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J Solution Chem (2012) 41:318–334

DOI 10.1007/s10953-012-9795-6

Modification of the Two-Point Scaling Theory

for the Description of the Phase Transition in Solution.

Analysis of Sodium Octanoate Aqueous Solutions

Henryk Piekarski ·Michał Wasiak ·Leszek Wojtczak

Received: 8 November 2010 / Accepted: 13 May 2011 / Published online: 2 February 2012

© The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract On the basis of conventional scaling theory, the two-point scaling theory was

modified in order to describe the influence of composition on the partial molar heat capacity

and volume during the micellization process. To verify the theory, isobaric heat capacities

and densities of aqueous sodium octanoate solutions were measured over wide composition

and temperature ranges and the modified approach was used to analyze the calculated partial

molar heat capacities and volumes of the surfactant in water. The results obtained indicate

that the micellization process is subject to the scaling laws. The results were compared with

those for other systems. Peculiar behavior of the critical indices was observed and correlated

with the structure of the micelles.

Keywords Scaling theory · Phase transition · Colloids · Heat capacity · Volume

1 Introduction

Research on colloidal systems is carried out using many different experimental techniques.

Interpreting the experimental data requires a model which combines a description of the

thermodynamics together with a description of the structure. One of such models is scaling

theory [1], particularly the two-point scaling theory introduced by Wojtczak et al. [2, 3] for

the description of paramagnetic–ferromagnetic phase transitions, which extends the theory

to the case of noncontinuous phase transitions. This idea came from the fact that some

analogies can be perceived between the phase transitions that occur in the two systems

Electronic supplementary material The online version of this article

(doi:10.1007/s10953-012-9795-6) contains supplementary material, which is available to authorized

users.

H. Piekarski (?) · M. Wasiak

Department of Physical Chemistry, University of Łód´ z, Pomorska 165, 90-236, Łód´ z, Poland

e-mail: kchfpiek@uni.lodz.pl

L. Wojtczak

Department of Solid State Physics, University of Łód´ z, Pomorska 149/153, 90-236, Łód´ z, Poland

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J Solution Chem (2012) 41:318–334319

(i.e. paramagnetic–ferromagnetic and regular solution–micellar systems): a dominant role

is played by surface effects and changes in degree of order of surface particles. Moreover,

the similarity between the shape of the curve describing the partial molar heat capacities at

constant pressure of surfactant on molality, and the shape of the temperature dependence of

the specific heat capacities under the constant external field in the case of magnetic systems,

gave us additional encouragement to consider whether phase transitions in these solutions

also follow scaling laws.

Our research showed indeed that the thermodynamic relations following from the scaling

theory can be used for the description of colloidal [4] and microheterogeneous systems

[5, 6]. The application of the theory, however, was completely “intuitive” and, in order to

interpret the results, a reformulation of the theory with respect to the original basis was

necessary.

2 Theoretical Study

In physics, critical phenomena is a term related with critical points, which are conditions

at which a phase boundary ceases to exist. For chemists, commonly known critical points

are the critical temperature (or pressure) at which vapor pressure curve terminates, or in

liquid systems the critical temperature of mixing. Another kind of critical phenomena are

the paramagnetic–ferromagnetic phase transitions, which are described by the conventional

scaling theory [1].

Scaling theory assumes that thermodynamic potentials are homogeneous functions with

respect to the external field and the reduced temperature ε. According to Stanley [1] the

static scaling hypothesis states that they are of more general form, namely: f(λax,λby) =

λf(x,y) rather then f(λx,λy) = λpf(x,y). The reduced temperature is defined as:

ε =|T −Tc|

where Tcis the critical temperature.

The external field term can have many meanings. In the conventional theory it was the

magnetic field [1] but it can be the pressure p when solutions are considered.

Application of scaling theory to the description of solutions requires the introduction of

one more variable, namely the reduced amount of solute μ defined as:

μ =|n2−nc|

where n2is the amount of the solute and ncis the critical amount of the solute; by analogy to

the conventional theory it is the amount of solute at which the critical phenomenon occurs.

