The direct determination of partial molar volumes and reaction volumes in ultra-dilute non-reactive and reactive multi-component systems using a combined spectroscopic and modified response surface model approach

Article abstract of DOI:10.1039/b515298d

Two experimental multi-component organometallic systems were studied, namely, (1) a non-reactive system consisting of [Mo(CO)₆], [Mn ₂(CO)₁₀], and [Re₂(CO)₁₀] in toluene under argon at 298.15 K and 0.1 MPa and (2) a reactive system consisting of [Rh₄(CO)₁₂] + PPh₃ → [Rh ₄(CO)₁₁PPh₃] + CO in n-hexane under argon at 298.15 K and 0.1 MPa. The mole fractions of all solutes were less than 140 × 10^-6 in system (1) and less than 65 × 10^-6 in system (2). Simultaneous in-situ FTIR spectroscopic measurements and on-line oscillatory U-tube density measurements were performed on the multi-component solutions. A newly developed response surface methodology was applied to the data sets to determine the individual limiting partial molar volumes of all constituents present as well as the reaction volume. The limiting partial molar volumes obtained for system (1) were 176.4 ± 2.5, 265.1 ± 2.4, and 276.8 ± 2.4 cm³ mol^-1 for [Mo(CO)₆], [Mn₂(CO)₁₀], and [Re₂(CO)₁₀], respectively and are consistent with independent binary experiments. The limiting partial molar volumes obtained for system (2) were 310.7 ± 2.7, 219.8 ± 2.2 and 461.5 ± 4.5 cm³ mol^-1 for [Rh₄(CO)₁₂], PPh₃ and [Rh₄(CO) ₁₁PPh₃], respectively. In addition, a reaction volume Δ_rV equal to -17.0 ± 5.7 cm³ mol^-1 was obtained. The present results demonstrate that both partial molar volumes and reaction volumes can be obtained directly from multi-component organometallic solutions. This development provides a new tool for physico-chemical determinations relevant to a variety of solutes and their reactions. The Royal Society of Chemistry 2006.

Full text of DOI:10.1039/b515298d

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PAPER
www.rsc.org/dalton | Dalton Transactions
The direct determination of partial molar volumes and reaction volumes in
ultra-dilute non-reactive and reactive multi-component systems using a
combined spectroscopic and modified response surface model approach†
Martin Tjahjono, Ayman D. Allian and Marc Garland*
Received 27th October 2005, Accepted 12th January 2006
First published as an Advance Article on the web 3rd February 2006
DOI: 10.1039/b515298d
Two experimental multi-component organometallic systems were studied, namely, (1) a non-reactive
system consisting of [Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10] in toluene under argon at 298.15 K and
0.1 MPa and (2) a reactive system consisting of [Rh4(CO)12] + PPh3 → [Rh4(CO)11PPh3] + CO in
n-hexane under argon at 298.15 K and 0.1 MPa. The mole fractions of all solutes were less than 140 ×
10−6 in system (1) and less than 65 × 10−6 in system (2). Simultaneous in-situ FTIR spectroscopic
measurements and on-line oscillatory U-tube density measurements were performed on the
multi-component solutions. A newly developed response surface methodology was applied to the data
sets to determine the individual limiting partial molar volumes of all constituents present as well as the
reaction volume. The limiting partial molar volumes obtained for system (1) were 176.4 2.5, 265.1
2.4, and 276.8 2.4 cm3 mol−1 for [Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10], respectively and are
consistent with independent binary experiments. The limiting partial molar volumes obtained for
system (2) were 310.7 2.7, 219.8 2.2 and 461.5 4.5 cm3 mol−1 for [Rh4(CO)12], PPh3 and
[Rh4(CO)11PPh3], respectively. In addition, a reaction volume DrV equal to −17.0 5.7 cm3 mol−1 was
obtained. The present results demonstrate that both partial molar volumes and reaction volumes can be
obtained directly from multi-component organometallic solutions. This development provides a new
tool for physico-chemical determinations relevant to a variety of solutes and their reactions.
The use of an appropriate in-situ spectroscopic measurement
Introduction
(possibly FT-IR, RAMAN, UV/VIS, etc.) together with a bulk
Many organometallic syntheses and homogeneous catalytic syn-
fluid property measurement would appear to provide sufficient
theses for fine chemicals and pharmaceuticals can be considered
experimental information for determining individual physico-
highly complex multi-component systems, possessing many prod-
uct(s) and/or intermediate(s).1 In-situ spectroscopic studies have
shown that many of the observable organic and organometallic
chemical solute properties from multi-component solutions. In
addition, a robust and general method of analysis involving
appropriate design of experiments, model development, and
species present are in fact non-isolatable transient intermediates.2–4
system identification, would be required to solve the correspond-
ing inverse problem,8 thereby obtaining the individual physico-
chemical quantities of interest.
Most of these reactions were conducted in a liquid phase, with a
very low concentration of the reacting species (in parts per million
levels). Recently, advanced signal processing techniques like band-
target entropy minimization (BTEM) have been successfully
deconvoluting, then identifying as well as quantifying the non-
isolatable reacting species.5–7 This information has been widely
used for kinetic modeling and process optimization. However, the
corresponding determination of the associated physico-chemical
solution properties of the individual solutes (non-isolatable as
well as isolatable) in multi-component systems has not advanced
to the same degree, due to combined numerical, analytical and
methodological difficulties.
Many bulk physico-chemical properties of solutions are of
common interest to physical chemists such as volume, dielectric
constant, refractive index and/or refraction, dipole moment etc.9
Among these, volumetric measurements are perhaps one of the
most straightforward thermodynamic measurements to obtain
and these lead to the determination of partial molar volumes,
volumes of interaction, as well as volume of reaction. Tradition-
ally, partial molar volumes have been determined from binary
data alone, in other words, from single-solute/solvent systems.10
Recently, a new and very general approach to the determination
of partial molar volumes from multi-component solutions alone
(without recourse to binary experiments) was developed and
applied to a non-reactive highly-associating system.11 This new
approach relies on the application of a response surface model
to the total molar volume of solution. Good experimental design
ensured that the composition space was adequately covered and
that optimal parameter estimation for the individual partial molar
volumes could be achieved.
