JOURNAL OF CHEMICAL PHYSICS
VOLUME 120, NUMBER 14
8 APRIL 2004
Towards an understanding of the heat capacity of liquids.
A simple two-state model for molecular association
Claudio A. Cerdeiriña, Diego González-Salgado, and Luis Romani
Departamento de Fı́sica Aplicada, Universidad de Vigo, Facultad de Ciencias del Campus de Ourense,
E-32004, Spain
Marı́a del Carmen Delgadoa) and Luis A. Torres
Departamento de Quı́mica, Centro de Investigación y Estudios Avanzados del I.P.N., Apdo. Postal 14-740,
México D.F. 07000, México
Miguel Costasb)
Laboratorio de Termofı́sica, Departamento de Fisicoquı́mica, Facultad de Quı́mica,
Universidad Nacional Autónoma de México, Cd. Universitaria, México DF 04510, México
共Received 1 December 2003; accepted 14 January 2004兲
A model for the temperature dependence of the isobaric heat capacity of associated pure liquids
C op,m (T) is proposed. Taking the ideal gas as a reference state, the residual heat capacity is divided
and associational C res,ass
contributions. Statistical mechanics is used to obtain
into nonspecific C res,ns
p
p
res,ass
by means of a two-state model. All the experimentally observed C op,m (T) types of curves in
Cp
the literature are qualitatively described from the combination of the ideal gas heat capacity C id
p (T)
o
(T).
The
existence
of
C
(T)
curves
with
a
maximum
is
predicted
and
experimentally
and C res,ass
p
p,m
observed, for the first time, through the measurement of C op,m (T) for highly sterically hindered
alcohols. A detailed quantitative analysis of C op,m (T) for several series of substances 共n-alkanes,
linear and branched alcohols, and thiols兲 is made. All the basic features of C op,m (T) at atmospheric
and high pressures are successfully described, the model parameters being physically meaningful. In
particular, the molecular association energies and the C res,ns
values from the proposed model are
p
found to be in agreement with those obtained through quantum mechanical ab initio calculations and
the Flory model, respectively. It is concluded that C op,m (T) is governed by the association energy
between molecules, their self-association capability and molecular size. © 2004 American
Institute of Physics. 关DOI: 10.1063/1.1667469兴
I. INTRODUCTION
In this work, we develop a model for the residual heat
capacity of a pure liquid, C res
p , based on a simple statistical
mechanical two-state model for its associational part,
. When C res,ass
is added to the ideal gas heat capacity
C res,ass
p
p
id
C p , the experimental C op,m (T) dependences 共i兲–共iii兲 are described. Furthermore, the existence of C op,m (T) curves with
共iv兲 a concave to convex inflection point 共as T increases兲 and
共v兲 a maximum, are predicted. Literature data for long chain
1-alcohols confirmed the existence of type 共iv兲 C op,m (T) dependence. Examples of type 共v兲 C op,m (T) curves are not
available in the literature, but their existence was experimentally corroborated here through the measurement of heat capacities for highly sterically hindered alcohols. Very recently,
C op,m (T) data at high pressures for n-alkanes6 –9 and for
1-alcohols10 have been reported. In the 1-alcohols case, the
C op,m (T) curve displays a shallow maximum followed by an
also shallow minimum as the temperature increases. This
behavior can also be described by the model presented here.
The model is applied to several series of substances allowing
us to evaluate its parameters, whose physical meaning and
reliability are examined in detail. It will be shown that the
qualitative and quantitative analysis presented here provides
a general framework for understanding all the experimentally
found C op,m (T) dependences for associated pure liquids.
The isobaric molar heat capacities, C op,m , for a large
number of pure substances in the liquid state have been assembled and critically reviewed in a series of compilations1,2
and comprehensive databases.3 An inspection of this information shows that, at atmospheric pressure, the temperature
dependence of the isobaric heat capacity, i.e., the experimental C op,m (T) curves, can take several forms,4 namely 共i兲
monotonically increasing curves, 共ii兲 curves with a shallow
minimum, and 共iii兲 curves with an inflection point 共convex to
concave, as T increases兲. In Ref. 4, an additional type of
temperature dependence was discussed, but this is outside
the scope of this work.5 Although for some substances molecular level rationalizations have been given for these
C op,m (T) dependences, no attempt to understand the variety
of C op,m (T) behaviors in a unified form has been reported. In
fact, it has been argued4 that there is no general relationship
for the description of the several experimental C op,m (T)
curves.
a兲
Present address: Unidad Queretaro del CINVESTAV, Apdo. Postal 1-708,
Queretaro 76001, México.
b兲
Author to whom correspondence should be addressed. Electronic mail:
costasmi@servidor.unam.mx
0021-9606/2004/120(14)/6648/12/$22.00
6648
© 2004 American Institute of Physics
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J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
Heat capacity of associated liquids
FIG. 1. 共a兲 Isobaric molar heat capacities C op,m for 1-hexanol 共䊐兲 from Ref.
2 and n-heptane 共䊏兲 from Ref. 1 and its contributions 共b兲 C id
p for 1-hexanol
o
id
共䊊兲 and n-heptane 共䊉兲 from Ref. 3, and C res
p ⫽C p,m ⫺C p for 1-hexanol 共䉭兲
res
and n-heptane 共䉱兲; dashed line is 关 C res
p 共1-hexanol兲-C p 共n-heptane兲兴.
6649
shows C op,m (T) for both liquids while Fig. 1共b兲 displays their
res
o
id
C id
p (T) and C p (T) 关obtained as C p,m (T)-C p (T)]. Since
res,ass
n-heptane is not an associated liquid, C p ⫽0 and
res,ns
, whereas for 1-hexanol both C res,ns
and
C res
p (T)⫽C p
p
res,ass
contribute to C res
.
Considering
that
association
is
the
Cp
p
main difference between 1-hexanol and n-heptane or, equivalently, that C res,ns
is approximately the same for both liquids,
p
for
1-hexanol
can be estimated as 关 C res
C res,ass
p
p 共1-hexanol兲res
C p 共n-heptane兲兴 and is shown in Fig. 1共b兲 共dashed line兲. It is
can be
clear then that, to a good first approximation, C res,ns
p
considered as temperature independent. This approximation
receives support from the observation that for substances devoid of any association such as branched alkanes, C res
p
changes very slightly with temperature,1–3 and from the fact
that successful theories for nonassociating liquids, such as
Flory theory,11 predicts that C res
p must be almost temperature
independent. For a nonassociated liquid the temperature dependence of C op,m is then dominated by C id
p (T), i.e.,
/dT.
In
contrast,
for
an
associated
liquid
dC op,m /dT⬵dC id
p
o
id
res,ass
dC p,m /dT⬵dC p /dT⫹dC p /dT. Hence, in order to describe the several experimentally found C op,m (T) curves it is
(T). This is done in
necessary to develop a model for C res,ass
p
the following subsection.
II. THE MODEL
B. A simple statistical mechanical
two-state model for C res,ass
p
A. Partitioning of the heat capacity of a liquid
Using the ideal gas as a reference state, the isobaric heat
capacity of a liquid C op,m can be expressed as
res
C op,m ⫽C id
p ⫹C p .
