42

EXPERIMENTAL

This chapter covers the details regarding the materials, their purification and

experimental techniques used for determination of various physicochemical properties

such as apparent molar volumes, viscosities and apparent molar adiabatic compressibilties.

The theoretical background of various physicochemical properties and uncertainties in

various quantities have also been included.

3.1 Materials

The various chemicals used for the present study are listed in Table 3.1 along with their

source, grade and molar mass.

3.1.1. Purification of the Chemicals

The saccharides and methyl glycosides of highest purity grade were used as

received without further purification; however these chemicals were dried over anhydrous

CaCl2 in a vacuum desiccator for 48 h before use. Potassium chloride was used after

drying for 24 h in an oven. Magnesium chloride and lithium chloride, were used as such

and kept stored in a vacuum desiccator. Conductivity water was prepared by distilling

deionised water first over acidic potassium permanganate and then over alkaline potassium

permanganate. It was further degassed before use by boiling for about 10 minutes to avoid

micro air bubbles in the solutions. The specific conductivity of the water used for studies

was less than 1.29 10-6

-1

cm-1

. All the solutions were prepared afresh on weight basis

using a Mettler balance with a precision of 0.01 mg.

3.2 Experimental Techniques

The apparent molar volumes, viscosities and apparent molar adiabatic

compressibilities have been obtained from the measurements of densities, flow time and

speed of sound using densimeter, Ubbelohde type capillary viscometer and ultrasonic

interferometer, respectively.

3.2.1 Vibrating-Tube Digital Densimeter

There are several methods of measuring density such as magnetic float density cell,

differential hydrostatic balance, pycnometer, vibrating-tube digital densimeter, etc.

The vibrating-tube digital densimeter (Model DMA 60/602, Anton Paar, Austria)

was used for density measurements and the schematic diagram for densimeter unit is

shown in Fig. 3.1.

43

Table 3.1: Specifications of chemicals used.

Compound Source Grade/

Mass fraction

Purity

Molar mass

(g mol-1

)

D-(+)-Xylose SRL* AR/(0.999) 150.13

D-(–)-Arabinose SRL AR/(0.999) 150.13

D-(–)-Ribose SRL AR/(0.999) 150.13

D-(+)-Mannose SRL AR/(0.999) 180.16

D-(–)-Fructose SRL AR/(0.994) 180.16

D-(+)-Galactose SRL AR/(0.997) 180.16

D-(+)-Glucose SRL AR/(0.994) 180.16

D-(+)-Melibiose SRL AR/(0.994) 342.30

D-(+)-Cellobiose SRL AR/(0.994) 342.30

D-(+)-Maltose monohydrate SRL AR/(0.994) 360.31

Sucrose Lancaster AR/(0.99) 342.30

D-(+)-Lactose monohydrate SRL AR/(0.999) 360.31

D-(+)-Trehalose dihydrate SRL AR/(0.998) 378.33

D-(+)-Raffinose pentahydrate SRL AR/(0.999) 594.53

(+)-methyl α-D-glucopyranoside S.D. Fine

Chem. Ltd.

AR/(0.989) 194.18

Methyl α-D-xylopyranoside S.D. Fine

Chem. Ltd.

AR/(0.98) 164.20

Methyl β-D-xylopyranoside S.D. Fine

Chem. Ltd.

AR/(0.98) 164. 20

Potassium Chloride(KCl) Qualigens AR/(0.999) 74.55

Magnesium Chloride

(MgCl2.6H2O)

Qualigens AR/(0.999) 203.31

Lithium Chloride

(LiCl)

CDH# AR/(0.999) 42.39

*Sisco Research Laboratories, India.

#Central Drug House Pvt. Ltd., India.

44

Fig

3.1

Sch

emati

c d

iagra

m o

f th

e d

ensi

tym

ete

r (D

MA

60/6

02)

45

(i) Principle

It is based on the principle1-2

of time-lapse measurement for a certain number of

oscillations of vibrating U-shaped sample tube, which is filled with liquid sample. The

oscillation device, which is a hollow glass oscillator, is electronically excited, in an

undamped fashion. The direction of oscillation is perpendicular to a plane passing through

the inlet and outlet opening of the sample tube. The natural frequency of the oscillating

device will be influenced only by the volume of the sample that fills it up to the nodal

points. Therefore, it is necessary to fill the oscillating sample tube at least past its nodal

points, which are fixed and not oscillating.