Department of Chemical and Biomolecular Engineering, 4 Engineer-
ing Drive 4, National University of Singapore, Singapore 117576. E-
mail: chemvg@nus.edu.sg; Fax: +65 6779-1936; Tel: +65 6874-6617
† Electronic supplementary information (ESI) available: Raw experimental
data for the multi-component non-reactive system. This includes (a) the
density differences between the pure solvent and the multi-component
solution, and (b) the volume differences Y defined by the left-hand-side of
eqn (11) or eqn (12), for each perturbation of the three semi-batch runs as
well as (c) compositional data. See DOI: 10.1039/b515298d
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In the present contribution, the approach is considerably
extended, to permit in-situ spectroscopic analysis and to encom-
pass ultra-dilute concentration regimes and reactive solutions.
In order to fully demonstrate the utility of the new approach,
two experimental multi-component organometallic systems were
studied in the present contribution, namely, (1) a non-reactive
system consisting of [Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10] in
toluene under argon at 298.15 K and 0.1 MPa and (2) a reactive
system of ligand substitution of CO with triphenylphosphine, PPh3
in tetrarhodium dodecacarbonyl, [Rh4(CO)12] using n-hexane as a
solvent and under argon at 298.15 K and 0.1 MPa.
controlled by a built-in Peltier thermostat to within 0.001 K and
a Perkin Elmer System 2000 mid-infrared FTIR spectrometer.
The fluid was pumped under isobaric (Argon, atmospheric
pressure) and isothermal conditions from the reactor through the
pump (Temp control Polyscience 9105, with temperature stability
0.05 K), DMA 5000, then high-pressure infrared cell (SS316)
with recycle back to reactor. Connections for vacuum and argon
were provided. A 2.000 piezo-transducer (PAA-27W, Keller AG,
Switzerland) was used throughout for pressure measurements.
The entire experimental setup was gas tight, and the approximate
liquid phase volume was 100 mL. In order to ensure quantitative
working conditions, and hence minimal decomposition of metal
complexes and phosphine, the system was thoroughly rinsed with
anhydrous solvent under argon and then evacuated to dryness
prior to each experiment. In addition, the reactor was fully covered
with aluminum foil, thus providing dark reaction conditions.17
Before performing density measurements, the DMA 5000 was
calibrated using Millipore quality de-ionized water and dry air.
Hamilton gas-tight 1.0 mL and 2.5 mL syringes were used for
introducing perturbation of solutions into the reactor through a
rubber septum. The amounts of solutes in stock solutions were
determined by using a balance (Analytical Plus, Ohaus, New
Jersey) with a precision of 10−5 g. The high-pressure infrared
cell was constructed at the ETH Zu¨rich of SS316 steel and could
be heated and cooled. The CaF2 and KBr single crystal windows
used (Korth Monokristalle, Kiel, Germany) had dimensions of
diameter 40 mm by thickness 15 mm. Two sets of Viton and
Silicone gaskets provided sealing, and Teflon spacers were used
between the windows. The construction of the flow through cell is a
variation on the design by Noack18 and differs in some respect from
other high-pressure infrared cells.19 The high-pressure cell was
situated in a Perkin-Elmer 2000 FTIR infrared spectrometer. The
cell chamber was purged with purified nitrogen (Soxal, Singapore,
99.999%). The resolution was set to 4 cm−1. A schematic diagram
of the experimental setup is shown in Fig. 1.
[Rh4(CO)12] + PPh3 → [Rh4(CO)11PPh3] + CO
(1)
The concentration regimes were ultra-dilute. The mole fractions
of all solutes were less than 140 × 10−6 in system (1) and less than
65 × 10−6 in system (2). The reaction (eqn (1)) is very fast.12–14
Simultaneous in-situ FTIR spectroscopic measurements and on-
line oscillatory U-tube density measurements were performed on
both multi-component solutions. A modified response surface
model for the ultra-high dilution multi-component solution was
proposed. Partial molar volume of each individual solute was
determined using the multi-component data alone. In addition,
the volume of reaction DrV for system (2) was determined. It is
important to note that this avoids the reliance on a large pressure
variation which is needed in DrV studies.15 To verify the accuracy
of the results as well as validate the models used, separate binary
studies were carried out in both systems.
The present contribution suggests
a wider use of the
present approach and more general applicability to
a
thermodynamic/physico-chemical system identification of very
dilute non-reactive as well as reactive systems, especially systems
involving non-isolatable but observable and quantifiable species.
Experimental
General information
In-situ spectroscopic and density measurement
All solution preparations were carried out under argon (Soxal,
Singapore, 99.999%) by using standard Schlenk techniques.16
The solvents n-hexane (Fluka, puriss, 99.6%+) and toluene
First, background spectra of the empty high-pressure infrared
cell were recorded. Then 100 mL solvent was transferred to the
reactor under argon. Pressure was set at 0.1 MPa of argon.
The stirrer was turned on and liquid was circulated throughout
the entire system. After the vapor–liquid equilibrium (VLE) was
established and confirmed by stable density measurements from
the DMA 5000, spectra of the argon–solvent solution in the cell
were recorded and the density of the pseudo argon–solvent liquid
was measured simultaneously.20 To reduce error, while actually
measuring the density with the DMA, the pump was turned off.
Thus density measurements were performed under static and not
flow conditions. A few minutes were typically needed for a DMA
5000 measurement. After density and absorbance measurement,
the pump was turned on and the solution was circulated back to
the reactor.
