Indian Journal of Chemistry
Vol. 43A, April 2004, pp. 743-747
A new approach to collision factor theory
applied to binary and ternary liquid mixtures
at different temperatures
Rita Mehra* & Rekha Israni
Department of Pure and App li ed Chemistry,
Maharshi Daya nand Saraswati University, Ajmer 305 009
Received 20 Decelllber 2002; revised 3 Decelllber 2003
Ultrasound velocities have been measured in pure hexadecane ,
heptadecane, I-butanol. I-pentano l, I-hexano l, I-heptanol and in
the binary mixtures of hexadeca ne/he ptadecane with l -alkanol s at
298 .1 5.308.15 and 318. 15 K. Similarly ultraso ni c velocity has
also been measured for the te rnary system s of hexadecane + 1butano l and hexadecane + I-heptano l with dime thyl sulphoxide
(DMSO), dimethylformamide (DMF) and tetrah yd rofuran (THF)
each as a third component at three te mperatures. The experimen!al
values of ultrasoni c velocities have been compared with those
predicted o n the basis of collisio n factor theo ry and the de merits
of thi s theo ry ha ve been overcome by apply ing a new approac h
inco rpo rating colli sion volume in stead of mo lar volume and the
number of mol ecu les involved in co lli sions. Thi s modificati o n has
also been app li ed o n the systems studied by other workers and th e
rel ative merits have been discussed. Further, the effect of
te mpe rature provides an oppo rtunity to study the nature and ex te nt
o f inte rac tion s betwee n co mpo ne nt mo lecules as interaction s are a
result of active colli sions bctwee n molccules.
There has been an increasing interest in the study of
molecular interaciions and a number of experimental
techniques have been used to investioate the
.
.
b
Interactions between the components of binary and
ternary liquid mixtures at different temperatures. The
ultrasonic velocity in conjunction with density and
thermodynamic data derived from it has been widely
used for this purpose. Successful attempts have been
made in the recent past! '') on theoretical evaluation of
ultrasonic velocity and its correlation with other
thermodynamic properties in binary and ternary liquid
mixtures using statistical and semi-empirical theories,
viz. collision factor theory , impedance dependence
relation , Nomo to relation , Van Dael Vanoael
ideal
b
mixing relation and Flory 's statistical mechanical
theory as applied to binary and ternary liquid mixtures
at different temperatures. It was concluded that
compared with other theori es, collision factor theory
results in larger deviation from experimental data for
polar and non-polar mixtures. Such workers seemed
to have ig nored th e necess ity of in corporating two
parameters in CFf: one to account for collision
volume instead of molar vo lume a nd the other to
incorporate the number of molecules involved in
active collisions. In the new approach both factors
were incorporated and close agreement was found
between experimental and theoretical values usi no
b
CFf. This new approach has been applied to eight
binary mixtures of hexadecane and heptadecane with
I-butanol, I-pentanol, I-hexanol and I-heptanol and
six ternaries of hexadecane + i-butanol! l-h eptanol +
DMSO/DMF/THF. Further this modification has also
been applied to binary mixtures (polar-polar) of N, Ndimethylacetamide (DMA) + l-hexanol , I-octan ol,
chlorobenzene and toluene as reported by Ali et 01. !o.
The increase in temperature (10 K) provides an
opportunity to study the effect of temperature on the
nature and extent of molecular interactions as
molecular interactions are a result of active collisions
between the component molecules. The present work
is an attempt to app ly col li sion factor theory to
different liquid mixtures viz. polar-polar and polarnon polar liquid mixtures.
Experimental
All the chemicals used were of Analar g rade
(E. Merck, India and SD Fine Mumbai , India) of
purity >99 % and were used as such. The purity of the
components was ascertained by comparing boiling
points, refractive indices and densities of pure
components with those reported in literature!! . The
c lose agreement between the ex perimental values
with those reported in literature ens ure th e reliability
of the present results. The binary and ternary liqui d
mixtures were prepared by mass measurements with a
precision of ±O.OOI g and they were kept in special
air-tight bottles to minimise evaporation losses . The
velocity of sound was measured using an ultraso nic
interferometer operating at 2 MHz frequency , the
percentage error in measurements being ±O.03 %. The
temperature of the test liquids and their mixtures was
maintained at ±O.l DC in a thermostatic water bath.