As a result the thermodynamic potential, in our case the Gibbs energy, can be expressed

as a function of three variables:

Tc

(1)

nc

(2)

G = G(ε,p,μ)

(3)

When the amount of solvent n1is constant and its mass is equal to 1000 g, the amount of

solute n2is equal to the molality of solution by means of:

n1=1000

M1

which allows use of the molality instead of the amount of solute, and equates the critical

amount of the solute with the critical micellar concentration c.m.c.

⇒

n2= m

(4)

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320 J Solution Chem (2012) 41:318–334

The conventional theory describes the behavior of the molar properties of pure sub-

stances. In the case of solutions, partial molar quantities should be considered instead. The

singularities in the behavior of thermodynamic functions related to the critical phenomena

in solutions occur with increasing molality, not temperature, as in the case of conventional

theory.

On the basis of the fundamental assumption of scaling theory about the homogeneity of

thermodynamic potentials, we can write:

λG(ε,p,μ) = G?λaε,λbp,λcμ?

for any λ and arbitrary a, b, c (non-vanishing simultaneously).

Differentiating both sides of Eq. 5 we obtain:

(5)

λ∂2G(ε,p,μ)

∂p∂n

=∂2G(λaε,λbp,λcμ)

∂p∂n

= ±1

ncλbλc∂2G(λaε,λbp,λcμ)

∂(λbp)∂(λcμ)

(6)

Taking into account the thermodynamic fundamentals (∂2G

to the homogeneous function:

?∂2G(λaε,λbp,λcμ)

we obtain:

±1

and:

∂p∂n2)T,n1= V2and applying it

∂(λbp)∂(λcμ)

?

T,n1

= V2

?λaε,λbp,λcμ?

(7)

ncλV2(ε,p,μ) = ±1

ncλb+cV2

?λaε,λbp,λcμ?

(8)

V2= λb+c−1V2

?λaε,λbp,λcμ?

λ = μ−1

(9)

Since Eq. 9 is valid for all values of λ it must hold also for the particular choice:

c

(10)

which is equivalent to:

λcμ = 1(11)

If we consider a constant (or zero, in conventional theory) field (the pressure in our

case), constant temperature, and the above (Eqs. 10 and 11) value for λ, then the expression

V2(λaε,λbp,λcμ) becomes constant and we denote it as Vo

2. This leads to:

V2= μ

1−b−c

c

Vo

2

(12)

Taking into account the above results, the equation for the partial molar volume, as a

function of solution molality, at constant pressure and temperature, can be derived in the

form of a power law as follows:

V2= Vo

2μβ?

(13)

with the critical index β?:

β?=1−b −c

c

(14)

A similar relation can be derived for the partial molar heat capacity:

?∂3G(λaε,λbp,λcμ)

∂(λaε)2∂(λcμ)

?

p,n1

= −1

TCp,2

?λaε,λbp,λcμ?

(15)

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J Solution Chem (2012) 41:318–334 321

with an expression Cp,2(λaε,λbp,λcμ) which becomes constant at constant temperature

and pressure:

Cp,2= μ−2a+c−1

Cp,2= C◦

c

C◦

p,2

(16)

p,2μ−α?

(17)

and the critical index α?:

α?=2a +c −1

c

(18)

Relations for other thermodynamic functions can also be derived.

For the partial molar isothermal compressibility we obtain:

κT,2= κ◦

γ?=2b +c −1

T,2μγ?

(19)

c

(20)

Analogously, the relation for the pressure dependence of the partial molar volumes in

terms of the scaling law has the form:

V2= V◦

δ?=

2p

1

δ?

(21)

b

1−b −c

(22)

Similar to the original theory [1] where the relation ξ ∝ μ−νcan be proven, we suspect

the relation for the “partial molar correlation length”, a property which characterizes only

the solute, is in the form:

ξ2= ξ◦

2μ−ν?

(23)

with the critical index ν?.