R
(Mallinckrodt Chem., ChromARꢀHPLC, 99.9%+) were refluxed
over sodium–potassium alloy under argon. The argon was further
purified prior to use by passage through a column containing
100 g of reduced BTS-catalyst (Fluka AG Buchs, Switzerland)
˚
and 100 g of 4 A molecular sieves to adsorb trace oxygen and
water, respectively. The metal complexes [Mo(CO)6], [Mn2(CO)10],
and [Re2(CO)10], and [Rh4(CO)12] with stated purity of at least
98% for all complexes were obtained from Strem Chemicals
(Newport, MA) and triphenylphosphine, PPh3 with stated purity
of greater than 99% obtained from Merck were used without
further purification.
Equipment
For the non-reactive system, both binary and multi-component
semi-batch experiments were performed. First, the apparatus was
filled with 100 mL toluene under argon, then stock solutions in
toluene of [Mo(CO)6] (ca. 49.9 mg in 10 mL for binary-1, 50.0 mg in
10 mL for binary-2, and 124.8 mg in 25 mL for multi-component
experiment), [Mn2(CO)10] (ca. 75.0 mg in 10 mL for binary-1,
The experimental apparatus for non-reactive and reactive sys-
tem studies consisted of a jacketed glass reactor (Aceglass)
equipped with a magnetic stirrer, a Teflon membrane pump (Cole-
Parmer) and an Anton-Paar DMA 5000 (precision of better
than 10−6 g cm−3) vibration tube densitometer thermostatically
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Fig. 1 Experimental setup for simultaneous in-situ FTIR measurements and density measurements. 1. Argon tank; 2. Argon purification column; 3.
Pressure transducer; 4. Jacketed continuous stirrer tank reactor (CSTR); 5. Hermetically sealed Teflon pump; 6. FTIR with high pressure flow through
cell; 7. Anton Paar DMA 5000 Densitometer; 8. Data acquisition.
75.2 mg in 10 mL for binary-2, and 187.7 mg in 25 mL for multi-
component experiment), [Re2(CO)10] (ca. 98.9 mg in 10 mL for
binary-1, 99.2 mg in 10 mL for binary-2, and 250.5 mg in 25 mL
for multi-component experiment) were prepared, and transferred
by injection into the reactor containing toluene–argon at 0.1
MPa. Typically, several perturbations using a single stock solution
were required for each binary study and optimally planned
perturbations using all three stock solutions were required for each
multi-component semi-batch experiment (see numerical section).
With respect to solution homogeneity, after each perturbation,
the solution was stirred and circulated for about 10 minutes before
simultaneous density measurements and spectral measurements
(in the range of 1000–4000 cm−1) were taken. With respect to
repeatability, at least 6 readings of density were performed during
each perturbation. Two semi-batch experiments of each binary
system consisting of [Mo(CO)6]–toluene, [Mn2(CO)10]–toluene,
and [Re2(CO)10]–toluene were performed, and for the multi-
component system [Mo(CO)6]–[Mn2(CO)10]–[Re2(CO)10]–toluene,
3 semi-batch experiments were performed.
hexane were studied with 2 semi-batch experiments and 1 semi-
batch experiment respectively. Two semi-batch experiments were
performed for the multi-component reactive system.
Determination of experimental moles
The simplest volumetric equation which represents the present
experiments takes the form eqn (2),
ꢀ
V(T, P, n) = nsVo +
niVi
(2)
∞
¯
s
solute-i
where V(T, P, n) is total volume of solution, Vos is the pure molar
∞
¯
volume of solvent, Vi is the partial molar volume of solute at
infinite dilution, ns is the mole of solvent and ni is the mole
of solute-i. At each perturbation of the system using the stock
solutions, each of the mole balances is updated by the increments
Dns and Dni. This multi-linear approximation implicitly assumes,
in particular, that the solvent volumetric properties do not vary in
the ultra-dilute solutions studied.
Experimental system
For the reactive system, both binary and reactive multi-
component semi-batch experiments were performed. The appa-
ratus was filled with 100 mL n-hexane under argon and stock
solutions of [Rh4(CO)12]–n-hexane (ca. 59.4 mg in 30 mL for
binary-1, 59.8 mg in 30 mL for binary-2) and PPh3–n-hexane
(ca. 102 mg in 10 mL for binary experiment, 31.7 mg in 20 mL
for reactive experiment-1 and 31.6 mg in 20 mL for reactive
experiment-2) were prepared, and transferred by injection into
the reactor containing n-hexane–argon at 0.1 MPa. Typically,
several perturbations using a single stock solution were required
for each binary study and many perturbations of PPh3 in sub-
stoichiometric amounts were required for each reactive multi-
component semi-batch experiment. With respect to solution
homogeneity, after each perturbation, the solution was stirred
and circulated for about 10 minutes before simultaneous density
measurements and spectral measurements (in the range of 400–
4000 cm−1 using KBr or in the range of 1000–4000 cm−1 using
CaF2) were taken.21 With respect to repeatability, at least 6
readings of density were performed during each perturbation. The
binary system consisting of [Rh4(CO)12]–n-hexane and PPh3-n-
An initial volume of solvent at ca. room temperature was added
to the experimental system thermostated at 298.15 K and the
experimental density measured qos . The experimental volume and
experimental density, together with molar mass, result in the initial
moles of solvent introduced nisnit and the molar volume Vso.
Perturbations using stock solutions
As mentioned, each stock solution was prepared by dissolving
a known mg solute msitock to a known quantity of solvent. The
maximum solute concentrations were ca. 2000 × 10−6. During
each perturbation, a known volume of stock solution Vinj was
injected into the system. This known volume Vinj was then used to
calculate the increments Dns and Dni according to eqn (3) and (4).
qos Vinj
Ms
∼
Dns
(3)
=
Vinjmsitock
VstockMi
∼
Dni
(4)
=
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The increments Dns and Dni are used to update the total moles of
solvent ns and solute ni.
liquid phase to a very good approximation. The procedure used
has been discussed in detail elsewhere.23–24
Since all the stock solutions are dilute, i.e. maximum ca. 2000 ×
10−6, eqn (3) and (4) are clearly very good first approximations to
the incremental quantities. But they are not exact, and accordingly,
The determination of the partial molar volumes of the species
present in the ultra-dilute multi-component solutions was achieved
using a simplified form of a newly developed response surface
approach to total molar volumetric data.11 The reaction volume
was calculated directly i.e. from the differences between limiting
partial molar volumes of the products and reactants.15,25
In the original multi-component response surface model for
dilute mixtures of highly interacting solutes (i.e. protic solutes
undergoing hydrogen bonding), the total molar volume Vm (T, P,
x) can be represented by 3 sets of terms, namely linear, bilinear
and higher interaction terms eqn (6).