To derive the mathematical relationship between sample density and natural

frequency or period it is possible to consider an equivalent system consisting of a hollow

body with the mass, M, which is suspended on a spring with a spring constant C. The

volume of hollow body is assumed to be filled by a sample of density, d and natural

frequency, f of such a system is thus given by equation (3.1)

VdM

Cf

2

1 (3.1)

Therefore, time period of oscillation, T is given as:

C

VdMT

2 (3.2)

By squaring and simplifying the equation (3.2):

C

Vd

C

MT

222 44

(3.3)

On substituting CVA /4 2 and CMB /4 2 into equation (3.3), it becomes:

BAdT 2 (3.4)

where A and B are the constants of the oscillator which have the contribution from the

spring constant of oscillator, empty oscillator's mass and volume of the sample which

participates in the oscillations. These constants are therefore instrument's constants for

each individual oscillator and can be determined by time calibration measurements with

sample of known density e.g. dry air and distilled water.

46

Therefore, the following relation can be used for the density difference of the

sample.

)( 2

2

2

121 TTKdd (3.5)

where K=1/A, d1 and d2, T1 and T2 are densities and time periods of sample 1 and 2,

respectively. Hence

)( 2

2

2

121 TTKdd (3.6)

To obtain a precision of + 10-n

in determining densities, it is necessary to measure a time

lapse at least 10n+1

times a chosen unit of time. If the resonance period of the oscillator is of

the order of 2-5 ms selected as the time unit, then a measurement time of 5-6 hours would

be required. Since this is impractical, the total time lapse is measured by independent

means in small but precisely defined units for a preset number of oscillations of the sample

holder. For reasons defined by electronic circulatory, the smallest time unit which can be

measured is 10 s. Thus precision of 10-5

-10-6

g cm-3

in density measurements is thereby

attained by measuring the time lapse in the range of 10 to 100 s.

(ii) DMA 602 Density Measuring Cell

DMA 602 density measuring cell is contained in its own separate housing, complete

with oscillator counter mass and thermostat connectors. It consists of a U-shaped

borosilicate oscillator sample tube fused into a dual wall glass cylinder. The space between

the U-shaped sample tube and the inner wall of the dual-wall cylinder is filled with gas of

high thermal conductivity to facilitate a rapid thermal equilibrium of the sample inside the

oscillator with the thermostat liquid which flows through the dual wall cylinder around the

sample tube. An additional shorter capillary tube inside the inner space of the dual wall

cylinder determines temperature by means of a temperature sensor. This capillary tube of

about 2 mm inside diameter has a very thin wall thickness of about 0.2 mm to assure good

heat transfer. The rest of the instrument consists of the electronic excitation system for the

oscillation and the electronics, which assure an interface-free transmission of the period

signal to the processing unit. Built-in air pump with a sintered glass filter for drying the

sample tube, as well as the sample tube illumination system have also been provided in the

densimeter.

47

(iii) DMA 60 Electronic Processing Unit

Analog signal generated in synchronism with the oscillating sample tube in the

density-measuring cell is transmitted into the electronic processing unit DMA 60. This

signal is amplified and displayed digitally through the light emitting diodes. Details of its

working have been described1-2

elsewhere.

(iv) Temperature Bath

An efficient constant temperature bath (Heto Birkerod/Denmark) with stability

within ± 0.01 K was used to control the temperature of the water circulating around the

densimeter cell. It has both cooling and heating arrangement. The water was made to flow

with the help of inbuilt water circulating pump. The length of the water circulating tubes

was reduced to minimum and these were insulated properly to avoid any heat loss.

Absoluteness of the temperature was checked by putting a Beckmann thermometer, which

was calibrated by determining the transition temperature of freshly crystallized sodium

sulphate with the help of 1°C (1/1000°C) Beckmann thermometer.