In the conventional theory some relations between the critical indices, e.g. Rushbrooke

relation α +2β +γ = 2 or Widom relation β(δ −1) = γ, can be proved [1]. The modified

theory causes some of these relations to have different forms:

α?+2β?+γ?=2a +c −1+2−2b −2c +2b +c −1

but they can be transformed into the original form for the case when the temperature, rather

than molality, is a variable and consequently the constant a is the denominator of the fraction

of Eq. 24:

c

=2a

c

?= 2(24)

c → a

⇒

2a

c

= 2 (25)

Some other relations remain unchanged:

β??δ?−1?=1−b −c

c

?

b

1−b −c−1

?

=2b +c −1

c

= γ?

(26)

In the conventional approach the so called geometrical relation can be assumed to be in

the form:

G

kTV∝ ξ−d

(27)

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322J Solution Chem (2012) 41:318–334

resulting from the fact that thermodynamic potential divided by kTV has the same dimen-

sion as the inverse of volume, which can be expressed, using the correlation length, as ξ−d.

Taking into account that heat capacity is the second derivative of the thermodynamic po-

tential, the behavior of the Gibbs energy can be described using the heat capacity critical

index α in the form of G ∝ ε2−α. Correlation length, in turn, in terms of scaling can be

written as ξ ∝ ε−ν. Comparison of the indices of both functions results in the relation which

links the critical index α (describing the behavior of heat capacity) with the dimensional-

ity of the system d, by means of the critical index ν which describes the behavior of the

correlation length [7], namely:

2−α = νd

(28)

The modification of the theory makes that we cannot be sure if this relation is true in the

case of a solution. However, assuming the relation:

2−α?= ν?d?

(29)

by analogy to Eq. 28, we can see that d?denotes the dimensionality with respect to solutions,

by analogy with the conventional approach, but a precise definition at this stage is difficult,

due the lack of appropriate experimental data.

Agreement of the relations between the critical indices indicates the possibility that our

treatment of the scaling theory could be applied to the description of solutions and phase

transitions induced by changes of solution composition.

The conventional scaling theory can be used for the description of the continuous phase

transitions(secondorder),inthecasewheretheorderparameterchangescontinuously.Itcan

be extended, however, to the case of noncontinuous phase transitions (first order). Such an

extension is described by the two-point scaling theory [2, 3], which is considered when the

system can appear in n phases confined by the stability points of each phase. The two-point

scaling is also based on the assumption that the thermodynamic potentials are generalized

homogeneous functions, although their singularities are related to the stability points of each

phase instead of the phase transition points which now are situated in the intervals between

stability points. This approach results in a different temperature scale for each phase. The

values of mspdenote the stability points for lower limits of the concentration of phase s

while the values msf represent the stability points for upper limits of the concentration of

the phase s. In our case it refers to the regular solution of surfactant monomers (s = 1) and

the phase of micellar structures (s = 2), respectively. In the case of a regular solution of

surfactant monomers we consider only the upper stability point, and therefore we denote

it simply as mf. Similarly for the micellar phase we consider only the lower limit denoted

as mp.

The crossing points of the curves for each phase indicate the critical molality mc, which

can be considered as the boundary between the concentration range in which the solution

predominantly exhibits properties characteristic for a regular solution (m < mc) and the

range in which the solution properties become typical for micellar systems (m > mc). This

point corresponds to the transition point in the conventional approach to scaling.

Considering the two-point version of scaling theory, the relations for the heat capacities

for each phase with respect to the molality variable m can be written as follows:

Cp,2(m ≤ mf) = C◦1

p,2

?

?m

1−

m

mf

?−α?

?−α?

1

(30a)

Cp,2(m ≥ mp) = C◦2

p,2

mp

−1

2

(30b)

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J Solution Chem (2012) 41:318–334 323

Taking into account that the apparent molar quantities can be determined directly from

experiment, the relations describing the concentration dependence of the apparent molar

heat capacities can be derived in terms of scaling:

?

?

For the calculation of the partial molar heat capacities Cp,2the derivative of the apparent

molar heat capacity Cp,?,2is needed:

Cp,?,2(m ≤ mf) = C◦1

p,2

1

1−α?

1

1−α?

