r i g h t hand s i d e o f equation (25) g i v e the c o n t r i b u t i o n t o d 0 / d t , due t o motion o f molecules, w i t h o u t t a k i n g i n t o account t h e e f f e c t o f c o l l i s i o n s .

Using equations (13) and (23) f o r bF and 0 we have

we a l s o have, using equation ( 3 ) f o r a ( = d v / d t ) ,

i t may be noted t h a t F i s independent o f v. Hence F/ v does n o t e x i s t : u s i n g equation (11) we g e t

Considering t h e l e f t hand s i d e o f equation (25), we have

Since i s independent o f time.

dF i s a small change i n t h e f o r c e t h a t i s a c t i n g on a p a r t i c l e d u r i n g a time d t . But, d u r i n g t h i s small i n t e r v a l o f time dt, t h e molecule i s displaced through an e x t e n t equal t o dx. The f o r c e a c t i n g on t h e p a r t i c l e , v a r i e s w i t h i t s p o s i t i o n also. Hence, t h e t o t a l change dF i s g i v e n by

where v ( = d x / d t ) i s t h e v e l o c i t y o f t h e p a r t i c l e . I t should be noted t h a t dx i s a small displacement o f t h e p a r t i c l e and n o t t h e d i s t a n c e t r a v e l l e d by t h e wave. We a l s o have,

s i m i l a r l y ,

2

t ^{= v}

⁹

6x

⁺

^a

*

6v (36) S u b s t i t u t i o n of equations (33), (35) and (36) i n equation (31) gives us,

S u b s t i t u t i n g equations (27), (28) and (29) i n equa- t i o n (25), using equation (30) and comparing t h e r e s u l t i n g equation w i t h equation (37) we have

Using equation (21) f o r F and equating t h e c o e f f i - c i e n t s o f F and F', we have

Hence we get,

and

now, using equation (23), we have, w i t h t h e h e l p o f equation ( 5 )

The i n t e g r a l on t h e l e f t hand s i d e represents t h e time-dependent molecular d e n s i t y ( o r number o f mo- l e c u l e s p e r u n i t volume) d u r i n g t h e establishment o f equi 1 i b r i um, i .e. under dynamic c o n d i t i o n s . L e t t h i s be denoted by NFt. Hence, we have,

6 and y are now known f u n c t i o n s o f x and v, accor- d i n g t o equations (41) and (42). However, we have

since go i s an even f u n c t i o n i n v. Using equation (15), we a1 so have

since go vanishes b o t h a t x = +m and x = ^-m. Thus u s i n g equations ( 5 ) , (11) and (21) and (41) and some p r o p e r t i e s o f F o u r i e r transforms, we have

+== i K a Po e i w t

I

^{BF dx dv =}

^{( r )} ₊

_{l w T}

-

_{2 exp}

[- <]

-m wo 2m wn

Again we have, u s i n g F o u r i e r transforms, s i n c e go i s even i n x and v,

where

S u b s t i t u t i n g equations (47) t o (49) i n equation (43) we have f o r w

2

wo

~ 8 - 2 2 8 JOURNAL DE PHYSIQUE

-14

.

and kT i s = 5 x 10

.

8 13

and w ( = 10 ) << wo ( = 10 ) . Hence, we have kTK /m wo2 << 2 1.

Hence equation (50) can be w r i t t e n as

where p = p exp ( i w t ) and represents t h e v a r i a -

0

b l e pressure. No i s t h e molecular d e n s i t y i n t h e absence o f the wave and i s independent o f p.

D i f f e r e n t i a t i n g w i t h respect t o p, we have

(K a l s o v a r i e s s l i g h t l y w i t h pressure. But

p ( d K/d p ) << K. Pressure v a r i a t i o n o f T a l s o can be neglected upto n e a r l y 100 atmosph@res, Referen- ce : J . Frenkel

,

K i n e t i c theory o f l i q u i d s ) .

dNFt i s a small change i n t h e molecular densi- t y due t o t h e a c t i o n o f t h e wave ( i . e . e x t e r n a l pressure p) and No i s t h e molecular d e n s i t y i n t h e absence o f t h e wave. Since t h e d e n s i t y o f the li- q u i d can be expressed as number o f molecules per u n i t volume x mass o f a molecule, we have t h a t t h e r e l a t i v e change i n molecular d e n s i t y = r e l a t i v e change i n a c t u a l d e n s i t y (p)

Hence,

where dV', i s t h e change i n t h e volume o f a mass (M) o f t h e l i q u i d and V; i s t h e volume i n t h e ab- sence o f e x t e r n a l pressure and V ' i s t h e volume o f t h i s mass o f l i q u i d i n t h e presence o f a p p l i e d pressure p.

where X+ i s t h e complex c o m p r e s s i b i l i t y o f t h e me- d i um

.

Hence,

Thus the c o m p r e s s i b i l i t y contains a term propor- t i o n a l t o t h e a p p l i e d pressure. This i s a r e s u l t o f t h e existence o f second order term i n t h e d i s - t r i b u t i o n f u n c t i o n . Consideration o f s t i l l h i g h e r o r d e r terms i n equation (12) leads t o s t i l l more dependance o f X on pressure. However, t h i s e f f e c t i s n e g l i g i b l e s i n c e Fx/kT << 1.

The f i r s t o r d e r and second order terms are compa- r a b l e t o each o t h e r and n e i t h e r o f them can be ne- glected, s i n c e K ( = w/V) can be n e a r l y equal t o 10 a t t h e present u l t r a s o n i c frequencies and 3 ( a p/kT) >

lo4.