共1兲
In Eq. 共1兲, the first contribution is the ideal gas heat capacity,
C id
p , which depends on molecular degrees of freedom but not
on molecular interactions. C id
p always increases with temperature 共with a small curvature兲 and with molecular size
whereas dC id
p /dT increases with molecular size. The second
contribution, C res
p , is the residual heat capacity, which can be
modeled starting from the following exact thermodynamic
relation:
冋 冉 冊 册冉 冊
C op,m ⫺C id
p ⫽⫺ 共 d⌬H vap /dT 兲 sat⫹ V⫺T
⳵V
⳵T
p
dp
dT
. 共2兲
sat
Away from critical conditions, the second term in Eq. 共2兲 is
negligible and thus C res
p equals, to a very good first approximation, the temperature derivative of the enthalpy of vaporization. Since ⌬H vap is a configurational property, C res
p can
be split into two contributions as
res,ass
C res
⫹C res,ns
,
p ⫽C p
p
C res,ass
p
共3兲
arises from the self-association of molecules in
where
the pure liquid—a definition of what is meant in this work by
association is detailed in the next section—and C res,ns
arises
p
from nonspecific contributions i.e., from weak London dispersion forces, long-range electrostatic forces and free voland C res,ass
can
ume effects. The relative importance of C res,ns
p
p
be explored using data for two liquids whose ideal gas heat
capacity are similar but differ markedly in their association
ability, namely a strongly associated liquid 共1-hexanol兲 and a
liquid where association is absent 共n-heptane兲. Figure 1共a兲
Steming from their simplicity and ability to capture the
essential elements of the physical behavior of the systems,
two state models have been widely employed to study many
phenomena such as protein unfolding,12 conformational
changes in liquid polymers,13 cluster aggregation,14 the hydrophobic effect in aqueous solutions,15 the anomalous properties of pure water,16 the behavior of rotational isomers,17
the effects of vacant sites in crystals,18 etc. In this context,
the novelty of the present work consists of the application of
a two-state model to the description of the isobaric heat capacity of pure liquids. The employment of a two-state model
to study the heat capacity behavior of associated mixtures
was embodied, but just qualitatively explored, in early
reports.19 More recently, it was also employed in order to
qualitatively explain the temperature dependence of the excess molar isobaric heat capacity of alkanol⫹alkane
mixtures.20
In order to model the associational part of the residual
heat capacity, it is first necessary to define what is understood by association. In this work, we consider that the molecules in the bulk of the liquid are associated when the intermolecular forces are of such magnitude, beyond a certain
critical lower limit, that an aggregate or collective of molecules is formed. This molecular aggregate must be such that
a new ‘‘species,’’ constituted by two or more of the individual molecules 共monomers兲, can be detected through the
measurement of some property of the system, spectroscopic
or thermodynamic. This is an operational definition since it
depends on the present detection limits of the experimental
techniques that, clearly, are continuously improving. If there
is not experimental evidence of the existence of such new
species, the liquid is regarded as nonassociated. On the other
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6650
Cerdeiriña et al.
J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
hand, if such evidence is present the liquid is considered as
associated. There are varying strengths of association in pure
liquids, from the weak association between molecules in
pure long chain linear alkanes19,21 共termed orientational order
or correlation of molecular orientations兲 to the strong association between alcohol molecules 共hydrogen bonds兲.
In microscopic terms, association can be modeled considering that individual molecules in the liquid are in different states depending on whether they are or not part of a
molecular aggregate. This is the basis of the simple model
presented here. We assume that each molecule in the liquid is
capable of two energy levels corresponding to the associated
state (A i ) with energy ⑀ A i and the dissociated monomer 共A兲
with energy ⑀ A (⌬ ⑀ ⫽ ⑀ A ⫺ ⑀ A i ⬎0). Volume and enthalpy
changes (⌬ v and ⌬h) between the two levels are also considered, such that ⌬ ⑀ ⫽⌬h⫺ P⌬ v . We further assume that
there are many molecular geometries 共position of atoms,
bond angles, etc.兲 which are compatible with each of the two
states. We describe this with the degeneracy numbers g A and
g A i , which represent the number of different accessible 共or
possible兲 molecular arrangements or geometries with energies ⑀ A and ⑀ A i , enthalpies h A and h A i and volumes v A and
v A i , respectively. In congruence with their significance, g A
Ⰷg A i , i.e., the number of accessible geometries for a molecule abruptly decreases in going from the dissociated or
monomeric state to the associated state, owing to the directional character of the intermolecular interactions producing
association. With this consideration, the isothermal–isobaric
. We consider N molensemble was used to obtain C res,ass
p
ecules in the liquid as independent identical objects. The
partition function for this system Y ass is related to that of an
individual object through Y ass⫽(Y mol) N where
Y mol⫽g A exp共 ⫺H A /RT 兲 ⫹g A i exp共 ⫺H A i /RT 兲 ,
共4兲
where the enthalpies for each level are expressed in a molar
basis. The Gibbs energy G⫽⫺RT ln Y and the isobaric heat
capacity C p ⫽⫺T( ␦ 2 G/ ␦ T 2 ) P is obtained from Eq. 共4兲 as
⫽R
C res,ass
p
冉 冊
⌬H
RT
2
r exp共 ⌬H/RT 兲
关 r⫹exp共 ⌬H/RT 兲兴 2
,
共5兲
with the association enthalpy ⌬H⫽H A ⫺H A i and r
⫽g A /g A i being considered temperature independent. It is
important to mention that, in Ref. 17, an expression, analogous to Eq. 共5兲, for the isochoric heat capacity was derived
and that when r⫽1 Eq. 共5兲 reduces to the equation for the
isobaric residual heat capacity reported many years ago in
Ref. 18. The occupation numbers of the two levels, in terms
of mole fractions, are
x A ⫽N A /N⫽
r
,
r⫹exp共 ⌬H/RT 兲
x A i ⫽N A i /N⫽
exp共 ⌬H/RT 兲
.
r⫹exp共 ⌬H/RT 兲
共6兲
共7兲
According to Eqs. 共6兲 and 共7兲 and owing to the restriction
g A Ⰷg A i , for strong association or low temperatures (⌬H
ⰇRT) N A ⰆN A i , and for weak association or high tempera-
FIG. 2. 共a兲 Associational part of the residual heat capacity, C res,ass
, from Eq.
p
共5兲 for different arbitrary sets of ⌬H 共in J mol⫺1兲 and the ratio r⫽g A /g A i
values: 25 000 and 1000 共A兲, 10 000 and 1000 共B兲, 25 000 and 10 000 共C兲
and 10 000 and 10 000 共D兲, 共b兲 fraction of molecules, from Eqs. 共6兲 and 共7兲,
on the monomer 共full lines兲 and the associated 共dashed lines兲 enthalpy levels
for 共A兲 and 共C兲 ⌬H and r values.
tures (⌬HⰆRT) N A ⰇN A i . Figure 2共a兲 shows the temperafrom Eq. 共5兲 for several arbitrary
ture dependence of C res,ass
p
sets of ⌬H and r values. The trends are as follows: 共i兲 As
⌬H decreases 共at fixed r兲 or as r increases 共at fixed ⌬H), the
maximum is displaced towards lower temperatures and 共ii兲
at the maximum increases
as r increases, the value of C res,ass
p
in magnitude and the peak becomes narrower. These trends
indicate that r can be considered as a measure of the selfassociation capability of the molecules. This is seen in Eq.
共7兲 that indicates that at constant T and ⌬H, as r decreases
the fraction of molecules on the associated state is increased,
as illustrated in Fig. 2共b兲. In other words, small r values
correspond to a high self-association capability. An increase
of the r value might occur as a result of a decrease in g A i
owing to either a sterical hindrance or an increase in molecular size, that impose a reduction in the number of ‘‘successful’’ geometries or arrangements in which two or more molecules can associate.