1

?mf

?mp

m

?

?

q◦

f−

?

?m

1−

m

mf

?1−α?

?1−α?

1?

2?

(31a)

Cp,?,2(m ≥ mp) = C◦2

p,2

2

m

q◦

p+

mp

−1

(31b)

Cp,2= Cp,?,2+mdCp,?,2

dm

(32)

and this quantity can be also expressed in terms of scaling:

dCp,?,2

dm

(m ≤ mf) = C◦2

p,2

?

1

1−α?

1

?

?mf

m2

×

?

−q◦

f+

1−

m

mf

+?1−α?

1

? m

mf

??

1−

m

mf

?−α?

1?

(33a)

dCp,?,2

dm

(m ≥ mp) = C◦2

p,2

?

1

1−α?

2

?m

?mp

m2

×

?

−q◦

p−

mp

−1−?1−α?

2

? m

mp

??m

mp

−1

?−α?

2?

(33b)

Similar equations can be written for each phase in the case of partial molar volumes:

V2(m ≤ mf) = V◦1

2

?

?m

1−

m

mf

?β?

?β?

1

(34a)

V2(m ≥ mp) = V◦2

2

mp

−1

2

(34b)

as well as for the apparent molar volumes:

V?,2(m ≤ mf) = V◦1

2

?

?

1

1+β?

1

1+β?

1

?mf

?mp

m

?

?

q◦

f−

?

?m

1−

m

mf

?1+β?

?1+β?

1?

2?

(35a)

V?,2(m ≥ mp) = V◦2

2

2

m

q◦

p+

mp

−1

(35b)

and their derivatives:

dV?,2

dm

(m ≤ mf) = V◦2

2

?

1

1+β?

1

?

?mf

m2

×

?

−q◦

f+

1−

m

mf

+?1+β?

1

? m

mf

??

1−

m

mf

?β?

1?

(36a)

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324J Solution Chem (2012) 41:318–334

Fig. 1 Qualitative influence of

molality on the partial molar heat

capacity and on the partial molar

volume according to the

two-point scaling theory

dV?,2

dm

(m ≥ mp) = V◦2

2

?

1

1+β?

−q◦

2

?m

?mp

m2

×

?

p−

mp

−1−?1+β?

2

? m

mp

??m

mp

−1

?β?

2?

(36b)

The shape of the curve for the partial molar heat capacity (and for the partial molar

volume) versus molality following from the two-point scaling theory is the same as the

shape of the curve for the heat capacity (and magnetization, respectively) versus temperature

following from the conventional scaling theory. The graphical representation of the obtained

results is showed in Fig. 1.

Intheoriginaltheorythemagnetizationdenotestheorderparameter, thevalueofwhichis

equal to zero after the transition point, and differs from zero before the critical point. In our

case the partial molar volumes can also correspond to an order parameter whose behavior is

characterized as follows: the parameter has a more or less constant value before the c.m.c.

while it starts to increase after the c.m.c.

To determine if our approach is applicable and if the phase transition in a solution obeys

the scaling laws, comparison with experimental data is necessary. We decided to study rep-

resentative compounds of a few different types of surfactants and test our approach.

In our previous research we investigated solutions of the cationic surfactant de-

cyltrimethylammonium bromide (C10TAB) [4], and the microheterogeneous aqueous so-

lutions of 2-butoxyethanol [5] and 2-(2-hexyloxyethoxy)ethanol [6].

Equations 30a, 30b to 33a, 33b, which describe the concentration dependence of heat

capacity, obtained as the result of the theory modification, are the same as those used pre-

viously [5, 6], Thus, we can see that our “intuitive” approach was correct, and the results

obtained can be discussed in terms of the modified theory, presented now in a more rigorous

form.

In order to extend the group of systems on which we test our approach, in the present

paper we study the heat capacity of the solutions of a representative of another group of the

surfactants, the anionic surfactant sodium octanoate (OctNa).