I t i s found t h a t t h e t h i r d order, f i f t h o r d e r e t c . , terms can be neglected i n compa- r i s o n w i t h f i r s t order term, w h i l e t h e f o u r t h o r - der, s i x t h o r d e r etc., terms can be neglected i n comparison w i t h second o r d e r term. Hence, t h e two s i g n i f i c a n t terms t h a t apear i n t h e expression f o r t h e complex c o m p r e s s i b i l i t y a r e as g i v e n i n equa- t i o n (55) and no more terms need be included.

Thus, t h e c o m p r e s s i b i l i t y o f t h e l i q u i d i n t h e presence'of an u l t r a s o n i c wave i s complex and has two terms

-

one i s independent o f t h e a p p l i e d pressure and t h e o t h e r term v a r i e s l i n e a r l y w i t h pressure, and they have a phase d i f f e r e n c e o f 90'.

Separating t h e r e a l and imaginary p a r t s o f equation (55) we get,

and

L

,-

The complex compressi b i 1 i t y can a1 so be expressed as,

where w and V a r e t h e frequency and v e l o c i t y o f t h e wave i s t h e amplitude a b s o r p t i o n c o e f f i c i e n t . Both a and V v a r y w i t h w. Separating t h e r e a l and imaginary p a r t s i n equation (58), we g e t

Comparing equations (56), (57) and (59), we g e t

and

D i f f e r e n t i a t i o n o f equations (60 a) and ( 6 1 a) w i t h respect t o p, a f t e r m u l t i p l y i n g b o t h sides w i t h Y, gives

and

Hence we have,

We a l s o have, according t o equation (62), i f w2 T 2 << 1

where t h e s u f f i x e s 1 and 2 r e f e r t o two d i f f e r e n t pressures.

Table I gives some a v a i l a b l e u l t r a s o n i c data and c a l c u l a t e d values o f T, wo and t h e r e a l and imaginary components o f compressi b i 1 i t y . I t may be seen t h a t T and wo a r e q u i t e reasonable i n t h e i r magnitude

,

showing t h e v a l i d i t y o f t h e present theory. Values o f X1 (w), are n e a r l y 10 t o 10 3 4 times l a r g e r than values o f X2 (w).

Table I : Calculated values of Molecular Vibrational frequency (W )» relaxation time ( T ) and compressibility (X = X, - i X„) from available data of pressure variation of ultrasonic Absorption and velocity and density for some organic liquids.

Liquid

Water Methyl alcohol Ethyl alcohol n-popyl alcohol n-butyl alcohol cci4

Hexane Toluene Liquid Methane cs2

Eugenol

T° A

273 303 303 303 303 303 303 303 145.6 244 303

Pressure i n dynes/cm xlO Pi

8.624

?2 50 22.75 24.1 24.2 23.32 18.42 49.0 49.0 18.228 43.414 12.94

Absorption i n cm":sec2x 10"1 7

(«/w2h

1.42 0.80 1.29 1.57 1.935 13.85

1.3375 2.275 0.4675 138.00

2.825

<a/w2)2

1.40 0.775 1.257 1.545 1.907 13.58

1.284 2.218 0.4475 134.8

2.835

Velocity i n cm/sec x 10 5

Vl V2

1.406 1.094 1.115 1.190 1.2340 0.907 1.060 1.290 0.97 1.314 1.482

1.412 1.106 1.129 1.2035 1.2454 0.9143 1.0905 1.3135 0.98 1.328 1.487

Density i n qms/c.c.

" 1 u2

1.00 0.782 0.781 0.798 0.800 1.57 0.651 0.857 0.369 1.300 1.058

1.002 0.784 0.783 0.7997 0.8018 1.573 0.6556 0.8603 0.372 1.305 1.0589

Angular frequency of the wave

w x 106

141.5 198.2 195.4 141.7 141.6 84.5 125.7 113.1 264.0 6.16 86.34

Liquid

Water Methyl alcohol Ethyl alcohol n-popyl alcohol n-butyl alcohol cci⁴ Hexane Toluene Liquid Methane

CS2 Eugenol

Relaxation time ( T ) x 10"1 2

equation (65)

2.802 1.807 2.665 2.506 3.513 27.95

4.535 3.983 1.264 279.4

4.333

Molecular w t .

(M)

18 32 46 60 74 154 86 92 16 76 164

Molecular mass (m) x 10 gms. -23

3 5.3 7.63 9.95 12.63 25.63 14.31 15.31 2.63 12.63 27.31

Cross sectional area (a) x 10" sq.cm.

7.575 13.09 16.69 19.64 22.56 23.46 28.61 24.91 13.61 16.60 31.82

Molecular v i b r a t i o n a l frequency (w0) x 10-12

equation(66) 4.544 3.579 3.636 4.334 5.018 3.510 3.776 5.554 3.431 12.500 6.125

x^x x io"^{1 3} equation (56)

at press 12.29 13.70 15.92 11.95 8.932 7.676 47.04 15.40 54.26 1.796 3.057

X2 x 1 0 "1 6

equation (57) ure P,,

2.515"

25.03 12.07 3.891 8.151 30.05 38.71 9.304 8.22

—

0.746

Experimental Data of Ultrasonic absorption and velocity for the various liquids, reported in Table I is obtained from the following references.

/I/ Hawley, J., Allegra, S. and Holton, G., Journal of the Acoustical Society of America, 47^ (1970) 137.

Ill Hawley, J., Allegra, S. and Holton, G., Journal of the Acoustical Society of America, 47 (1970) 158.

131 Litovitz, T.A., Carnevale, E.H. and Kendall, P.A., Journal of Chemical Physics, 26_ (1957) 465.

74/ Singer, J.R., Journal of Chemical Physics, ^1 (1969) 4729.