C. Qualitative framework for the analysis of C po , m „ T …
Figure 3 shows a schematic representation of the isobaric heat capacity of a liquid as a function of the dimensionless parameter RT/⌬H. In Fig. 3共a兲, the three compores,ns
, and C res,ass
are shown,
nents of C op,m , namely C id
p , Cp
p
o
while in Fig. 3共b兲 the total C p,m is displayed. In Fig. 3共a兲,
has been taken as temperature independent and C id
C res,ns
p
p is
shown for large 共case 1兲 and small 共case 2兲 molecular sizes.
Clearly, as indicated in Fig. 3共b兲, the five different types of
C op,m (T) curves 共i兲–共v兲 are described by the present model.
Figure 3共b兲 indicates that the temperature dependence of
and
C op,m is the result of the relative importance of C res,ass
p
.
Note
that
type
共i兲
curves
can
occur
either
at
low
T
and
C id
p
large ⌬H values or at high T and small ⌬H values. Ex-
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J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
FIG. 3. Schematic representation of the isobaric heat capacity of a liquid
C op,m as a function of the dimensionless parameter RT/⌬H. 共a兲 Three com共dot–dashed兲,
ponents of C op,m according to Eqs. 共1兲 and 共3兲, namely C res,ass
p
id
C res,ns
共dashed兲 and C id
p
p 共full line兲; C p is shown for two cases, namely for
large 共1兲 and small 共2兲 molecular sizes. 共b兲 Total C op,m for cases 共1兲 and 共2兲;
the five different types of experimentally observed C op,m curves 共i兲–共v兲 are
indicated.
amples of types 共i兲–共iv兲 C op,m (T) curves 共from the literature兲
are shown in Fig. 4. Examples of type 共v兲 C op,m (T) curves
have been experimentally found for the first time in this
work22 using a highly accurate differential scanning calorimeter and are shown below 共Fig. 7兲.
For an homologous series of substances where ⌬H is
expected to be constant, as molecular size increases C op,m (T)
Heat capacity of associated liquids
6651
must change following case 1 in Fig. 3共b兲, i.e., change from
a type 共i兲 behavior to 共iii兲, then to 共iv兲 and, finally, for very
large size members return to being type 共i兲. This is the case
of the series of 1-alcohols in Fig. 4共a兲 where the association
is strong and takes place through O–H¯O hydrogen bonds.
If ⌬H is small, as for the homologous series of 1-thiols
where association takes place through weak S–H¯S hydrogen bonds, a type 共ii兲 behavior is expected to occur. This is
illustrated for three short chain 1-thiols in Fig. 4共b兲. For substances where r is large, the importance of C res,ass
increases
p
关compare curves A and C in Fig. 2共a兲兴 producing a maximum
in C op,m (T); this is the case of highly sterically hindered
alcohols 共Fig. 7兲.
It appears that the particular C op,m (T) displayed by a
given substance depends on the association enthalpy ⌬H and
r, the self-association capability that itself depends on molecular geometry and size. On the other hand, C id
p (T) depends on molecular size that, in turn, influences the temperature range where the substance is in the liquid state. An
examination of the available data in the literature1–3 and the
present experimental results shows that at atmospheric pressure the majority of substances belong to only one of the five
types of C op,m (T) dependences. In other words, at atmospheric pressure the continuous change from one type of
curve to another is interrupted by the melting and boiling
points. However, since application of pressure increases the
temperature range for the substance to be in the liquid state,
the C op,m (T) behavior under those conditions might include
more than one region in Fig. 3共b兲. The appearance of a multiregion C op,m (T), which is far from being a common occurrence at atmospheric pressure, should be more widespread at
elevated pressures.
III. DISCUSSION
A. Data treatment and literature sources
The two state model described above can be quantitatively applied to reliable experimental data. To perform this
analysis, from Eqs. 共1兲, 共3兲, and 共5兲 the residual heat capacity
is expressed as
o
id
C res
p ⫽C p,m 共 T 兲 ⫺C p 共 T 兲
⫽C res,ns
⫹R
p
FIG. 4. Experimental C op,m from Refs. 1 to 3 illustrating type 共i兲–共iv兲 temperature dependences. 共a兲 1-propanol 关 C3 OH, type 共i兲兴, 1-hexanol 关 C6 OH,
type 共iii兲兴, 1-decanol 关 C10OH, type 共iv兲兴, 1-tridecanol 关 C13OH, type 共iv兲兴
and 1-octadecanol 关 C18OH, type 共iv兲兴, 共b兲 methanethiol 关 C1 SH, type 共ii兲兴,
ethanethiol 关 C2 SH, type 共ii兲兴 and 1-propanethiol 关 C3 SH, type 共ii兲兴. Examples of type 共v兲 curves are shown in Fig. 7 and in Ref. 22. Full lines are
calculated C op,m (T) using the proposed model.
冉 冊
⌬H
RT
2
r exp共 ⌬H/RT 兲
关 r⫹exp共 ⌬H/RT 兲兴 2
,
共8兲
being the adjustable parameters. For
with ⌬H, r and C res,ns
p
C op,m (T), no direct experimental data from the original literature sources were used but rather we employed polynomials that correlate the experimental data.1–3 To construct these
polynomials, all the available data were carefully evaluated
using several criteria and classified according to their accuracy. In all cases, we used the most recently reported correlating polynomial. For C id
p (T), we used the polynomials
given in Ref. 3. In some cases, these polynomials were used
below the lower temperature bound were they are reported as
valid 共sometimes, a hundred degrees below兲; according to
the researchers that created Ref. 3, this is a valid and safe
extrapolation.23 For the sterically hindered alcohols 3-ethyl3-pentanol and 5-methyl-5-nonanol the polynomials for
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6652
J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
C id
p (T) are not given in Ref. 3; hence, they were estimated
using a group contribution method.24 Both C op,m (T) and
C id
p (T) were evaluated every degree.
In Refs. 1–3 there is a large number of substances for
which C op,m data are available. Unfortunately, for many of
them C op,m is only reported at a single temperature. From the
subset where C op,m (T) data are accessible, in this quantitative
analysis we have considered some individual liquids as well
as several homologous series of substances, covering from
weakly to strongly associated liquids and from atmospheric
to elevated pressures. Our main selection criteria to include a
given substance in this analysis were the accuracy of the data
and the size of the temperature interval where data are available. The later criteria is very important since unless the
temperature range is large, there is not enough information in
the input C res
p data to have reasonable confidence in the output values of the adjusted parameters. Also, whenever possible we preferred to study liquids belonging to a homologous series rather than individual substances. This allowed
us to disregard data for a given member共s兲 of the series
whose C res
p (T) data were clearly outside the general trend, to
test the internal consistency and physical significance of the
fitted parameters within the series, and to reach general conclusions regarding that particular family of compounds. The
in Eq. 共8兲 reported and disparameters ⌬H, r, and C res,ns
p
cussed below were obtained using the software Origin 5.0.