(v) Experimental Procedure for Density Measurements

Initially the instrument was switched on for about 30 minutes. The sample tube was

washed several times with deionised water and then with double distilled water. It was

then dried. The liquid whose density was to be determined, was injected continuously into

the lower opening of the U-shaped vibrating tube, until the excess liquid flows out of the

upper part. Liquid should be filled in the tube very slowly to enable the liquid to properly

wet the walls of the sample tube. Care was taken to avoid trapping of micro air bubbles.

Disposable, polyethylene syringe about 2 cm3 in volume was used for introducing the

liquid into the U-tube. After filling the sample tube completely, its openings were closed

with teflon stoppers. Then, the illumination was switched off and start point was pressed

on the DMA 60 processing unit. After observing few read out cycles, a constant value of

the time period, T was noted. Each observation is the mean of triplicate measurements.

Instrument constant, K was calculated after measuring the T values for dry air and

double distilled degassed water in the U-tube, which were taken as standards. The density

of dry air, dT,P (g cm-3

), at temperature T(C) and atmospheric pressure, P (torr) was

calculated3 by using following equation:

dT,P = [0.0012390/1 + 0.00367 T] [P/760] (3.7)

48

Density of the air free pure water was taken from the Kell's data4. The K values

were checked regularly. The densimeter was tested by measuring the densities of the

aqueous solution of sodium chloride at different concentrations at 298.15 K. The results

obtained agreed well with the literature values5-8

within the combined uncertainties of

0.1%.

3.2.3. Multifrequency Ultrasonic Interferometer

The sound velocity, u was determined using Multifrequency Ultrasonic

Interferometer (Model: M-82, Mittal Enterprises, India) which is a direct and simple device

for the measurement of the sound velocities of the liquids with a high degree of accuracy.

The schematic diagram for multifrequency ultrasonic interferometer unit and the measuring

cell are shown in Figs. 3.2-3.3.

(a) Principle

The principle used in the measurement of sound velocity, u is based on the accurate

determination of the wavelength, of sound waves in the medium. Sound waves of known

frequency, f are produced by a quartz crystal fixed at the bottom of the cell. These waves

are reflected by a movable metallic plate kept parallel to the quartz crystal. If the

separation between these two plates is exactly a whole multiple of the sound wavelength,

standing waves are formed in the medium. This acoustic resonance gives rise to an

electrical reaction on the generator which excites the quartz crystal and the anode current

of the generator becomes maximum.

If the distance is now increased or decreased, and the variation is exactly one half

of the wavelength, /2 or multiple of it, anode current becomes maximum. From the

knowledge of wavelength, the sound velocity can be obtained as:

u = f (3.8)

where f is frequency of waves.

(b) Design and Description of Interferometer

The ultrasonic interferometer consists of the following parts:

(i) The high frequency generator.

(ii) The measuring cell.

49

Fig. 3.2 Multifrequency Ultrasonic Interferometer (Model: M-82)

50

MICROMETER

REFLECTOR

EXPERIMENTAL

LIQUID

TO CONST. TEMP

WATER BATH

QUARTZ CRYSTAL

R.F.INPUT

THE MEASURING CELL

Fig. 3.3 The Measuring Cell

51

(i) The High Frequency Generator

It is meant to excite the quartz crystal fixed at the bottom of the measuring cell at its

resonant frequency to generate ultrasonic waves in the experimental liquid filled in the

measuring cell. The panel of the body having high frequency generator is provided with a

microammeter to observe the changes in current and two controls marked 'Adj' and 'Gain'

are also provided, which are used for initial adjustment of the needle of the ammeter and

sensitivity regulation, respectively.

(ii) Measuring Cell

It is a specially designed, double walled cell, which is made of steel with gold plating

at the bottom covering the quartz crystal and chromium plating on inner walls of the cell

for maintaining the temperature of the liquid during the experiment. A fine micrometer

screw has been provided at the top that can lower or raise the reflector plate in the cell

through a known distance.