∼
this is made explicit in the equation with , the approximately
=
equal to symbol. In eqn (3), it is assumed that the experimental
Vinj is a good approximation for the incremental solvent volume
and that the solvent molar volume can be approximated by its
pure molar volume Vos . In eqn (4), it is assumed that the volume
of solvent used in the stock solution is equal to the volume of
stock solution Vstock. In either case, the errors in the incremental
moles Dns and Dni are anticipated to be considerably less than
1%. As seen in the results section, the error bounds for partial
molar volumes are a little larger than 1%. Therefore, there are
other sources of error in this study which are greater than those
introduced by the approximations eqn (3) and (4).
N
N−1
N
ꢀ
ꢀ ꢀ
Vm(T, P, x) =
aixi +
aij xixj
i=1
i=1 j>i
N−1
N
ꢀ ꢀ
Since it is difficult to introduce very small and precise quantities
of solutes (accuracy of 10−4 g or better) directly into the present
system without introducing further mass balance and other errors,
the present method of preparing stock solutions was used. This
issue will be addressed further in the discussion.
+
c ij xixj(xi − xj)
(6)
i=1 j>i
Notation T, P, x and N represent temperature, pressure, mole
fraction and number of components respectively. Given a set of
appropriate experimental values for Vm, the coefficients ai, aij, and
c ij can be solved, and finally, the infinite dilution partial molar
volumes and volumes of interaction can be determined.11
For the case of ultra dilute multi-component solutions of weakly
or negligibly interacting solutes, changes to eqn (6) should be
made. First, the bilinear and higher interaction terms can probably
be neglected (eqn (7) and (8)).
Total molar volumes and mole fractions
The primary quantities used in this study for modeling purposes
are the total molar volume Vm and mole fractions. The total molar
volume is defined by eqn (5)
ꢁ
ꢁ ꢁ
(nisnit
+
Dns)Ms +
(
Dni)Mi
ꢃ
N
N−1
N
ꢀ
ꢀ ꢀ
i
ꢂ
Vm
=
(5)
Vm(T, P, x) =
a xi +
a xixj
ij
(7)
i
ꢁ
ꢁ ꢁ
init
i=1
i=1 j>i
q ns
+
Dns+
(
Dni)
i
N
ꢀ
where the experimental solution density q appears in the denomi-
nator. Note that the small incremental errors in moles, i.e. Dns and
Dni, appear in both the numerator and denominator. This reduces
somewhat the accumulation or propagation of error in the value
of Vm. The same situation, namely, presence of incremental errors
in both numerator and denominator of mole fractions also occurs.
Vm(T, P, x) =
a xi
i
(8)
i=1
Secondly, eqn (7) and (8) can be re-arranged, in order to avoid ill-
conditioning26,27 and to make the computation more numerically
stable (see eqn (9a)–(9c) or eqn (10), resectively)
ꢀ
ꢀ
Y = Vm(T, P, x) − asxs =
aixi +
aisxixs
solute-i
solute-i
Numerical aspects
ꢀ
ꢀ
+
aij xixj
(9a)
For the multi-component [Mo(CO)6]–[Mn2(CO)10]–[Re2(CO)10]–
toluene semi-batch experiments, a recently developed experimen-
tal design11 was used to obtain an optimal distribution of data
points throughout the multi-component composition space.
Band-target entropy minimization (BTEM) was used to recover
a normalized observable pure component spectrum of species-i
at the wavenumber range m, aˆi×v present in the binary and multi-
component solution, for both the non-reactive and the reactive
systems.22
The concentrations or moles of the species present in the
reactive multi-component solutions were determined using i) the
raw experimental FTIR absorbance data, Ake×v, where k denotes
the number of spectra taken in one step of semibatch experiment,
e denotes the number of semibatch steps, ii) the BTEM normalized
reconstructed spectra of total species s, aˆs×v, iii) information on
the initial reagent quantities used in the experiment Noke×s and iv)
the mole balances for Rh and PPh3, which are conserved in the
solute-i solute-j>i
ꢀ
Y = Vm(T, P, x) − asxs =
(ai + aisxs)xi
solute-i
ꢀ
ꢀ
+
aij xixj
(9b)
(9c)
(10)
solute-i solute-j>i
ꢀ
ꢀ
ꢀ
Y = Vm(T, P, x) − asxs =
a,xi +
a xixj
ij
i
solute-i
solute-i solute-j>i
ꢀ
Y = Vm(T, P, x) − asxs =
a xi
i
solute-i
The difference between total molar volume of solution and the
solvent that represents the total molar volume of contributing
solutes is defined by Y.