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J Solution Chem (2012) 41:318–334 325

3 Experimental

According to De Lisi et al. [8] the alkyltrimethylammonium bromides undergo micellize in

a different way than the carboxylates since their apparent molar heat capacity goes through

a maximum at the c.m.c., while for carboxylates a “hump” at c.m.c. can be observed. These

authors have analyzed the influence of composition on the apparent molar heat capacity, the

apparent molarrelative enthalpies andtheapparentmolarvolumesofsodiumdecanoate with

a “phase-separation” model, and they attributed the “hump” at the c.m.c. to the “relaxation”

contribution.

Sodium octanoate in aqueous solution has quite a high c.m.c. [9], which is crucial from

the point of view of calorimetric experiments.

Sodium octanoate micelles have a small aggregation number [10–14]. It is known from

the computer simulation [10–14] that, in the case of small micelles such as those of OctNa,

large thermal fluctuations are present.

González-Pérez et al. investigated sodium octanoate aqueous solutions in the molality

ranges 0.04–0.9 mol·kg−1[9] and 0.4–2.0 mol·kg−1[15]. From the speed of sound and

density versus molality isotherms the authors calculated c.m.c. values which decrease from

0.3828 mol·kg−1at 298.15 K to 0.3589 mol·kg−1at 318.15 K [9]. The same authors ob-

served also the point called the critical micelle transition (c.m.t.) [15] at which the transition

from spherical to nonspherical micelles occurs. The obtained c.m.t. values have a maximum

value of 0.95 mol·kg−1at 308 K [15]. The values were calculated from the density and

conductivity data but it should be pointed out that the changes in these functions were very

small and the criteria for determination of the c.m.t. are doubtful. In the case of adiabatic

compressibilities the authors did not observe a change of slope but only an inversion of the

temperature behavior of compressibilities at the point corresponding to c.m.t. The authors

calculated the apparent molar volumes and compressibilities of the surfactant in water but

did not present the appropriate values or graphs.

Apparent molar volumes and adiabatic compressibilities at 298.15 K in the molality

range up to 2 mol·kg−1were investigated also by Huang et al. [16, 17], who observed that

the apparent molar volume of surfactant increases slightly up to 0.4 mol·kg−1after which it

starts to increase rapidly. At higher concentrations it remains almost constant. Huang et al.

[16, 17] determined aggregation numbers equal to 10–15 [17], but did not observe the c.m.t.

A similar value, equal to 16 at 293.15 K was obtained by D. Adair et al. [18] from viscosity

measurements.

Aqueous solutions of sodium octanoate were extensively investigated by Ekwall et al.

[19–25]. By analyzing the influence of composition on partial molar volumes, the au-

thors indicate the existence of three critical points of micellization. Rapid increases of ap-

parent molar volumes were observed at 0.38–0.52 mol·dm−3, 1.15–1.41 mol·dm−3, and

2.4–3.1 mol·dm−3intervals; however, the increase in the second region is much smaller than

in the first and the change in the third region is not pronounced. On the basis of viscosity

measurements, Ekwall et al. [20, 21] found that spherical micelles existed in the solution in

the concentration range up to 1.9 mol·dm−3. Above this concentration cylindrical micelles

were observed.

Hayter and Zemb [26] investigated OctNa aqueous solutions using small angle neutron

scattering (SANS) in the concentration range between the c.m.c. and c.m.t. values suggested

by other authors. They observed a linear increase of the aggregation number with increasing

concentration. The aggregation number obtained changed from 15 at the concentration of

0.60 mol·dm−3up to 23 for a 1.20 mol·dm−3solution. An extrapolation gives the value

13±1 for a solution at the concentration equal to c.m.c. On the basis of the results obtained,

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326J Solution Chem (2012) 41:318–334

the authors [26] suggest a continuous increase of the micelle size rather than a transition to

spherical structure.

Apparent molar heat capacities and volumes were investigated by Rosenholm et al. [27,

28] who observed a maximum in the apparent molar heat capacity versus molality curve

as well as a rapid increase of the apparent molar volumes in the concentration range above

0.37 mol·dm−3. These measurements were only at 298.15 K and not over a large concentra-

tion range, and this was also a reason for us to investigate this system.