On the basis of molecular kinetic theori ", Schaff
developed the following formula for th e velocity of
sound in pure liquids:
USB
U =U ~ Srr = V-
... ( I )
744
INDIAN J CHEM, SEC A, APRIL 2004
where U~ = 1600 ms· I , S is the collision factor, rr =
8/V is the space filling factor, 8 the actual volume of
the molecules per mole of the liquid and V is the
molar vo lume.
Nutsch Kuknkies 13 extended this concept to the
binary mixtures and formulated the following:
... (2)
This may be written as:
Z in equation (6) represents the total number of
collisions per unit time per volume and 2Z denotes the
number of molecules entering into collision. For a
bimolecular collision, one collision involves two
molecules and Z is therefore multipli ed by a factor 2
in Eq. 6.
Collision diameter (cr) in Eqs (6) and (7) is the
distance between the molecules when the rapidly
increasing repulsive forces just batance the very
gradually increasing attractive forces and it can be
evaluated using the following relation 14 (8)
... (3)
... (8)
The suffixes 1 and 2 represent the components I and
2 respectively. The values of 8 for the pure
components can be calculated using Eq. (4):
... (4)
The prior values of collision factor (S) and those
computed using the modified approach (S,) are given
in Table 1. The collision factor by modified approach
(S,) can be evaluated using Eq. (9)
where N is the Avagadro Number and rill is the
molecular radius which can be computed from the
expression due to Schaffs
... (5)
.. . (9)
So the modified approach of collision factor theory
can be summarized as:
u . =U
mi X
(xI81+X282)(XIS'+X2S')
~
XI VI (efl)
... (10)
+ X 2V2(e fJ)
Results and discussion
and
y, Rand T stands for ratio of principle specific heats,
gas constant and absolute temperature respectively.
In the present paper the theory as revised by Mehra
reports that if cr and Z represent the collision diameter
and number of molecules entering into collisions
respectively, then the effective volume or collision
volume and the number of molecules entering into
collision will be given by:
... (6)
... (7)
The experimental values of ul trasound velocity
EXP
(U ) along with those predicted from Schaff's
collision factor theory (USCFr ) and Mehra-Schaff s
collision factor theory (UMSCFJ) for representative
binaries at (298.15,308.15 and 318.15) K are given in
Figs I and 2.
Table I-Collision factor for pure liquids at 298.15
K with (5') and without modification (5)
Hexadecane
Heptadecane
I-B utanol
I-Pentanol
I-Hexanol
I-Heptanol
DMSO
DMF
THF
5
5'
0.01753
0.01699
0.03061
0.02795
0.02593
0.02429
0.02842
0.02413
0.01814
0.0599
0.0555
0.3336
0.2572
0.2064
0.1 563
0. 1842
0. 1618
0.1429
745
NOTES
1600 .0
1 4 00 .0 . - - --
- - -- - --
- - --
-------,
Heptadecane + 1- butanol
Hexadecane + 1 - hexanol
1350 .0
1500 .0
1 300 .0
1400 .0
:> 1300 .0
~L=='"==*=----==::=E::~~
1200 .0 t=::::::===---=~=----
1250 .0
:l
1200 .0
1150 .0
1100 .0
1100 .0
1 000 .0
1050 .0
+--.,-----,.-------.----r--r---.--.----,---r-~
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
O.B
0 .9
1 .0
1450 .0 - r - - -- - - -- -- - - - -- -- ---,
1400 .0
Hexadecane + I - heptanol
1 200 .0
0 .0
0 .1
0 .2
0 .3
1450 .0
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
Heptadecane +1- pentanol
1400 .0
1350 .0
1350 .0
1300 .0
:> 1250 .0
100 0 .0
~L~~;;:;:~==::~~:~~::~
!,b~:::::;t==t--
1300 .0
:> 1250 .0
1200 .0
1150 .0
1150 .0
1100 .0
1100 .0
1050.0
1050.0