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Finally, further physical significance can be attached by recog-
nizing that eqn (9c) and eqn (10) are equivalent to eqn (11) and
eqn (12),
ꢀ
ꢀ
ꢀ
∞
Y = Vm(T, P, x) − xsVo =
xiVi
+
a xixj
ij
(11)
¯
s
solute-i
solute-i solute-j>i
ꢀ
∞
Y = Vm(T, P, x) − xsVo =
(12)
¯
xiVi
s
solute-i
respectively, where Vos (cm3 mol−1) is the molar volume of solvent
and Vi (cm3 mol−1) is the limiting partial molar volumes of
∞
¯
solutes-i. The resulting equations, namely eqn (11) represents a
model with linear and bilinear terms which accounts for slight
interactions and eqn (12) represents a model with linear terms only
(strictly no interaction). It can be noted that precedence exists for
such substitutions and re-arrangements, and this basic approach
(resulting in a truncated form somewhat similar to eqn (10)) was
used previously by Young and Smith for simple 2 solute systems.28
Results
Non-reactive binary system
Three different semi-batch binary experiments, namely
[Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10] in toluene under argon,
were carried out at 298.15 K and 0.1 MPa. Density as well as FTIR
spectra of these binary systems were measured. Two experimental
runs for each of these binary systems were performed.
From FTIR absorbance spectra, the pure component spectra of
[Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10] were obtained by using
BTEM from these binary experiments, and the results are shown
in Fig. 2. The BTEM analysis did not indicate the presence of
any further organometallic impurities. The spectra compare very
well with literature references, where maxima at 1983 cm−1 for
[Mo(CO)6],29,30 1976, 2010, 2045 cm−1 for [Mn2(CO)10]31 and 1970,
2011, 2070 cm−1 for [Re2(CO)10]32 are observed. These spectra were
later used for comparison with pure component spectra obtained
using BTEM from the multi-component solutions.
Fig. 2 The pure component spectra of (1) [Mo(CO)6], (2) [Mn2(CO)10] and
(3) [Re2(CO)10] in toluene obtained from experimental binary absorbance
data using BTEM.
Table 1 Data for binary experiments and limiting apparent molar
volumes in toluene at T = 298.15 K and P = 0.1 MPa
Range of mole
a
Solute-i
Exp
fraction (×10−6
)
φoi /cm3 mol−1
[Mo(CO)6]
[Mn2(CO)10]
[Re2(CO)10]
1
2
1
2
1
2
19.9–149.2 (8)
20.0–149.3 (8)
20.3–151.7 (8)
20.3–151.9 (8)
16.0–119.5 (8)
16.0–120.0 (8)
175.5 1.8
176.6 0.5
264.9 0.8
266.3 3.3
278.8 1.3
276.9 0.9
The average density measurements (from at least six measure-
ments) for each perturbation were used to determine the apparent
molar volumes of [Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10] in
toluene at 298.15 K and 0.1 MPa. The apparent molar volume of
solute-i, φi33 was determined using eqn (13),
a Values in parentheses show number of semi-batch perturbations (data
points).
Vm(T, P, x) − xsVso
reproducibility, and (c) for comparison purposes with the partial
molar volumes obtained from the multi-component solutions and
thus to confirm the validity of the utility of the presently developed
multi-component approach.
␾ =
(13)
i
xi
where Vm(T, P, x) was simply calculated from the experimental
solution density and total mass in the solution. The molar volume
of toluene Vos was determined from the experimental density of
pure toluene and molar mass.
Non-reactive multi-component system
The concentration ranges of the studies (in mole fractions)
and the limiting apparent molar volume (extrapolated to zero
concentration using a linear regression of the apparent molar
volume versus mole fraction),34 φio of those three solutes are
tabulated in Table 1. As shown in Table 1, it was possible to
determine these quantities with typically about 1% error, and
inter-experimental reproducibility is also very high. These binary
systems were measured (a) in order to establish the sensitivity and
accuracy of the measurement method in the ultra dilute solute
region (xsolute-i less than 152 × 10−6), (b) in order to establish
It has been shown that experimental design is a crucial factor
in utilizing the response surface model for direct determination
of partial molar volume of each individual species from multi-
component systems.11 Therefore, a similar approach is taken in the
present study. Three sets of experiments consisting of 12 different
compositions each were pre-designed prior to the experimental
work. All experiments were started with pure toluene as a major
component (solvent) under argon, and subsequently adding one
stock solution at a time to change the solution composition.
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The constraints imposed for the dilute solution region in the
current study are shown in eqn (14), (15) and (16),
can be obtained reliably, even from these highly overlapping
spectral data, using a least squares fitting procedure. Fig. 4(b)
shows the concentration profiles obtained from the corresponding
perturbations in a multi-component experiment (run I). BTEM
analysis did not indicate the formation of any new additional
carbonyls during these experiments.
0 ≤ xi ≤ 0.000140 for i = 1, 2, 3
0.9997 ≤ x4 ≤ 1
(14)
(15)
N
ꢀ
xi = 1
(16)
i=1
where N is the total number of components (in this study, N =
4), subscript-i refers to the solutes [Mo(CO)6] (1), [Mn2(CO)10] (2),
[Re2(CO)10] (3) and the major component, toluene (4).
The raw experimental results of the multi-component experi-
ments are shown in Table S1 (ESI†). The mole fractions of the three
organometallic solutes [Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10]
were calculated from the actual experimental mass balance of
the perturbations used in the experiments.35 Also, (a) the density
differences between the pure solvent and the multi-component
solution, and (b) the volume differences Y defined by the left-
hand-side of eqn (11) or eqn (12), for each perturbation of the three
semi-batch runs are tabulated in Table S1.† The 3-dimensional
plots generated by the composition data and the volume difference
Y data constitute the response surface maps for these systems, and
form the basis for the analytical approach.
For every perturbation/injection, the absorbance spectra of the
mixtures were measured. An example of the multi-component
absorbance spectra (taken from step 12 of semi-batch run-1)
before and after preconditioning (e.g. background and solvent
subtraction36 as well as baseline correction37) is given in Fig. 3.
The pure component spectra reconstruction results via BTEM
are shown in Fig. 4(a). The inner product(s) between unit
vectors of reconstructed pure component spectra of [Mo(CO)6],
[Mn2(CO)10], and [Re2(CO)10] and their references (Fig. 2) are
0.9993, 0.9920, and 0.9970, respectively. Since the pure component
spectra are accurate, the relative concentrations of these solutes
Fig. 4 (a) Pure component spectra of (1) [Mo(CO)6], (2) [Mn2(CO)10] and
(3) [Re2(CO)10] in toluene from a multi-component study using BTEM. (b)
Relative concentration profile of pertinent component of run I.