The quartz crystal, which generates the ultrasonic waves, is fixed at its bottom. The

temperature of the experimental liquid in the cell was maintained at constant temperature

by circulating the water from constant temperature circulator (Model: Julabo F 25) with

an accuracy of 0.01 K.

(c) Working of Ultrasonic Interferometer

Instrument was operated in the following manner to measure the sound velocities:

1. The cell was inserted in the square base socket and clamped to it in the square base

socket with the help of a screw provided on one of its side.

2. Unscrewed the knurled cap of the cell and lifted it away from double walled

construction of the cell. In the middle portion of it, experimental liquid was poured

and cap was screwed again.

3. Two chutes in double walled construction were provided for circulation of water to

maintain the desired temperature.

4. The high frequency generator was connected with cell by co-axial cable provided

with the instrument.

5. Initially temperature bath was switched on for about 30 minutes.

Now switched on the generator and the position of needle on the ammeter was

adjusted with the knob marked as „Adj‟ and knob marked „Gain‟ was used to increase the

52

sensitivity of the instrument for greater deflection, if desired. The ammeter was used to

notice the number of maximum deflections while micrometer was moved up and down in

liquid.

After the equilibration, the micrometer was slowly moved upward till the anode

current was maximum. The initial reading on micrometer scale was noted. Similarly 20

maxima readings on the ammeter were passed and micrometer reading at twentieth

maximum was again noted. The difference between initial and final reading gives the

distance, 'D' travelled by reflector. The average of 10 readings was treated as final value of

the distance travelled by micrometer. The generator was switched off after measurements

were completed and liquid was taken out of the cell. The upper part of cell was removed

and cell was cleaned with distilled water and then rinsed with double distilled water and

filled with the next experimental liquid.

The following relation relate the distance and wavelength of the ultrasonic sound

waves

D = n /2 (3.9)

The sound velocity, u was calculated using equation (3.9). The uncertainties in

sound velocities were ± 0.5 ms-1

, while these were precise within 0.1 ms-1

. The measured

value for u in water at 298.15 K (1496.66 ms-1

) agrees well with the literature9-10

value

(1496.69 ms-1

).

3.2.2. Ubbelohde Type Capillary Viscometer

There are ranges of viscometers available commercially for the measurement of

viscosity. The selection of viscometer depends on a number of criteria i.e. magnitude of

viscosity to be measured, whether the liquid or solution are elastic, whether they are

transparent or opaque and the temperature dependence of viscosity. Some types of

viscometers are: capillary, rotational and moving body,11

etc. In the present work, the

viscosities of the solutions have been measured using Ubbelohde type capillary

viscometer as shown in Fig. 3.4.

(i) Design of Viscometer

The viscometer consists of three arms A, B, C and bulb D of capacity about 60 cm3.

Arm A is fixed to one side of the bulb D. To the upper part of bulb D, another small bulb G

53

Fig. 3.4 Ubbelohde type capillary viscometer

54

is fused to one side of which the arm C is fixed. The lower end of the bulb G is drawn into

a jet just touching the bottom of bulb E. To the upper part of bulb G, arm B of viscometer

is fixed. The arm B contains a fine capillary H, a small bulb I, again a capillary J with the

wide bore, bulb K of capacity of about 10 cm3, again a capillary L of the same bore as J

and another small bulb M, extending into the wide tube T. On the capillaries J and L the

marks X1 and X2 were etched, within which the flow times of the solutions were

determined.

The purpose of bulbs I and M and capillary is to ensure exact noting of the time of

start and finish and thus ensure reproducible times. The absence of the bulb I gives rise to

the bubble formation on the exhaustion of the liquid in the bulb K and thus do not allow

the finish to be noted accurately.

(ii) Calibration of the Viscometer

The viscometer was calibrated using deionised, doubly distilled and degassed water.

The flow time of water between reference marks X1 and X2 was noted with the help of a

stopwatch at different four temperatures. The flow times along with literature values of

densities and viscosities were fitted to the equation (3.10)

η = ρ (at-b/t) (3.10)

where a and b are the constants of the viscometer, ρ and t are the density and flow time of

water, respectively.