B. The series of n-alkanes
Depolarized Rayleigh scattering and thermodynamic
measurements have shown that there are short-range correlations of the molecular orientations 共c.m.o.兲 in n-alkanes
(n-Cn ) liquids.21 These correlations are between segments
and require an anisotropy of shape. This weak association is
then present in the linear n-Cn , particularly in long chain
ones, and absent in branched alkanes. Because of their weak
nature, the c.m.o. have a small effect on the heat capacity of
the pure n-Cn , but they are clearly revealed in the excess
heat capacities when they are mixed with a globular molecule such as cyclohexane.21 It is then interesting to test the
performance of the present model for these weak associations. The weak character of the c.m.o. implies that the
C op,m (T) for pure n-Cn liquids are type 共i兲 curves located in
the right-hand-side of curve 1 in Fig. 3共b兲, and their associational residual heat capacities belong to the high temperature
peak, as shown in Fig. 5共a兲 for n-octane
side of the C res,ass
p
and n-hexadecane. For these two n-alkanes Figs. 5共b兲 and
5共c兲 shows the partitioning of the heat capacity according to
Eq. 共1兲 and the experimental C op,m (T), respectively. Table I
reports the model parameters obtained using Eq. 共8兲 for nine
n-Cn liquids. Both, ⌬H and r are seen to increase with molecular size. As the chain length of the n-Cn increases, the
number of geometries in which n-Cn molecules can associate decrease 共respect to those available as monomer兲 and
hence r increases. The increase of ⌬H values is due to the
segmental character of the c.m.o., i.e., as the chain length
increases these associations are more numerous. Compared
to the values obtained for stronger associations such as
H-bonding 共see below兲, the ⌬H and r values in Table I are
Cerdeiriña et al.
FIG. 5. 共a兲 Associational residual heat capacities 共䊊兲, C res,ass
, for hexadep
from Eq. 共5兲 using
cane (n-C16) and octane (n-C8 ). Solid curves are C res,ass
p
⌬H and r values in Table I; although the fitting to C res
p was performed using
data every degree, here for clarity data are shown every five degrees. 共b兲
res
Ideal gas (C id
p ) and residual (C p ) contributions to the total heat capacity
id
according to Eq. 共1兲; C p for n-C16 共䉭兲 and for n-C8 共䉱兲 were taken from
Ref. 3; C res
p for n-C16 共䉮兲 and for n-C8 共䉲兲 were calculated using Eq. 共8兲.
共c兲 Total isobaric molar heat capacities, C op,m (T), for n-C16 and for n-C8
from Ref. 1.
small. This is consistent with the weak character of the association between n-Cn molecules; in fact, for n⫽6 – 8 the
⌬H values in Table I indicate that for these short alkanes the
c.m.o. are only marginally present.
increase linearly with the
Figure 6 shows that C res,ns
p
n-Cn chain length. It also displays the results of calculations
using Flory theory,11 that directly provide the residual heat
capacities. These calculations were done in temperature intervals that are close to those used to fit the n-Cn C res
p data
共see Table I and Fig. 6 caption兲, using the thermal expansion
coefficients, isothermal compressibilities and molar volumes
given in Ref. 25. It was found that the residual heat capacities from Flory theory change very little with T as indicated
by Fig. 6 where only the results for the lowest and highest
temperature used are shown. In Flory theory only weak dispersion forces, not able to produce association, are taken into
account. In fact, the discovery of the correlations of molecular orientations in long n-Cn liquids was mainly based on the
contrast between the experimental results and the predictions
from Flory theory that, in the absence of those association, is
able to give good predictions of the excess thermodynamics
functions.21 The excellent accordance between the fitted
and the Flory theory calculations seen in Fig. 6 imply
C res,ns
p
that, as assumed when developing the two state model, most
of C res,ns
is effectively due to weak dispersion forces and free
p
volume effects. It also implies that the partitioning of the
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J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
Heat capacity of associated liquids
6653
TABLE I. Model parameters for n-alkanes.a
n
T min /K
T max /K
⌬H/J mol⫺1
r
/J K⫺1 mol⫺1
C res,ns
p
␹2
6
7
8
10
11
12
14
16
17
255
198
248
248
258
268
293
295
299
300
330
360
340
423
380
358
395
370
3000⫾30
4100⫾11
4424⫾9
6594⫾6
7803⫾11
8647⫾11
10 522⫾1
11 490⫾12
11 064⫾5
7.1⫾2.2
13.7⫾0.6
11.7⫾0.8
59.7⫾0.6
63.3⫾1.2
63.1⫾1.3
134.4⫾0.2
135.7⫾1.5
138.6⫾1.4
49.85⫾0.23
54.33⫾0.12
60.23⫾0.13
68.56⫾0.03
71.26⫾0.06
72.90⫾0.17
78.45⫾0.02
85.61⫾0.18
99.42⫾0.13
0.000 23
0.024 78
0.006 26
0.001 06
0.018 19
0.031 75
0.000 03
0.026 07
0.002 83
Fitted using Eq. 共8兲; T min and T max are the lower and upper bounds, respectively, of the temperature interval
o
id
used in the fitting process (C res
p data calculated every degree兲; C p,m (T) were taken from Ref. 1 and C p (T) from
Ref. 3.
a
heat capacity proposed in Eqs. 共1兲 and 共3兲 is reasonable.
From this agreement and from the very good fit of C res
p seen
is
in Fig. 5共a兲, it is concluded that the assumption that C res,ns
p
temperature independent is also quite good.
C. The series of 1-alcohols
For the series of linear alcohols (Cn OH) accurate heat
capacity data are available in wide temperature intervals.
However, for n⫽9, 11, 14, and 15 the calculated C res
p clearly
do not to follow the general trend observed for the other
twelve members of the series studied here, and hence they
were not used in the quantitative analysis. For these four
1-alcohols, this result might be due to the inaccuracy of the
o
id
starting data to obtain C res
p , i.e., C p,m (T) and C p (T), the
former being the most probable source of their nonconsonant
behavior. As shown in Fig. 4共a兲, for short chain 1-alcohols
type 共i兲 C op,m (T) curves are observed, but as the chain length
increases types 共iii兲 and 共iv兲 are found. For the Cn OH series,
FIG. 6. Contribution to the heat capacity of n-alkanes from weak dispersion
, from Table I against the n-Cn
forces, and free volume effects 共䊐兲, C res,ns
p
carbon number; line is the least squares linear fit of C res,ns
共correlation cop
for several n-Cn using
efficient of 0.98兲; Flory theory calculations of C res,ns
p
the data reported in Ref. 25 at the following low 共䉱兲 and high 共䉲兲 temperatures 共in K兲: 共228, 336兲 for n⫽6, 共253, 395兲 for n⫽8, 共253, 442兲 for n
⫽10, 共273, 388兲 for n⫽12, 共298, 353兲 for n⫽14, 共300, 390兲 for n⫽16 and
共304, 396兲 for n⫽17.
the associational residual heat capacities belong to the low
temperature side 共small n values兲 or to the maximum region
共medium and large n values兲 of the C res,ass
peak, as shown in
p
Fig. 7共a兲.
The three model parameters were preliminarily fitted in
an individual manner, i.e., to each 1-alcohol. Two observations from these fittings are relevant: First, all the obtained
⌬H values were very close implying that it is possible to
perform a global fitting for the whole series of Cn OH using a
common, shared, ⌬H value. Second, for the long chain
FIG. 7. 共a兲 Associational residual heat capacities 共䊊兲, C res,ass
, for 1-butanol
p
(C4 OH), 1-heptanol (C7 OH), 1-decanol (C10OH) and 3-ethyl-3-pentanol
from Eq. 共5兲 using ⌬H and r values in Tables II
共3E3P兲. Curves are C res,ass
p
and IV; although the fitting to C res
p was performed using data every degree,
here for clarity data are shown every five degrees. 共b兲 Ideal gas (C id
p ) and
residual (C res
p ) contributions to the total heat capacity according to Eq. 共1兲;
C id
p for C7 OH 共䉱兲 were taken from Ref. 3 and for 3E3P 共䉭兲 were calculated
using the group contribution method in Ref. 24; C res
p for C7 OH 共䉲兲 and for
3E3P 共䉮兲 were calculated using Eq. 共8兲. 共c兲 Total isobaric molar heat capacities for C7 OH from Ref. 2 and for 3E3P from Ref. 22.