3.1 Experimental

Sodiumoctanoate(Sigma-Aldrich ≥ 98%) wasusedasreceivedwithoutfurtherpurification.

Deionized water was triply distilled in an argon atmosphere and degassed under vacuum. All

solutions were prepared by weight.

The heat capacities under constant pressure were measured by means of a high sensitivity

differential calorimeter Micro DSC III (Setaram, France) based on the Calvet principle. The

cpmeasurements were carried out within the temperature range 285.15–358.15 K using the

“continuous with reference” mode. In this method the differential heat flow, between a cell

filled with the investigated liquid and a reference, occurring during a continuous increase

of calorimeter temperature is determined. In the temperature range under investigation, the

scanning rate was 0.35 K·min−1. For measurements weused a batch-type cell of about 1cm3

volume. The cpvalues for each temperature were calculated from cp= f(T) function, by

interpolation. As a reference substance of known heat capacity, water was used. Using the

procedure developed in our laboratory and described widely by Góralski et al. [29], the

uncertainty in the cpvalues can be estimated to be smaller than 0.25% with an error in the

absolute temperature determination of 0.05 K.

The densities of all surfactant solutions were measured using a flow densimeter (Sodev,

model 03, Sherbrook, Quebec). The densimeter was calibrated with reference to pure wa-

ter and nitrogen gas (absolute 1 atm). The density of water was taken from Kell [30] and

that of nitrogen was calculated from the van der Waals equation of state. The reproducibil-

ity of the density measurements was 5 × 10−6g·cm−3and the uncertainty was estimated

as 2 × 10−5g·cm−3. The stability of the temperature in the densimeter (±0.001 K) was

achieved with a closed loop thermostat and controlled using a special device calibrated with

a PT 100 thermometer that allowed estimation of the absolute uncertainty in the temperature

as 0.02 K. The measurements were performed under static conditions.

All solutions were prepared by weight with a mean uncertainty in the molality of 2 ×

10−5mol·kg−1.

3.2 Results

3.2.1 Heat Capacity

From the calorimetrically determined cp(listed in Table S1 of supplementary material) data

the apparent molar heat capacities were calculated from the following equation:

Cp,?,2= M2cp+1000(cp−c∗

The partial molar heat capacities were then calculated according to Eq. 32. This pro-

cedure was previously used for the determination of partial molar heat capacities of other

colloidal systems and satisfactory agreement was found with data available in the litera-

ture [4, 5]. The values of the apparent molar heat capacities of sodium octanoate in aqueous

p,1)

m2

(37)

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J Solution Chem (2012) 41:318–334 327

Fig. 2 Influence of composition

on the apparent molar heat

capacity of sodium octanoate in

aqueous solutions. Symbols

represent experimental results at

P 293.15 K, ! 313.15 K,

1 333.15 K and e 353.15 K

solution, obtained as a function of molality at four temperatures, are shown in Fig. 2. For

each temperature a maximum can be observed at a salt molality of about 0.3–0.5 mol·kg−1,

which shifts to the higher values with increasing temperature. At the lowest temperature the

maximum is most pronounced. The locations of these maxima correlate qualitatively with

the c.m.c. values available in the literature [9, 15]. The shape of the curve for the partial

molar heat capacity versus molality, and its values, correspond well with those investigated

by Rosenholm et al. [27, 28]. At low molality, the apparent molar heat capacities increase

with increasing temperature. In the salt molality range 0.6–1.0 mol·kg−1this relation starts

to change and, at high molality, inverse behavior of Cp,?,2is observed. Within the same

composition range González-Pérez et al. observed similar behavior for the adiabatic com-

pressibilities and, on the basis of conductometry, the authors [15] also identified the c.m.t.

within that range.

To analyze the data, the two-point scaling approach was applied. The apparent and partial

molar heat capacities as well as the derivatives of the apparent molar heat capacities were

fitted simultaneously to Eqs. 31a, 31b, 30a, 30b and 33a, 33b, respectively, for both phases.