(iii) Working of Viscometer

The viscometer was cleaned using freshly prepared chromic acid. The cleaned

viscometer was washed with distilled water, rinsed with acetone and finally dried. Then

the sample solution was transferred into the viscometer and it was immersed into the

thermostat and clamped vertically. After equilibrium for about 45 minutes at the desired

temperature, the arm B of the viscometer was connected to a vacuupet and the solution

from bulb E was filled to arm B when the solution has just crossed the mark X1 on the

capillary L, the vacuupet was released and solution was made to fall from mark X1 to mark

X2. The flow time was recorded with the stopwatch of the resolution of ± 0.01s. The

average of at least four-five readings reproducible within 0.l s was used as final flow time.

The constant temperature water bath (Model: INSREF/India) was used to control the

temperature within 0.01 K. The working of viscometer was checked by measuring the

55

viscosities of aqueous solution of sodium chloride and results were found to agree well

with the literature values12

.

3.3 Theoretical Background of Various Thermodynamic and Transport Properties

The theoretical background of various thermodynamic and transport properties have

been discussed as follows:

3.3.1. Partial Molar Volume

The partial molar volume of a solute, 2V can be defined, as the change in the

volume of solution that results on the addition of one mole of solute to such a large

quantity of the solution that there is no appreciable change in the overall concentration of

the solution when the temperature, T, pressure, P, and number of moles of the other

components if present, remains unchanged.

Mathematically, the partial molar volume of a solute, 2V can be represented as:

2V =

.......n,P,T21

n

V

(3.11)

The total volume of solution, V can be expressed as:

(3.12)

where 1V is the partial molar volume of solvent and n1 and n2 are the number of moles of

solvent and solute, respectively. 2V is generally determined from the apparent molar

volume, V2, which may be defined by the relation:

V2, = 2

011

n

VnV (T and P are constants) (3.13)

or

011,22 VnVnV (3.14)

where 0

1V is the molar volume of pure water (solvent) under identical conditions of T and

P. The relationship between V2, and 2V can be obtained by using the equations (3.12) and

(3.14):

2211 VnVnV

56

2V =

........,,2

,2

2,2

,,211 nPTnPT

n

VnV

n

V

(3.15)

....,,2

,22

2

0

11

11

221

1

1

nPTn

VnVn

nn

VnVV

(3.16)

In terms of the experimentally measured densities, d0 and d of the solvent and

solution, and the molar masses, M1 and M2 of solvent and the solute, the V2, is given by

the equation:

(3.17)

when the concentration is expressed in molality, then n2 = m, the molality of the solution

and n1 is equal to the number of moles of solvent in 1000 g of the solvent, then V2,

becomes:

0

2,2

100010001

dd

mM

mV (3.18)

or

0

02,2

1000)(

mdd

dd

d

MV (3.19)

when the molar concentration is used, then n2 = c, the molarity, then V2, becomes:

0

02,2

)(1000

cd

dd

d

MV

(3.20)

To obtain reliable values of apparent molar volumes, V2, it is necessary to measure

the densities with a great precision. At infinite dilution, the apparent molar volume and the

partial molar volume become equal (V0

2, =V0

2).

The extrapolation of apparent molar volume to infinite dilution has been made by

the following equations:

1. Masson equation13

2. Redlich-Meyer equation14

3. Owen-Brinkley equation15

0

112211

2

,21

Vnd

MnMn

nV

57

Masson13

found that V2, of electrolyte varies linearly with the square root of their

molar concentration, c as follows:

V2, = cSV v0

2 (3.21)

where Sv is the experimental slope.

Redlich and Rosenfeld16

on the basis of Debye-Huckel limiting law predicted a

constant slope for a given ionic charge. On this basis, Redlich and Meyer14

suggested the

following extrapolation function

V2, = cbcSV vv 0

2 (3.22)

where Sv is the theoretical limiting slope and bv is an empirical constant determined from

experimental results.