Downloaded 14 Apr 2004 to 132.248.175.160. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
6654
TABLE II. Model parameters for 1-alcohols (Cn OH). a
⌬H⫽23124⫾27/J mol⫺1
␹ 2 ⫽5.99063
n
T min /K
T max /K
r
/J K⫺1 mol⫺1
C res,ns
p
1
2
3
4
5
6
7
8
10
12
13
18
180
159
153
188
205
227
275
290
283
298
304
332
337
350
370
390
400
430
448
468
504
486
525
545
89.0⫾5.1
158.3⫾4.2
242.1⫾3.7
264.4⫾3.5
333.4⫾4.1
315.0⫾4.1
423.0⫾7.7
558.6⫾7.6
627.4⫾6.9
687.6⫾7.8
729.4⫾7.2
736.6⫾6.9
31.26⫾0.27
37.49⫾0.23
43.88⫾0.22
54.35⫾0.26
55.31⫾0.30
67.18⫾0.36
65.84⫾0.55
65.72⫾0.36
74.54⫾0.24
80.11⫾0.27
91.71⫾0.22
115.05⫾0.22
a
Cerdeiriña et al.
J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
Fitted using Eq. 共8兲; T min and T max are the lower and upper bounds, respectively, of the temperature interval used in the fitting process (C res
p data
calculated every degree兲; C op,m (T) were taken from Ref. 1 for n⫽1, 2, 4, 5,
and 13 and from Ref. 2 for n⫽3, 6, 7, 8, 10, 12, and 18; C id
p (T) were taken
from Ref. 3. For n⫽1 – 8 C res
p was calculated using Eq. 共8兲 and for n
o
id
res,ass
共c.m.o.兲 共see text兲.
⫽10– 18 using C res
p ⫽C p,m (T)⫺C p (T)⫺C p
members of the series (n⭓10) the fittings were not as good
as those obtained for the smaller members. We postulate here
that these less adequate fittings are due to the fact that molecules in long chain alcohols associate not only via strong
H-bonds, but also via correlations of molecular orientations
共c.m.o.兲 as those described above for the n-Cn liquids. The
treatment of such combined associations could be accomplished assuming that these two contributions to C res,ass
are
p
res,ass
共H-bonds兲
and
C
共c.m.o.兲,
independent, namely C res,ass
p
p
each given by the same expression 关Eq. 共5兲兴 but with different ⌬H and r values. For 1-alcohols with n⭓10, the indio
id
res,ass
共c.m.o.兲
viduals fittings to C res
p ⫽C p,m (T)⫺C p (T)⫺C p
2
were significantly better 共decrease in ␹ values by factors of
共c.m.o.兲 was not considered;
two to four兲 than when C res,ass
p
共c.m.o.兲
contribution
for each Cn OH was
here, the C res,ass
p
calculated using the parameters in Table I for the corresponding n-alkane. These improved fittings are taken as clear evidence of the presence of c.m.o. in long chain 1-alcohols. For
共c.m.o.兲 did not produce
n⫽6 – 8, the employment of C res,ass
p
better fittings and hence it can be considered that the c.m.o.
are clearly only present for 1-alcohols with n⭓10. Given this
situation, we performed a global fitting for the Cn OH series
where C res
p for n⫽1 – 8 was obtained using Eq. 共8兲 and for
共c.m.o.兲 contribution was considered as den⭓10 the C res,ass
p
scribed above. For 1-tridecanol and 1-octadecanol, the ⌬H
and r values for the corresponding n-alkanes were estimated
from linear regressions of each of these parameters against n,
using the data in Table I. The results from this global fitting
are given in Table II and illustrated, for the total heat capacity C op,m (T), in Fig. 4共a兲.
The common ⌬H value for 1-alcohols in Table II corresponds to strong association via hydrogen bonding. It is satisfactory that this ⌬H value is in agreement with many other
estimates for the enthalpy of H-bond formation in
1-alcohols,26 including that obtained from the statistical associating fluid theory 共SAFT兲.27 It is also in agreement with
the hydrogen bond energies estimated in this work using
quantum mechanics second-order Møller–Plesset 共MP2兲 ab
initio calculations.22 The r values in Table II increase linearly
with the alcohol chain length and are much bigger than those
for the corresponding n-alkanes 共see Table I兲. As in the n-Cn
series, the increase of r values with 1-alcohol chain length is
due to a decrease in the number of possible molecular arrangements in which the molecules can associate 共respect to
those available as monomer兲. On the other hand, the increase
of r values in going from the n-Cn to the Cn OH can be
understood considering that for a given n value while the g A
values remain essentially constant, the g A i values decrease
significantly owing to the directional character of the
H-bonds producing the associated multimers in the alcohol.
contriAs discussed above in a qualitative form, the C res,ass
p
butions in Fig. 7共a兲 are responsible for the temperature dependences of the total heat capacities seen in Fig. 4共a兲. In
maximum for
Fig. 7共a兲, the low temperature C res,ass
p
1-decanol is due to the existence of the c.m.o. in this long
chain alcohol 共see previous subsection兲. The results in Figs.
7共b兲 and 7共c兲 will be discussed in detail below, in the context
of the obtained results for branched alcohols. As expected, in
increase linearly with the alcohol chain
Table II the C res,ns
p
length. The slope of the straight line through these data, 4.6
⫾0.3 J K⫺1 mol⫺1, represents the contribution of a CH2
group to the heat capacity arising from weak dispersion
forces, long-range interactions and free volume effects. In
the case of the n-alkanes, this same contribution 共slope of the
straight line in Fig. 6兲 is evaluated to be 3.9⫾0.3
J K⫺1 mol⫺1. The reasonable consistency between these two
values is another indication of the soundness of the partitioning of the heat capacity expressed by Eqs. 共1兲 and 共3兲.
D. The series of 1-thiols
For the series of linear 1-thiols (Cn SH) accurate heat
capacity data are available in wide temperature intervals for
n⫽1 to 6, 7, and 10. Due to the weak S–H¯S hydrogen
bonding present in these substances, they exist in the liquid
state only at low temperatures. The short-chain thiols
C op,m (T) display a minimum, i.e., a type 共ii兲 curve, as shown
in Fig. 4共b兲. The three model parameters were preliminary
fitted in an individual manner, i.e., to each 1-thiol, finding
that the ⌬H values increase with n, from ⬃2600 J mol⫺1 for
methanethiol to ⬃8100 J mol⫺1 for 1-decanethiol; the r values also increase with n, their magnitude being much smaller
than for the corresponding 1-alcohols. The small values for
⌬H were expected for these substances where association
takes place through weak S–H¯S hydrogen bonds. However, if association were to be the dominant factor determining C op,m (T), a common ⌬H would have been found in these
individual fittings for the 1-thiols, as it was the case for the
1-alcohols. Under these conditions, i.e., small ⌬H and r values reflecting weak association between 1-thiol molecules, it
is possible that the assumption of a temperature independent
is no longer adequate. We postulate that for 1-thiols
C res,ns
p
res,ass
and C res,ns
, contribute
both contributions to C res
p , i.e., C p
p
significantly to the temperature dependence of C op,m . We
⫽C 1 ⫹C 2 T and
tested this hypothesis considering that C res,ns
p
performing again individual fittings, this time with four pa-
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J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
Heat capacity of associated liquids
6655
TABLE III. Model parameters 1-thiols (Cn SH). a
⌬H⫽3756⫾21/J mol⫺1
␹ 2 ⫽0.02415
/J K⫺1 mol⫺1
C res,ns
p
n
T min /K
T max /K
r
C 1 /J K⫺1 mol⫺1
C 2 /J K⫺2 mol⫺1
1
2
3
4
5
6
155
131
168
161
201
201
271
308
314
314
321
371
8.55⫾0.21
7.16⫾0.15
6.53⫾0.15
9.27⫾0.16
6.27⫾0.31
16.63⫾3.25
23.95⫾0.41
39.18⫾0.34
50.16⫾0.31
59.45⫾0.28
69.95⫾0.35
83.86⫾1.08
0.03947⫾0.00150
0.00501⫾0.00110
⫺0.01702⫾0.00099
⫺0.03226⫾0.00087
⫺0.04818⫾0.00122
⫺0.06799⫾0.00236
Fitted using Eq. 共8兲; T min and T max are the lower and upper bounds, respectively, of the temperature interval
o
id
used in the fitting process (C res
p data calculated every degree兲; C p,m (T) and C p (T) were taken from Ref. 3.