Since these multi-component experiments were conducted in
the ultra-dilute range, the full response surface model eqn (6)
was reasonably truncated. Two truncated models, namely models
with linear and bilinear terms (linear-bilinear model), eqn (11),
and models with linear terms only (linear model), eqn (12),
were used to describe the response surface map in this ultra-
dilute range for the multi-component organometallic solutions.
The parameters describing the response surface map contain the
volumetric information of each solute in the infinite dilution. Thus,
the individual limiting partial molar volume of each species can
be determined simultaneously from the multi-component system.
The data in Table S1,† with the exclusion of the binary
composition data, were used in the analysis, and a least squares
∞
¯
method was applied to determine the parameters, Vi in linear
∞
¯
model eqn (12) and V as well as aij in the linear-bilinear model
i
eqn (11). The limiting partial molar volumes of the solutes-i are
∞
¯
provided by the parameters Vi for both models as shown in eqn
(11), (12). A comparison of the values resulting from the binary
and multi-component studies can be seen in Table 2.
As can be seen in Table 2, the limiting partial molar volumes for
[Mo(CO)6] and [Mn2(CO)10] obtained from the multi-component
solutions (linear model) are in agreement with those obtained
from the binary solutions, but the data for [Re2(CO)10] is slightly
outside the confidence limits. This indicates the limitations of
determining the partial molar volumes from this multi-component
solution using a linear model alone. Table 2 also shows that the
limiting partial molar volumes of all the solutes in the multi-
component solutions (linear–bilinear model) are in agreement
with those obtained from the binary solutions. This indicates
that the determination of the partial molar volumes from this
multi-component solution is possible using the more robust
linear–bilinear model which correctly captures the curvature of
Fig. 3 Absorbance spectra of multi-component non-reactive system of
[Mo(CO)6], [Mn2(CO)10], and [Re2(CO)10] in toluene (i) before and (ii) after
preconditioning (run I, step 12).
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∞
Table 2 Comparison of limiting partial molar volume, Vi (cm3 mol−1
)
¯
Table 3 Data for binary experiments and limiting apparent molar volume
φoi (cm3 mol−1) in n-hexane at T = 298.15 K and P = 0.1 MPa
in toluene at T = 298.15 K and P = 0.1 MPa determined from binary and
multi-component studies
Range of mole
a
Solute-i
[Rh4(CO)12]
PPh3
Exp
fraction (×10−6
)
φoi /cm3 mol−1
Limiting partial molar volume, V∞i /cm3 mol−1
¯
313.2 4.6b
310.8 6.4b
219.8 2.2
Multi-component
1
2
30–65 (5)
30–65 (5)
40–380 (8)
Solute
Binarya
Linear
Linear and bilinear
a Values in parentheses show the number of semi-batch perturbations (data
points). b These limiting apparent molar volumes were obtained from
average values of the apparent molar volume within the concentration
range.
[Mo(CO)6]
[Mn2(CO)10]
[Re2(CO)10]
176.1 1.9
265.6 3.4
277.9 1.6
178.7 1.1 (1.48) 176.4 2.5 (0.17)
261.9 1.1 (1.39) 265.1 2.4 (0.19)
269.9 1.1 (2.88) 276.8 2.4 (0.40)
a Average value from binary experiments 1 and 2 as listed in Table 1. Values
in parentheses show the percentage deviations from binary studies results.
that the free PPh3 had vibrations at 696 and 742 cm−1 in the
range of 550–850 cm−1 as well as 1808, 1879, 1893, 1948 and
1963 cm−1 in the range of 1800–2200 cm−1.40 It did not indicate
other species, including the most probable degradation product
OPPh3 (triphenylphosphine oxide) with bands at 698, 725, 753
and 759 cm−1.41
The average density measurements (from at least six measure-
ments) for each perturbation were used to determine the apparent
molar volumes φi of [Rh4(CO)12] and PPh3 in n-hexane from the
binary experiments under argon at 298.15 K and 0.1 MPa using
eqn (13).
The concentration ranges of these binary studies (given in mole
fractions) and the limiting apparent molar volume φoi are tabulated
in Table 3. As shown in Table 3, the limiting apparent molar
volumes of both binary systems could be determined with typically
less than 1% error, and inter-experimental reproducibility for two
binary runs of [Rh4(CO)12] in n-hexane is considered very high.
the response surface. The average uncertainty in the determined
partial molar volumes is ca. 1%, and the values obtained from
binary and multi-component data differ by only ca. 0.3%. This
is further verified by analysis of the response surface curvature
coefficients aij, determined to be a12 = 33183 31990 (cm3 mol−1),
a13 = −57561
30913 (cm3 mol−1), a23 = −89926
31609
(cm3 mol−1). From these coefficients, it is clear that the presence
of component 3, the [Re2(CO)10], gives rise to non-negligible
non-linearities in the multi-component data. In straightforward
physical-chemical terms, association between some of the solutes
is occurring.
Multi-component reactive organometallic system
Binary reference experiments. Two different semi-batch binary
experiments, with either [Rh4(CO)12] or PPh3 in n-hexane under
argon, were carried out at 298.15 K and 0.1 MPa. Density as
well as FTIR spectra of these binary systems were measured.
The purpose of these binary studies was (i) to identify and/or
quantify the impurities that might be present, (ii) to obtain pure
component reference spectra and (iii) for later comparison with
the partial molar volumes obtained from the multi-component
solutions. Two experimental runs and one experimental run were
carried out for the binary studies of [Rh4(CO)12] and PPh3 in n-
hexane, respectively.