Owen-Brinkley‟s equation15

can be used in the extrapolation and to represent

concentration dependence of V2, as:

V2, = cKcKaWcKaSV vvv 21)()(0

2 (3.23)

where Kv is an empirical constant and „a‟ is the distance of closest approach. The ideal

method to determine V2, value is to make the measurements for very dilute solutions,

where the deviation from limiting law is very small.

In case of non-electrolytes and zwitterionic solutes, V2, versus molality, m plots

are linear and V0

2 values are obtained by least squares fitting of the following equation:

V2, = V0

2+SvmA (3.24)

For the present study, the equation (3.24) has been used to obtain V0

2 values.

3.3.3. Adiabatic Compressibility

The measure of a change in volume of liquid as a response to a change in pressure,

relative to volume at constant temperature is called isothermal coefficient of

compressibility (KT).

58

Mathematically,

KT = TP

V

V

1 (3.25)

where V is volume of solution, P and T are the pressure and temperature, respectively. The

minus sign indicates that volume decreases with increase of pressure and vice-versa. As KT

is the coefficient of compressibility of the bulk solutions, it is not particularly informative

with regard to either solute-solvent or solute-solute interactions. Therefore, it is better to

determine partial molar quantities to have information on these interactions17-18

. Hence,

partial molar isothermal compressibility of a component in a solution is defined as:

T

i

iTP

VK

, (3.26)

It is generally difficult to calculate precise values of partial molar compressibilities

from a direct determination of the derivative (Vi /P)T. A more convenient method is to

determine the apparent molar compressibility from which the partial molar property can be

derived. The apparent molar isothermal compressibility of a solute in a solution is given by

the equation:

T

T

P

VK

,2, (3.27)

where V2, is the apparent molar volume of solute (equation 3.19).

In order to obtain accurate apparent molar compressibilities, it must be possible to

measure the isothermal compressibilities of solution and solvent with great precision. It is

not an easy task to measure directly isothermal coefficient of compressibilities with the

precision necessary to give reliable apparent molar compressibilities at low pressure.

However, it is possible through speed of sound measurements to obtain very precise values

of a related quantity termed as isentropic coefficient of compressibility

KS = -(1/V)(V/P)S (3.28)

The sound velocity through a fluid, u is related to adiabatic compressibility, KS and

the density, d by the following equation:

59

KS = 1/u2d (3.29)

The relationship between KT and KS is given by equation (3.30):

KT = KS + 2 T/ (3.30)

where α is the coefficient of thermal expansion, = (1/V)/(V/T)P and is the volumetric

heat capacity. The apparent molar isentropic compressibilities, KS,2, can be obtained from

the coefficient of compressibilities and densities of solution and solvent by using the

relation:

0

0

0

2,2,

mdd

dKdK

d

MKK SSS

S

(3.31)

where KS and K0

S are the coefficients of compressibilities of solution and solvent,

respectively and M2, d, d0 and m are the same parameters as used in the previous section.

At infinite dilution, the apparent molar adiabatic compressibility is equal to partial

molar compressibility of the solute (K0

S,2, = K0

S,2).

3.3.2. Viscosity

Viscosity is the measure of the internal friction of a fluid. This friction becomes

apparent when a layer of fluid is made to move in relation to another layer. The greater the

friction, greater the amount of force required to cause this movement, which is called

“shear”. Shearing occurs whenever the fluid is physically moved or distributed, as in

pouring, spreading, mixing, etc.

In case of liquids, viscosity is defined as the ratio of the shearing stress, F' to shear rate,

S as follows:

= F'/ S (3.32)

where is the coefficient of viscosity. The force per unit area required to produce

the shearing action is termed as shear stress i.e. F' = F/A, whereas shear rate can be

described as the measure of change in speed at which the intermediate layers move with

respect to each other i.e.

S=dV/dx (3.33)

60

According to Newton‟s law, the ratio of F'/ S is a constant and is independent of the

shear rate. This law is true only for ideal or “Newtonian” solutions, whereas for non-

Newtonian solutions the ratio of F' / S is not a constant and shear rate is varied.