The results of this fitting for the four chosen temperatures are shown in Figs. 3, 4, 5, 6

together with the experimental points. The values of the critical indices obtained and other

fitting parameters are given in Tables 1 and 2.

3.2.2 Molar Volumes

From the measured density data (listed in Table S2 of supplementary material) the apparent

molar volumes at 293.15 K were calculated from the following equation:

+1000(d −d∗

where d∗

as follows:

V2= V?,2+m2dV?,2

The values of the apparent molar volumes of sodium octanoate in aqueous solution, as a

function of molality at 293.15 K, are shown in Fig. 7.

The two-point scaling approach was applied to the analysis of the data. The apparent,

partial molar volumes and the derivatives of apparent molar volumes were fitted simulta-

neously to Eqs. 35a, 35b, 34a, 34b and 36a, 36b, respectively, for both phases. The results

are shown in Fig. 8 together with the experimental data. The values of the critical indices

obtained and other fitting parameters are given in Tables 3 and 4.

V?,2=M2

d

1)

m2dd∗

1

(38)

1is density of the pure solvent (water). The partial molar volumes were calculated

dm2

(39)

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328J Solution Chem (2012) 41:318–334

Fig. 3 Influence of composition

on the apparent, partial and the

derivative of apparent molar heat

capacities of sodium octanoate in

aqueous solution at 293.15 K.

Curves represent the best fit of

the two-point scaling equations

Table 1 The values of the critical exponents α?

critical molality mcof the considered system

1, α?

2, stability points mf, mp, constants C◦1

p,2,C◦2

p,2, and the

T

/K

α?

1

α?

2

mf

/mol·kg−1

mp

/mol·kg−1

C◦1

p,2

/J·mol−1·K−1

C◦2

p,2

/J·mol−1·K−1

mc

/mol·kg−1

293.15

313.15

333.15

353.15

0.125

0.125

0.125

0.125

0.125

0.125

0.125

0.125

0.446

0.448

0.490

0.312

0.440

0.381

0.338

0.263

582

589

560

525

269

303

357

438

0.4400

0.3813

0.3408

0.2711

4 Discussion

As one can see from Figs. 3, 4, 5, 6 and 8, the influence of composition on the partial molar

heat capacities as well as the partial molar volumes is subject to the scaling laws. The values

of the critical indices obtained are temperature independent as was observed previously

for all of the examined systems [4–6]. The parameter α?

0.125 as in all the previously examined systems and the parameter α?

as the parameter α?

decyltrimethylammonium bromide. The values of critical molalities and their changes with

the temperature are close to the values determined by means of different techniques given

in literature [9, 15–17, 19–25]. Also the values of stability limits for each phase determined

1has also the same value α?

2has the same value

1=

1. The same situation was observed previously for aqueous solutions of

Page 12

J Solution Chem (2012) 41:318–334329

Fig. 4 Influence of composition

on the apparent, partial and the

derivative of apparent molar heat

capacities of sodium octanoate in

aqueous solution at 313.15 K.

Curves represent the best fit of

the two-point scaling equations

Table 2 Values of the two fitting

parameters q◦

the description of the

experimental behavior of the

Cp,?,2and (dCp,?,2/dm)

functions

fand q◦

pused for

T

/K

q◦

f

q◦

p

293.15

313.15

333.15

353.15

1.0007

1.0010

1.0046

1.0000

2.1197

1.7454

1.3920

1.0586

Table 3 Values of the critical exponents β?

critical molality mcof the considered system

1, β?

2, the stability points mf,mp, constants V01

2, V02

2, and the

T

/K

β?

×103

1

β?

×103

2

mf

/mol·kg−1

mp

/mol·kg−1

V◦1

2

/cm3·mol−1

V◦2

2

/cm3·mol−1

mc

/mol·kg−1

293.15

−0.570.68 0.4380.416 162164 0.4180

fromthepartialmolarheatcapacitybehaviorandthepartialmolarvolumebehaviorareclose

to each other. Similar observations in the case of the critical molality additionally confirm

that the micellization process is subject to the scaling laws.