a
rameters (⌬H, r, C 1 , and C 2 ). The results were that for n
⫽1 – 6, the ⌬H values were very close implying that, as with
the 1-alcohols, it is possible to perform a global fitting for
the series of Cn SH using a common ⌬H value. The results
from this global fitting are given in Table III and illustrated,
for the total heat capacity C op,m (T), in Fig. 4共b兲. For n⫽7
and 10, the four parameter fitting produced larger ⌬H values
than for the smaller members of the series, and for this reason they were not used for the global fitting with a shared
⌬H. It is possible that this failure is due to the fact that for
these 1-thiols, especially for n⫽10, the correlations of molecular orientations modeled above for the n-alkane series
can be relatively important. In this context, it appears that
1-decanethiol can be considered a borderline case. For this
thiol, C op,m (T) must be the result of a delicate balance between several weakly temperature dependent contributions,
that the simple partitioning expressed by Eqs. 共1兲 and 共3兲 is
not able to capture satisfactorily.
The parameters in Table III indicate that the association
enthalpy ⌬H for the 1-thiols is much smaller than that for
the 1-alcohols. This is also the result obtained for the hydrogen bond energies from quantum mechanics MP2 ab initio
calculations.22 The weakness of the association produced by
S–H¯S hydrogen bonds is also reflected in the small r values. The C 2 values are small indicating a weak temperature
that, however, is necessary to consider
dependence for C res,ns
p
in order to reach an adequate description of C op,m (T). For the
C n SH series, the associational residual heat capacities belong
peak. This is
to the high temperature side of the C res,ass
p
for
1-propanethiol
is conshown in Fig. 8共a兲 where C res,ass
p
trasted with that for 1-propanol that, as discussed above, bepeak. For
longs to the low temperature side of the C res,ass
p
these two substances, Fig. 8共b兲 displays the two contributions to the heat capacity according to Eq. 共1兲, indicating that
while the temperature dependences of C id
p are similar, those
are
opposite,
producing
entirely
different temperafor C res
p
ture dependences for their total heat capacities, C op,m , shown
in Fig. 8共c兲. The results in Fig. 8 stress again the capability
of the present model to describe different C op,m (T) in a concerted form 关in this case, types 共i兲 and 共ii兲 curves兴.
E. Branched alcohols. The C po , m „ T … maximum
Apparent molar heat capacity measurements for secondary and tertiary alcohols mixed with inert solvents, per-
formed at very low alcohol concentrations and at 298 K,
have shown that the sterical hindrance over the hydroxyl
group of the branched alcohols reduce their association capability while their ⌬H values remain practically constant
and close to the value for linear alcohols.28 In terms of the
two state model, a reduction on the association ability must
be reflected in larger r values. Under these conditions, i.e., a
large r value and a ⌬H equal or close to that for 1-alcohols,
(T) will display a
for an sterically hindered alcohol C res,ass
p
sharper peak 共with its maximum at lower temperatures兲 than
for its corresponding isomeric 1-alcohol 关see Fig. 2共a兲兴.
Hence, for such an alcohol it is more probable that C op,m (T)
FIG. 8. 共a兲 Associational residual heat capacities 共䊊兲, C res,ass
, for
p
from Eq.
1-propanethiol (C3 SH) and 1-propanol (C3 OH). Curves are C res,ass
p
共5兲 using ⌬H and r values in Tables II and III; although the fitting to C res
p
was performed using data every degree, here for clarity data are shown
id
res
every five degrees. 共b兲 Ideal gas (C p ) and residual (C p ) contributions to
the total heat capacity according to Eq. 共1兲; C id
p for C3 SH 共䉭兲 and for C3 OH
共䉱兲 were taken from Ref. 3; C res
p for C3 SH 共䉮兲 and for C3 OH 共䉲兲 were
calculated using Eq. 共8兲. 共c兲 Total isobaric molar heat capacities for C3 SH
from Ref. 3 and for C3 OH from Ref. 2.
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6656
Cerdeiriña et al.
J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
TABLE IV. Model parameters for branched alcohols.a
nb
T min /K
T max /K
⌬H/J mol⫺1
r
/J K⫺1 mol⫺1
C res,ns
p
␹2
3, 2Pr
4, 2B
4, 2M1P
5, 2M2B
5, 3M2B
5, 2Pe
6, cC6
6, 3M3P
7, 3E3P
8, 2E1H
10, 5M5N
185
188
180
275
218
298
298
278
278
204
278
355
345
380
347
370
367
433
368
368
353
368
22768⫾53
21819⫾35
22598⫾27
20380⫾80
19851⫾21
21220⫾25
21498⫾9
20863⫾20
21168⫾63
22241⫾137
24992⫾38
353.5⫾4.5
454.7⫾3.8
382.8⫾2.2
358.8⫾6.2
544.4⫾4.7
525.3⫾8.6
271.9⫾1.2
1077.8⫾9.3
2000.2⫾48.7
447.0⫾13.9
1545.5⫾14.7
42.10⫾0.06
45.51⫾0.06
45.53⫾0.05
71.36⫾1.00
44.65⫾0.21
45.23⫾0.52
58.26⫾0.11
47.28⫾0.22
54.37⫾0.62
85.18⫾0.27
80.48⫾0.49
0.1502
0.1462
0.1683
0.5609
0.9275
0.1042
0.0525
0.0928
0.9949
1.9971
0.6591
Fitted using Eq. 共8兲; T min and T max are the lower and upper bounds, respectively, of the temperature interval
used in the fitting process (C res
p data calculated every degree兲. Code used is: 2-propanol 共2Pr兲, 2-butanol 共2B兲,
2-methyl-1-propanol 共2M1P兲, 2-methyl-2-butanol 共2M2B兲, 3-methyl-2-butanol 共3M2B兲, 2-pentanol 共2Pe兲, cyclohexanol 共cC6兲, 3-methyl-3-pentanol 共3M3P兲, 3-ethyl-3-pentanol 共3E3P兲, 2-ethyl-1-hexanol 共2E1H兲 and
5-methyl-5-nonanol 共5M5N兲; C op,m (T) were taken from Ref. 2 for 2M1P, 2M2B, 3M2B, 2Pe, cC6, and 2E1H,
from Ref. 1 for 2Pr and 2B and Ref. 22 for 3M3P, 3E3P and 5M5N; C id
p (T) were taken from Ref. 3 for all
substances except 3E3P and 5M5N where the contribution method in Ref. 24 was used.
b
Number of carbon atoms in the alcohol.
a
will display a maximum within the temperature interval
where it is a liquid. In order to test this prediction by the two
state model, in this work C op,m (T) were measured for three
branched alcohols, namely 3-methyl-3-pentanol 共3M3P兲,
3-ethyl-3-pentanol 共3E3P兲 and 5-methyl-5-nonanol 共5M5N兲.