In case of liquids, decreases with the increase in temperature. The viscosity of

liquid is mainly dependent upon intermolecular forces. In order to flow, the molecules in a

liquid need energy to escape from their neighbours. Evidently this energy is more freely

available at higher temperature than at low temperature.

Relative viscosity, r may be defined by the relation, = /0, where and 0 are

the viscosities of solution and solvent, respectively.

Development of Jones-Dole Equation

Arrhenius19

gave the following relationship between the relative viscosity, r and

concentration, c for moderately dilute solution:

r= Ac (3.34)

where A is a constant for given salt and temperature. Various workes have shown negative

curvature for the salt solutions instead of straight at lower concentrations and at low

temperature. Rabionovich20

have concluded that deploymerization of water molecules must

be responsible for negative viscosity.

Jones and Dole21

from the special behaviour of salts concluded that there must be

some effect, which is of relatively greater importance and is responsible for curvature in

viscosity vs concentration plots. This effect always tends to increase whether the overall

effect of addition of salt is to increase or decrease the viscosity. The decrease in viscosity

was attributed to interionic forces.

Earlier Debye and Huckel observed that the effect of interionic forces in opposing

the motion of ions is proportional to the square root of the concentration in very dilute

solutions. Thus, they gave the equation:

BccA 11

(3.35)

where is the fluidity and A and b are constants, where A has negative values for the

strong electrolytes, which tend to stiffen the solutions or decrease the fluidity and zero for

61

the non-electrolytes. The B has positive values for the liquids with high fluidity and

negative for those with low fluidity.

Later Jones and Talley22

measured the viscosities of urea and sucrose solutions and

further confirmed that the values of A for non-electrolytes like urea and sucrose is zero and

the equation (3.35) reduced to:

= 1 + Bc (3.36)

This equation was extended to represent the data for solutions in the following form:

r = /o = 1 + Bc (3.37)

where B is viscosity B-coefficient, c is the molarity.

3.4 Preparations of Solutions

The solutions were prepared in standard joint flasks (50 cm3). The flasks were

cleaned with freshly prepared chromic acid, followed by washing with distilled water and

then after rinsing with acetone these were dried. The weight of dried flask was noted and

required amount of solute was added into it and then weighed. The required amount of

solvent was then added and again weighed. The molality of the resulting solution was

calculated as follows:

Weight of empty flask = w1g

Weight of empty flask + solute = w2g

Weight of empty flask + solute + solvent = w3g

Weight of solute = (w2-w1) = „a‟g

Weight of solvent = (w3-w2) = „b‟g

Molality, m = 1000 „a‟/ (M „b‟)

where M is molecular weight of solute, a and b are the weights of solute and solvent,

respectively.

3.5 Estimation of Uncertainties

(A) Uncertainty in apparent molar volume, V2,

The uncertainty in V2, may arises from (i) density (ii) molality measurements.

(i) Uncertainty in density measurements

The density has been calculated by using equation (3.6):

62

d1 = d2 + K (T12 – T2

2)

The uncertainty in density „d2‟ determination may arise due to uncertainties in calibration

constant K and T values.

By rearranging the equation (3.6):

(3.38)

where densities, d1, d2 are constants (taken from literature) and time periods T1 and T2 are

experimentally determined quantities for water and dry air, respectively.

Uncertainty in „K‟ due to uncertainty

In

122

2

2

1

12111 .

2.dT

TT

TddUT

(3.39)

in

222

2

2

1

22122 .

2.dT

TT

TddUT

(3.40)

Total uncertainty in K= 2

2

2

1 UU

For dT1 = dT2 = 1 10-6

, the estimated uncertainty in „K‟ comes out to be 4.15 10-5

.

From equation (3.6),

Uncertainty in „d2‟ due to uncertainty

in T1 = U1 = 2 KT1.dT 1 (3.41)

in T2 = U2 = 2 KT2.dT2 (3.42)

in K = U3 = (T12 – T2

2) dK (3.43)

The estimated uncertainty in „d2‟ due to uncertainty in temperature (0.01K), U4 =

0.0000003.

Thus total uncertainty in density measurement becomes

(3.44)

The estimated uncertainty in density measurements comes out to be 2.67 10-6

g cm-3

.