FTIR absorbance spectra of the binary system of [Rh4(CO)12] in
n-hexane were investigated in the metal carbonyl vibrational range
1800–2200 cm−1. BTEM was applied and maxima for [Rh4(CO)12]
were observed at 1886s, 2044s, 2070vs and 2074vs cm−1.6,12,13,38 In
addition, the presence of the high nuclearity cluster [Rh6(CO)16],
an impurity, was indicated by vibrational bands at 1819s and
2075vs cm−1.6,39 Quantitative analysis shows that the maximum
concentrations of [Rh6(CO)16] throughout experiment-1 and
experiment-2 are less than 0.2 × 10−6 and 0.15 × 10−6 mole
fractions, respectively. The extremely low amount of [Rh6(CO)16]
is, in part, the result of good experimental procedure, in order to
minimize oxygen/water/light present in the reactor system. In this
study, in both binary and multi-component reactive systems, the
amount of impurity present was very small, and its presence will
not significantly alter the volumetric analysis. Further details of
the numerical analysis are provided in the following section.
In the binary experiment with PPh3 in n-hexane, the spectro-
scopic analysis was performed by focusing on both the far-infrared
region in the range of 550–850 cm−1 and in mid-infrared region in
the range of 1800–2200 cm−1 using KBr windows. BTEM showed
Multi-component reactive experiments. The ligand substitu-
tion reaction eqn (1) was conducted in n-hexane solvent at
298.15 K and 0.1 MPa under argon. The reaction was run in semi-
batch mode by introducing perturbations PPh3 into the reactor
in sub-stoichiometric amounts. The reaction is very fast.12–14 Free
PPh3 was not observed in any of the in-situ FTIR reaction spec-
tra. The mono-substituted product [Rh4(CO)11PPh3] was totally
soluble in n-hexane in the range of concentrations used (less than
65 × 10−6 mole fraction). During the experiments, both density
and FTIR spectroscopic measurements were performed. The
changes in the metal carbonyl vibrational bands in the range 1800–
2200 cm−1 during one of the semi-batch runs is shown in Fig. 5.
Spectral preconditioning, namely subtraction of background
and n-hexane36 and baseline drift correction37 were initially per-
formed before further analysis (e.g. reconstructed pure component
spectra by BTEM and quantitative analysis). As an example, one
selected absorbance spectra of one reaction mixture before and
after preconditioning are presented in Fig. 6.
Next, BTEM was applied to the preconditioned spectra to
reconstruct all the pure component spectra involved in the
reaction. The spectra of the major pure components, namely
reactant [Rh4(CO)12] and the product [Rh4(CO)11PPh3], as well
as the minor pure components [Rh6(CO)16] and the di-substituted
product [Rh4(CO)10(PPh3)2] were successfully recovered with high
signal to noise ratio and are presented in Fig. 7. The reconstructed
spectra of minor products [Rh6(CO)16] and [Rh4(CO)10(PPh3)2]
have lower signal to noise ratios as they exist in trace amounts
in solution. BTEM analysis did not indicate the presence of any
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Fig. 7 Recovered pure component spectra obtained by BTEM: (i)
[Rh4(CO)12], (ii) [Rh4(CO)11PPh3], (iii) [Rh4(CO)10(PPh3)2] and (iv)
[Rh6(CO)16].
Fig. 5 In-situ FTIR reaction spectra of the ligand substitution reaction
of [Rh4(CO)12] with PPh3 at 298.15 K and 0.1 MPa (under argon) in the
range 1800–2200 cm−1 during a 12 perturbation semi-batch experiment.
The nominal rhodium concentration is ca. 65 × 10−6 mole fraction.
agreement with those reported in the literature.12,13 The spectral
deconvolution of the latter species is slightly sub-optimal due
to its very low concentrations and the limited variation in its
concentrations (variance of signals).
Further in-situ quantitative spectroscopic analysis was carried
out by using spectroscopy and mass balances based on injected
mass. The results of these quantitative analyses for both the
binary and reaction systems are presented in Fig. 8(a) and (b),
respectively. It can be seen that satisfactory results for [Rh4(CO)12]
are observed in both binary and reaction systems. However, a
noticeable discrepancy between the two approaches is observed
in Fig.8(b) for [Rh4(CO)11PPh3] at high concentrations. The
calculation based on injected mass alone, neglects the influence
of the side-reaction, and thus overestimates the real moles
of [Rh4(CO)11PPh3] in the solution. Whyman also noticed the
formation of di-substituted product, [Rh4(CO)10(PPh3)2] when he
added mole ratio PPh3 : [Rh4(CO)12] = 1 : 1.12 These deviations
became more pronounced at mole ratios PPh3 : [Rh4(CO)12] greater
than 0.7 : 1 where the second substitution product becomes
non-negligible. Throughout experiment-1 and experiment-2, the
maximum concentrations of [Rh4(CO)10(PPh3)2] were less than
2.2 × 10−6 and 3.3 × 10−6 in mole fractions, respectively.
Another species which can be considered as an unavoidable
impurity, namely [Rh6(CO)16], was also monitored in the present
study. Quantitative analysis showed that the maximum concentra-
tions of [Rh6(CO)16] throughout experiment-1 and experiment-2
were less than 0.23 × 10−6 and 0.14 × 10−6 in mole fractions,
respectively.
Quantitative analysis of the first few perturbations, where
mole ratio PPh3 : [Rh4(CO)12] = 0.0–0.7 : 1 showed that the
maximum concentration of [Rh4(CO)10(PPh3)2] in experiment-
1 and experiment-2 were less than 1.4 × 10−6 and 2.4 × 10−6
mole fractions, respectively. These concentrations are considered
sufficiently low so that the assumption that the contribution of
the minor component is negligible can be reasonably applied.
Fig.