1 000 .0 +--.-------.---r--.----r---.---,--...---.,------j
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
O.B
0 .9
1 .0
1 000 .0
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0.7
0 .8
0 .9
M o le fri c ti on
Mo le fricti o n
298.15 K t--" ) 308. 15 K (""* ) and 3 18.15 K ( .&- )] and MehraSchaffs Collision Factor Theo ry [at 298.15 K ( ..... ) 308.15 K
(...-J and 318.15 K .IL)] wi th ex pe rim ental values o f ultraso und
velocity [at 298.15 K t--+ ) 308.15 K E-*) and 3 18. 15 K f-I-) in
ms- I for hexadecane + I-hexano l and hexadecane + I-he ptanol
Fig. 2 - Comparison of Schaaff's Co llisio n Factor Theory [at
298.15 K t--) 308.15 K t*) and 318. 15 K ( ..&-)] and MehraSchaffs Collision Facto r Theory [at 298.15 K ( ...... ) 308.15 K
(......) and 318.15 K ---)] with experimental values of ultraso und
velocity [at 298. 15 K f-+) 308. 15 K E-* ) and 3 18. 15 K# ) in
ms- I fo r heptadecane + I-butanol and he ptadecane + I-pentanol
binary mixtures
binary mixtures
The respective values for ultrasound velocity for
representative ternaries are given in Fig. 3.
Figure 4 depicts sound velocity values and those
calculated using CFf and modified approach for data
reported by Ali et a/.
Perusal of Figs 1 to 4 reveal s that the new approach
is more applicable to the systems at different
temperatures. Th e original CFf developed by Schaff
was specifically meant to be used for non-associated
liquids . n-alkanol s are all associated and so CFf is
not applicab le to these systems at all. The reason for
deviation of the velocities calculated using CFf lies
in this fact, which is its limitation . Moreover, in
collision factor theory , molecules are treated as real
and non-elastic and molecular interactions are
considered to be the result of active coll isions
between the molecules . In the present work, the
concept of collision factor theory is still same, but the
coll isions factors SI and S2 which are in fact
correction terms (as also proposed by Schaff) are
adjusted by incorporating coll ision volume, instead of
molar volume and number of molecules involved in
active collision to give better results6 . In the case of
polar-polar system, both the components are polar
organic liquids which have strong molecular forces of
attraction. Hence, their collisions are likely to be less
elastic resulting in greater extent of interactions. But
in the case of polar-non polar (Figs I and 2) liquid
mixtures, the interaction s are less as compared to
polar-polar solvent resulting in more elastic collisions,
so the percentage error between the experimental and
theoretical values (U SCFf ) are greater. As the
temperature is rai sed, the number of molecules
entering into collision increases and ultimately their
collision volume increases so the ultrasound velocity
decreases and hence the interactions between the
component molecules decrease. This IS also
confirmed by the decrease 111 deviation from
Fig. 1 -
Compari so n of Schaaff' s Colli sio n Facto r Theory [at
INDIAN J CHEM, SEC A, APRIL 2004
746
1450 .0
Hexadecane + 1- butanol + THF
1400.0
1350 .0
1300 .0
1250 .0
::> 1200 .0
1150 .0 ·
1100 .0 ·
1050 .0
1000 .0·
0 .0
,
0 .1
0.2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
experimental values of ultrasound velocity. The
average percentage deviations of Schaff's collision
factory theory and Mehra-Schaff' s collision factor
theory for all the systems at differem te mperatures are