(ii) Uncertainty in molality

Molality, m of a solution has been calculated using the relation:

m = w2 1000/(w1 M2) (3.45)

2

2

2

1

21

TT

ddK

2

4

2

3

2

2

2

1 UUUUUd

63

where w2 is the weight of solute, w1 is the weight of solvent and M2 is the molecular weight

of solute which is constant, so uncertainty in „m‟ may arise from the uncertainty in w2 and

w1, which can be estimated as follows:

Uncertainty in „m‟ due to uncertainty

in w2 = U1 = 12

21000

wM

dw

(3.46)

= 1.07 10-6

mol kg-1

in w1=U2 = 2

12

121000

wM

dww

(3.47)

=1.22 10-7

mol kg-1

Total uncertainty in „m‟ becomes

Um = 2

2

2

1 UU (3.48)

The estimated uncertainty in „m‟ comes out to be 1.08 10-6

mol kg -1

.

Uncertainty in V2, due to uncertainty

in m = U m = dmmdmd

22

0

10001000 (3.49)

= 3.55 10-3

cm3 mol

-1

in d = Ud = ddd

M

dm

2

2

2

1000 (3.50)

= 0.020 cm3 mol

-1

Total uncertainty in V2, becomes

U(V2.) = 22

dm UU (3.51)

The estimated uncertainty in V2, comes out to be 0.020 cm3 mol

-1.

(B) Uncertainties in adiabatic compressibilities and apparent molar adiabatic

compressibilities

The uncertainties in adiabatic compressibility arise due to uncertainty in (i) sound velocity

(ii) density (iii) temperature variation.

Uncertainty in „KS‟ due to uncertainty,

in U = Ux = dUdU

..

23

(3.52)

where dU = 0.5 s

= 0.00160 10-10

kg-1

s2

64

in d = Uy = ddUd

..

122

(3.53)

= 7.31 10-18

kg-2

m4s

2

in T = Uz = dTT

K s . (3.54)

= 0.000090 10-10

kg-1

m s2 K

-1

where dT = 0.01 K.

Total uncertainty in „KS‟ becomes

U(Ks,2, ) = 222

zyx UUU (3.55)

The estimated uncertainty in „KS‟ comes out to be 0.00160 10-10

kg-1

ms-2

.

Uncertainty in apparent molar adiabatic compressibility, KS.2, arises due to the uncertainty,

in KS = U1 = (M2 + 1/m) 1/d dKS (3.56)

in d = U2 = (M2 + 1/m) KS/d2.dd (3.57)

in m = U3 = (K0

S/d0 – Ks/d) 1/m2 dm (3.58)

Total uncertainty in apparent molar adiabatic compressibility

U(Ks,2.) = 2

3

2

2

2

1 UUU (3.59)

Estimated uncertainty in KS.2. comes out to be 0.063 10-15

m3 mol

-1 Pa

-1.

(C) Uncertainty in viscosity

The uncertainty in viscosity, may arise due to uncertainty in (i) constants a and b of the

equation (3.10) i.e. = d

t

bat , (ii) measured flow time and (iii) density.

Uncertainty in „a‟ due to uncertainty

in t = Ua = (/dt2 + 2b/t

3) dt (3.60)

where dt = 0.01

= 2.43 10-7

Uncertainty in „b‟ due to uncertainty

in t = Ub = (2 at - /d) dt (3.61)

= - 0.0257

Uncertainty in „‟ due to uncertainty

in a = U1 = t d.da (3.62)

= 9.86 10-5

65

in b = U2 = d/t.db (3.63)

= -9.78 10-4

in t = U3 = a d.dt + bd/t2 dt

= dt

2t

bdad (3.64)

= 5.128 10-8

in d = U4 = at.dd – b/t .dd

= dd

t

bat

where dd = 2.667 10-6

= 3.4590 10-8

Thus, total uncertainty in viscosity becomes

U = 2

4

2

3

2

2

2

1 UUUU (3.65)

The estimated uncertainty in viscosity comes out to be 0.00014 mPa s.

66

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