Both the obtained data as well as experimental procedures
can be found in Ref. 22. The experimental C op,m (T) results
corroborate the existence of such a maximum for 3M3P and
3E3P, the latest being displayed in Fig. 7共c兲. To the best of
our knowledge, this is the first time that such C op,m (T) behavior has been clearly observed.
The model parameters for 3M3P, 3E3P, and 5M5N as
well as for another eight branched alcohols are given in
Table IV as obtained from individual fittings. In Table IV,
the ⌬H values 共average is 21 763 J mol⫺1兲 are seen to be
close to the value for 1-alcohols; this is also observed with
the hydrogen bond energies from quantum mechanics MP2
ab initio calculations.22 In contrast, most of the r values in
Table IV are bigger for the branched alcohol than for the
corresponding linear isomers 共see Table II兲. The increase of r
values is due to the sterical hindrance over the hydroxyl
group that, in terms of the present model, produces a decrease in the number of molecular arrangements in which the
branched alcohol molecules are able to associate 共respect to
those for the linear isomer兲. This is notorious for the highly
branched tertiary alcohols 3M3P and 3E3P whose r values
are 3.4 and 4.7 times bigger than for 1-hexanol and
1-heptanol, respectively. For a constant ⌬H, Eq. 共5兲 indicates that the increase of r displaces the maximum on C res,ass
p
towards lower temperatures 关compare curves A and C in Fig.
2共a兲兴. This is the displacement seen in Fig. 7共a兲 in going
from the linear 1-heptanol to its branched isomer 3E3P. As
shown in Fig. 7共b兲, while for these two isomeric alcohols
res
their C id
p (T) are essentially the same, their C p (T) are strikingly unlike, producing completely different temperature dependences for their total heat capacities, C op,m , shown in Fig.
values are similar to
7共c兲. As expected, in Table IV the C res,ns
p
those in Table II for the corresponding isomeric linear alco-
hols. The results in Fig. 7 emphasize the ability of the
present model to rationalize different temperature dependences of the heat capacity in a unified form 关in this case,
type 共iii兲 and 共v兲 curves兴.
It is important to emphasize that for 3M3P and 3E3P the
appearance of a maximum in their C op,m (T) curves is the
result of a combination of quite specific factors, namely: 共1兲
A large r value, producing a large and narrow C res,ass
peak
p
owing to the restriction imposed on association by the sterical hindrance over the hydroxyl group, 共2兲 a medium-tolarge ⌬H value that avoids the displacement of C op,m to region 共iii兲 共larger ⌬H) or to region 共ii兲 共smaller ⌬H) in case
2 curve in Fig. 3共b兲, and 共3兲 a medium molecular size that
produce a relative small C id
p (T) avoiding the displacement of
C op,m from case 2 to case 1 in Fig. 3共b兲. It is then not surprising that, previous to this work, no other substance at
atmospheric pressure has been found to display a maximum
in C op,m . In this context, the C op,m (T) results22 for 5M5N,
i.e., a type 共i兲 curve in the range 278.15–368.15 K when 共due
to its geometry兲 a maximum might have been expected, can
be explained as due to its larger molecular size 共as compared
to 3M3P and 3E3P兲 that translates into a larger C id
p and hence
.
a reduced the effect of C res,ass
p
F. Other series
Two other series of substances can be preliminarily examined with the model, namely amines and acids. A detailed
treatment for these series is hampered by the lack of data for
several of their members and by the reduced temperature
intervals where the data are available for some others. For
the amines, there are data for the first five members of the
series, i.e., for methaneamine to 1-pentaneamine. However,
for 1-pentaneamine heat capacities are only measured in a
short temperature interval and the C res
p for 1-propaneamine
clearly do not follow the trend observed for methaneamine,
ethaneamine and 1-butaneamine. Using only these three
members of the series a global fitting was performed obtain-
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J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
Heat capacity of associated liquids
6657
TABLE V. Model parameters for n-alkanes (n-Cn ) and for 1-alcohols (Cn OH) at high pressures.a
n
P/MPa
⌬H/J mol⫺1
11
11
11
11
12
12
12
12
13
13
13
13
15
15
15
15
20
50
80
100
20
50
80
100
20
50
80
100
20
50
80
100
8526⫾1041
9506⫾1126
10373⫾744
10485⫾751
8863⫾166
9119⫾408
9922⫾462
9970⫾521
11 008⫾641
11 121⫾457
10 626⫾244
10 590⫾151
12 314⫾668
12 424⫾579
12 522⫾545
12 560⫾572
r
n-alkanesb
77⫾104
35⫾3
44⫾4
45⫾5
41⫾5
45⫾13
43⫾2
47⫾3
65⫾4
69⫾3
75⫾6
93⫾19
103⫾9
109⫾8
119⫾8
125⫾8
C res,ns
/J K⫺1 mol⫺1
p
␹2
66.0⫾ 3.9
52.6⫾6.2
47.3⫾4.1
46.5⫾4.2
66.7⫾1.5
64.0⫾3.9
58.2⫾2.8
58.0⫾3.4
61.9⫾4.2
59.7⫾3.1
63.2⫾2.3
65.1⫾2.7
67.6⫾4.8
66.0⫾4.3
65.3⫾4.2
65.2⫾4.6
0.1114
0.2008
0.1051
0.1099
0.0129
0.0925
0.0460
0.0737
0.1444
0.0812
0.0448
0.0718
0.2533
0.2094
0.2148
0.2676
1-alcoholsc
C res,ns
/J K⫺1 mol⫺1
p
n
r
P⫽2 MPa
6
7
8
10
P⫽10 MPa
6
7
8
10
P⫽30 MPa
6
7
8
10
⌬H⫽22 280⫾67/J mol⫺1
357.4⫾9.6
419.3⫾10.6
453.4⫾11.1
552.9⫾13.3
⌬H⫽21 985⫾120/J mol⫺1
318.7⫾12.4
362.0⫾13.7
409.8⫾15.8
534.9⫾19.3
⌬H⫽22 444⫾87/J mol⫺1
415.2⫾11.9
456.8⫾12.5
510.9⫾13.8
650.7⫾17.1
␹ 2 ⫽0.655 34
55.29⫾0.59
59.10⫾0.54
68.99⫾0.52
76.90⫾0.53
␹ 2 ⫽5.448 29
54.02⫾0.71
60.34⫾0.70
66.67⫾0.74
74.79⫾0.69
␹ 2 ⫽3.254 20
43.98⫾0.56
49.83⫾0.52
56.65⫾0.52
65.27⫾0.52
Fitted using Eq. 共8兲.
C op,m (T) were taken from Refs. 6 to 9 and C id
p (T) from Ref. 3.
res
c o
C p,m (T) were taken from Ref. 10 and C id
p (T) from Ref. 3. The total number of experimental C p points used
in the fittings were: 48 at 2 MPa, 104 at 10 MPa, and 102 at 30 MPa.
a
b
ing a common ⌬H value of 5395⫾21 J mol⫺1. This tentative
value is consistent with association through weak hydrogen
for
bonds between amine molecules; as such, the C res,ass
p
amines belong to the high temperature side of associational
residual heat capacity peak. For the acids, whose C res,ass
perp
tain to the low temperature side of the peak, it was possible
to perform only a few individual fittings. They produced ⌬H
values comparable or higher than those for the 1-alcohols,
indicating strong association via hydrogen bonding. In general, it can be concluded that more accurate heat capacity
data in wide temperature intervals are needed to further
evaluate the performance of the model. In this context,
amines and acids are clear candidates to future experimental
work.