given in Table 2. From Table 2 it is clear that the
average percentage deviations are greater for ternary
systems as compared to binary systems. This might be
due to the fact that the concept of triple collisions has
not been included. Triple collisions mean the number
of occasions on which three molecules meet or if a
third molecule approaches while a pair of molecules
1450 .0
Hexadecane +1- heptanol + DMSO
1400 .0
1450
1350 .0
1400
DMA +c hlorobenzenc
DMA +toluene
1300 .0
1350
::> 1250 .0
1200 .0
1300
1150 .0
1250
1100 .0
1200
1050 ,0
1000 .0
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1150+-______~------~-------.--.----,
' ------~,
0 .0
0 .2
O.B
1 .0
0.4
0 .6
1.0
Mole fraction
Mole fr.ction
Fig. 3 - Comparison of Schaafr's Colli sion Factor Theory [at
298 . 15 K p.- ) 308.15 K (""*) and 318.15 K ( ---- )] and MehraSchaffs Collision Factor Theory [at 298.15 K (-.-) 308. 15 K
(....-J and 3 18.15 K - )] with experimental values of ultraso und
ve locity [at 298.15 K t-+ ) 308. 15 K E-*) and 3 18.15 K f+) in
ms- I for hexadecane + I-butanol/heptanol + polar/nonponlar
ternary mixtures
Fig. 4 - Compari son of Schaffs Colli sion factor Theory 1 DMA
+ chlorobenzene (*) and DMA+toluene ( ~ )] and Mehra-Schaffs
Colli sion Factor Theory [DMA + chl orobenzene (0) and DMA +
) ] with experimental va lues of ultrasound velocity
toluene b
[DMA + chlorobenzene (x) and DMA + tolu,! ne ( ~ )] in ms- I as
reported by Ali e l. al. at 303. 15 K.
Table 2-Average percentage error for Schaffs Collision Factor Theory (SCFT) and Mehra Schaff's Collision Factor Theory (MSCFT)
for va ri ous binary and ternary systems at different temperatures
System
Hexadecane + I-butanol
Hexadeca ne + I-pe ntano l
Hexadecane + I-hexa nol
Hexadecane + I-hepta no l
Heptadecane + I-butano l
Heptadecane + I-pe ntan o l
Heptadeca ne + I-hexano l
Heptadecane + I-hepta no l
Hexadecane+ l -b utan o l+DMSO
Hexadecane+ l -b utanol+DM F
Hexadecane+ I-b utan ol+THF
Hexadecane+ l-hep tanol+DMSO
Hexadecane+ l -heptano l+DMF
Hexadecane+ I-hep tano l+ THF
DMA+I-hexanol *
DMA+ I-octano l*
DMA+ I-chlorobenzene
DMA+tolue ne*
*Reported by Ali el al. at 303 .1 5 K
at 298. 15 K
SCFT
MSCFT
at308 .1 5 K
SCFT
MSCFT
at 3 18.1 5K
SCFT
MSCFT
-12 .66
-8 .90
-3 .97
-4 .1 7
- 11.4
-8.96
-8.03
-7.86
10.86
12.89
12.8
\3.5
14.9
\3.5
0.68
0.52
0.1 8
0.12
- 10.73
-8.74
- 5.25
-5.06
- 10.2
-9.12
-9.08
-9. 18
10.14
10.6
10.2
11.7
12.5
13.1
-6.83
- 10.66
- 6.30
-4.38
-6.18
-5.94
-5 .90
-5 .08
10.26
7.48
7.46
6.18
7.66
7.94
- 0.42
-0.39
- 0.37
- 0.38
-0.44
-0.40
-0.39
-0.38
1.48
1.40
1.16
1.36
1.3
0.94
0.04
0.09
0.10
0.08
-0.38
-0.39
- 0.40
- 0.39
-0.42
-0.46
-0.36
-0.35
1.38
1.31
1.28
1.24
1.2
1.16
- 0. 39
--0. 36
- 0. 37
- 0.39
-·0.4 1
- 0.48
- 0.35
-·(1.34
1.46
1.4 1
1.l6
1. 14
1.09
Q l)1
NOTES
are still within the molecular distance of each other.
This concept is further under investigation .
Thus keeping in view the behaviour of the systems
under investigation, it can be said that the positive
deviations in velocity are attributed to molecul ar
association and complex formation whereas negative
deviations in velocity are attributed to molecular
dissociation of an assoc iated species caused by
addition of an inert solvent or an active solvent and
interstitial acco mmodation of the component
molecules . From the reported data it appears that the
latter is dominant than the former.
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