As mentioned above, the critical index describing the behavior of the heat capacity in the

conventional theory can be related to the dimensionality of the system by means of Eq. 28.

Page 13

330J Solution Chem (2012) 41:318–334

Fig. 5 Influence of composition

on the apparent, partial and the

derivative of apparent molar heat

capacities of sodium octanoate in

aqueous solution at 333.15 K.

Curves represent the best fit of

the two-point scaling equations

Table 4 Values of the two fitting

parameters qf

the description of the

experimental behavior of the

V?,2and dV?,2/dm functions

oand qp

oused for

T

/K

q◦

f

q◦

p

293.15 1.00010.9960

For a precise calculation of the dimensionality, values of the ν?

it can be observed that the values of α?

solutions of monomers (concentration range below c.m.c.), the values of α?

equal to 0.125 for each previously analyzed system [4–6] as well as for the present system.

For the micellar solutions, in which spherical micelles are formed (C10TAB [4], OctNa,

present work), the α?

(2-butoxyethanol [5], and 2-(2-hexyloxyethoxy)etanol aqueous solutions [6]), the value of

α?

iindices are necessary, but

iare related to the structure of the solution. For the

1obtained are

2index also takes the value 0.125. For microheterogeneous solutions

2is equal to 0.33 for both systems.

5 Conclusion

The two-point scaling theory was modified in order to describe the phase transition in so-

lution. Based on assumptions similar to those in the case of the conventional scaling theory

and using the same procedures, relations describing the influence of composition on the

partial molar heat capacities and volumes were derived.

Page 14

J Solution Chem (2012) 41:318–334 331

Fig. 6 Influence of composition

on of the apparent, partial and the

derivative of apparent molar heat

capacities of sodium octanoate in

aqueous solution at 353.15 K.

Curves represent the best fit of

the two-point scaling equations

Fig. 7 Influence of composition

on the apparent molar volumes of

sodium octanoate in aqueous

solution at 293.15 K

The form of the relations obtained is the same as in our “intuitive” approach [5, 6],

and the previously obtained results can be discussed in terms of the modified theory in the

present paper.

Our approach can be extended to the discussion of some other thermodynamic properties

such as the partial molar isothermal compressibilities. The appropriate equations can be

derived, but the lack of experimental data prevents our obtaining the other critical indices

and further discussion.

The values of the critical indices α?

same as for all the previously investigated systems, and the values of α?

the decyltrimethylammonium bromide solutions, which could indicate universality of the

1for the sodium octanoate monomer solution are the

2are the same as for

Page 15

332J Solution Chem (2012) 41:318–334

Fig. 8 Influence of composition

on the apparent, partial and the

derivative of apparent molar

volumes of sodium octanoate in

aqueous solution at 293.15 K.

Symbols represent experimental

results. Curves represent the best

fit of the two-point scaling

equations

micellization process. The value of the critical index α?

gregates formed. The stability points for each phase, and critical molalities determined from

the scaling analysis of data from two independent measurements, were close to each other.

The results obtained confirm that the aggregation process is subject to scaling laws and the

scaling analysis can provide much useful information. Our model, based on scaling laws,

seems to be more versatile than e.g. the phase separation model used by De Lisi et al. [8] for

the analysis of the micellization of the carboxylates, which, as the authors suggest, does not

work well for alkyltrimethylammonium bromides solutions. In our case we cannot distin-

guish the “relaxation” contribution. It seems to be incorporated in our model, as the scaling

laws consider the singular part of the thermodynamic potentials. This “relaxation” effect is

related to the shift of equilibrium when the temperature is changed [31]. This equilibrium

means that the change of thermodynamic properties at c.m.c. can be “slow” (occurs over

some concentration range). In our approach this fact is taken into account by the “phase

coexistence” region, whose width (span) changes with the temperature, and depends on the

type of surfactant. From this point of view our approach exhibits some advantages for ana-

lyzing the behavior of different types of surfactants.

2depends on the structure of the ag-

Open Access

which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the

source are credited.

This article is distributed under the terms of the Creative Commons Attribution License

Page 16

J Solution Chem (2012) 41:318–334 333

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