G. n-Alkanes at high pressures
Heat capacity data for four n-alkanes (n⫽11, 12, 13, and
15兲 at high pressures ( P⫽20, 50, 80, and 100 MPa兲 in the
temperature range 313.15–373.15 K have been recently
reported.6 –9 Unfortunately, only seven temperatures were
measured at each pressure and hence it is only possible to
obtain tentative values for the model parameters. Individual
fittings for the four n-alkanes at each pressure were performed, the results being reported in Table V. Some preliminary conclusions can be reached from the results in Table V
and their comparison with those at atmospheric pressure in
Table I. The ⌬H values are almost independent of P, their
values being slightly bigger than those at atmospheric pressure. The later is consistent with an enhancement of the correlations of molecular orientations at high pressures, where
these associations must be favored.29 As P is increased, no
clear trend is observed for r; since the r values are strongly
dependent on the number of experimental C op,m points used
in the fitting and the size of the temperature interval where
these data are available, it is not possible to draw any conclusion from the comparison with the r values at atmospheric
values in Table V are clearly
pressure in Table I. The C res,ns
p
smaller than those at atmospheric pressure in Table I and
appear to decrease with increasing P. Calculations using
Flory theory, as a function of P and at 298.15 K, for
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6658
Cerdeiriña et al.
J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
n-heptane and n-nonane showed that their residual heat capacities decrease with increasing pressure. These calculations were done using accurate thermal expansion coefficients, isothermal compressibilities and molar volumes data
( P) behavior
in Ref. 30. The fact that the qualitative C res,ns
p
found with the present model is the same as that produced by
Flory theory confirms that the partitioning of the heat capacity proposed in Eqs. 共1兲 and 共3兲 is satisfactory, i.e., most of
arises from weak dispersion forces not able to produce
C res,ns
p
association and, to a good first approximation, it can be considered temperature independent.
H. 1-Alcohols at high pressures
Very recently, heat capacity data at P⫽2, 10, and 30
MPa in the temperature range 325.7–570.7 K have been
reported10 for four members of the linear alcohols (Cn OH)
series (n⫽6, 7, 8, and 10兲. The most salient feature of these
data is that at 10 and 30 MPa the C op,m (T) curves display a
maximum followed by a minimum as the temperature increases. In other words, these C op,m (T) curves include three
regions 关curves types 共iii兲, 共v兲, and 共ii兲兴, following case 2
curve in Fig. 3共b兲. Although the difference between the heat
capacity values at the maximum and the minimum is smaller
than the quoted experimental error, in this work we assume
that it is likely that the above described behavior of C op,m (T)
is correct. This assumption receives support for the following
facts: 共1兲 As shown in detail in this work, C op,m (T) can go
through a maximum or a minimum and 共2兲 the two-state
model is able to satisfactorily explain 共see below兲 the existence of a maximum followed by a minimum; given the success of the model to describe all the experimentally observed
C op,m (T) curves at atmospheric pressure, it is likely that the
adequate description of data al high pressures is not fortuitous.
The preliminary individual fittings of the model parameters produced very similar ⌬H values. Hence, global fittings
using a common ⌬H at each pressure were performed. The
values of the parameters are given in Table V and the quality
of the fitting to C res
p is illustrated in Fig. 9共a兲 for the four
1-alcohols at P⫽30 MPa. These good fittings allow a very
reasonable description of the C op,m (T) at high pressures, as
depicted in Fig. 9共b兲 for two of the systems, 1-hexanol at 10
MPa and for 1-decanol at 30 MPa. Clearly, as indicated in
the qualitative discussion given above, the present model is
capable of producing C op,m (T) curves that include more than
one of the regions or types of curves in Fig. 3共b兲. According
to the model, these multiple-region behavior for C op,m (T) is
simply the result of the application of pressure that increases
the temperature range for the substance to be in the liquid
state, and not to some especial effect or additional contribution to the heat capacity.
values are seen to increase with
In Table V, the C res,ns
p
1-alcohol chain length and, for a given 1-alcohol, decrease
with pressure. The first variation is in agreement with that
found at atmospheric pressure in Table I. The second variation is consistent with the results discussed above for the
n-alkanes at high pressures. For the four 1-alcohols the r and
⌬H values decrease in going from 2 to 10 MPa and then
FIG. 9. 共a兲 Residual heat capacities, C res
p , for 1-hexanol 共䊊兲, 1-heptanol
共䉮兲, 1-octanol 共䉭兲 and 1-decanol 共䊐兲 at 30 MPa from Ref. 10; full lines are
the fitted C res
p using Eq. 共8兲, the model parameters being in Table V. 共b兲
Molar heat capacities C op,m for 1-hexanol at 10 MPa 共䉭兲 and for 1-decanol
res
at 30 MPa 共䊐兲 from Ref. 10; full lines are C op,m ⫽C id
p ⫹C p calculated using
the present model.
increase again at 30 MPa. On the other hand, at the three
pressures the r values increase with the 1-alcohol chain
length, as it was the case at atmospheric pressure. It appears
as if the C op,m (T) data at 10 MPa are inconsistent with those
at the other two pressures; this is also indicated by the ␹ 2
value obtained at 10 MPa as compared to that at 30 MPa,
both obtained with a comparable number of fitted experimental points. A possible reason for this is the proximity, at
10 MPa and high temperatures, to the ( P,T) conditions of
the gas–liquid critical point, where C op,m experiments a
strong divergence to infinity.
The pressure dependence of ⌬H can be used to estimate
the volume change upon association, ⌬V, through the relation ⌬H⫽⌬E⫹ P⌬V. Using the ⌬H values at 2 and 30
MPa, ⌬E⫽22 280 J mol⫺1 and ⌬V⫽5.9 cm3 mol⫺1 . This
⌬V value is in excellent agreement with other estimates obtained from excess volume data for 1-alcohol⫹inert solvent
binary mixtures, using chemical association equilibrium
models develop within the classical thermodynamic realm:
5.6 cm3 mol⫺1 in Ref. 31 using data for all 1-alcohols and 4.9
cm3 mol⫺1 in Ref. 26 using data from 1-hexanol to
1-decanol. The ability to obtain a very reasonable ⌬V value
and to describe adequately both the temperature and pressure
dependences of the heat capacities for 1-alcohols is another
indication that the present model is able to capture the essential elements determining the thermodynamic behavior of
these liquids.
I. Concluding remarks
The formation of molecular aggregates in pure liquids
modifies the entropy of the system. The structures that this
association process produce are sensible to temperature producing changes of entropy and heat capacity, C p
⫽T( ␦ S/ ␦ T) P . The heat capacity can then be used as an
approximate sensor of molecular structure in the liquid. The
two state model presented in this work appears to capture the
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J. Chem. Phys., Vol. 120, No. 14, 8 April 2004
essential elements determining the temperature and pressure
dependences of the heat capacity of pure liquids, providing
an insight into the macroscopic manifestation of the association phenomena that occur at the molecular level. Using this
model, the effect of association on other thermodynamic
properties such as the isobaric thermal expansivities and the
isothermal compressibilities, as well as an extension aimed at
the description of excess properties of mixtures, are currently
under evaluation in our laboratories.
ACKNOWLEDGMENTS
We thank Dr. Milton Medeiros and Dr. Ana Martı́nez for
useful and stimulating discussions, and Dr. Ángel Piñeiro
and Dr. Ernesto Carrillo-Nava for their comments to the
manuscript. This work was supported by the Consejo Nacional de Ciencia y Tecnologı́a de México 共CONACyT兲 research project program 共Grant No. 41328-Q兲.
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