6
Absorbance spectra of reaction (i) before and (ii) after
preconditioning.
other metal carbonyls nor free PPh3 during these experiments
using mid-infrared spectroscopy. Additional experiments were
carried out, focusing on the far-infrared region 550–850 cm−1
using a cell with KBr windows, and these experiments further
confirmed the absence of free PPh3. The FTIR spectra of mono-
substituted product [Rh4(CO)11PPh3] at 1856m, 1875m, 1906vw,
2011w, 2022ms, 2031s, 2054vs, 2057vs, and 2087s cm−1 and
di-substituted product [Rh4(CO)10(PPh3)2] at 1822mw, 1859mw,
2004m, 2016ms, 2021m, 2044s, and 2069s cm−1 are in good
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Fig. 8 Mole(s) comparison determined from spectra prediction (solid symbols) and mass injection (open symbols) of (a) binary system of [Rh4(CO)12]
and (b) reaction system. Symbols: ᭹, [Rh4(CO)12]; ꢀ, [Rh6(CO)16]; ꢁ, [Rh4(CO)11PPh3]; ᭜, [Rh4(CO)10(PPh3)2].
Accordingly, the error induced by this assumption becomes
imbedded in the determination of limiting partial molar volume
of [Rh4(CO)11PPh3].
density of pure n-hexane and molar mass. Two major solutes
are considered in the modified response surface model, namely
[Rh4(CO)12] and [Rh4(CO)11PPh3]. In order to retain a tractable
problem and avoid unnecessary propagations in error, these
two quantities were treated as lumped parameters, whereby the
mole fractions of [Rh4(CO)12] and the impurity [Rh6(CO)16] were
combined and the mole fractions of [Rh4(CO)11PPh3] and the
impurity [Rh4(CO)10(PPh3)2] were combined. All mole fractions
used for this analysis are based on Fig. 8(b). The present study
did not attempt to include minor trace species explicitly in a 4
solute problem since their concentrations are very low and thus
the inverse problem becomes severely ill-conditioned.
The bulk density of the solutions increased noticeably as a func-
tion of mole ratio PPh3 : [Rh4(CO)12] as shown in Fig. 9. Especially
at mole ratio PPh3 : [Rh4(CO)12] = 0.0–0.7 : 1, the reaction system
can be approximated/modeled as a 2 solute system, namely the
reactant [Rh4(CO)12] and the major product [Rh4(CO)11PPh3] in
the pseudo multi-component solution consisting of n-hexane–
argon with dissolved CO42 and in the presence of two trace
components [Rh6(CO)16] and [Rh4(CO)10(PPh3)2]. In principle,
either eqn (11) or eqn (12) can be applied to determine the
limiting partial molar volume of [Rh4(CO)12] and [Rh4(CO)11PPh3]
in this multi-component reactive solution. However, due to the
more limited concentration range used (65 × 10−6 mole fraction
compared to 140 × 10−6 mole fraction in the non-reactive multi-
component), and few data points measured, the use of eqn (12)
with linear terms only was considered more appropriate.
A comparison of the resulting limiting partial molar volume
determined from multi-component reactive solutions with sep-
arate binary study of [Rh4(CO)12] in n-hexane–argon alone, are
summarized in Table 4.
It can be seen from Table 4 that the two multi-component
reaction experiments are quite consistent. The limiting partial
molar volumes of [Rh4(CO)12] obtained from binary and multi-
component are very close, especially for experiment-1. The values
are statistically overlapping within the 95% confidence limit.
Limiting partial molar volumes of [Rh4(CO)11PPh3] obtained
from multi-component reaction solutions are slightly different.
The discrepancy is probably due to the fact that the maximum
concentration of the trace impurity [Rh4(CO)10(PPh3)2] is slightly
higher in experiment-2 (ca. 2.4 × 10−6) compared to experiment-1
∞
¯
Table 4 Mole fraction range and limiting partial molar volume, Vi
(cm3 mol−1) of [Rh4(CO)12] and [Rh4(CO)11PPh3]
Mole fraction
¯
Solute-i
Exp
range (×10−6
)
V∞i /cm3 mol−1
[Rh4(CO)12]
1
2
20–65
20–65
310.7 2.7
303.3 2.2
312.0 5.4a
461.5 4.5
473.8 2.3
Fig. 9 Density of reaction mixture at T = 298.15 K and P = 0.1 MPa.
Total molar volume Vm(T, P, x) was calculated from the
experimental solution density and total mass in the solution (e.g.
total mass of solvent, major and minor components). The molar
volume of n-hexane Vos was determined from the experimental
[Rh4 (CO)11PPh3]
1
2
0–42
0–42
a Average value from two binary experiments given in Table 3.
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(ca. 1.3 × 10−6). This fact suggested that the limiting partial molar
volume of pure [Rh4(CO)11PPh3] must be slightly smaller than the
values obtained from this study.
The reaction volume for eqn (1) can be directly determined
by using the differences of the limiting partial molar volumes of
products and limiting partial molar volumes of reactants as shown
in eqn (17).15,25
limiting partial molar volumes of each solute. This conclusion was
based on comparison with separate determinations of limiting
partial molar volume performed in binary experiments. The
average error in the linear model was about 2%. We consider this
error to be quite acceptable, indeed very small, when the sensitivity
of organometallics and their very high dilution are taken into
consideration. Moreover, the average error in the linear–bilinear
model was considerably lower at 0.3%. This suggests that genuine
non-linearities in the data are present, and that intermolecular
interactions might exist.
The above analysis of the “non-reactive” multi-component
system would be null-and-void if some degradation/reaction of
the species present had occurred. Therefore, sensitive on-line
FTIR measurements were made in order to confirm that no new
complexes had been formed. Formation of any new complexes
would definitely cause the misinterpretation of the volumetric
results. The FTIR measurements did not indicate the presence
of any new species. In addition, band-target entropy minimization
(BTEM) was also applied to the data set in order to identify any
new spectroscopic features. BTEM did not indicate the presence
of any complexes other than the three solutes used.
∞
∞
∞
∞
¯
3
¯
¯
¯
DrV = VRh (CO) PPh + VCO − VRh
− VPPh
(17)
(CO)
12
4
11
3
4
The more reliable limiting partial molar volumes of [Rh4(CO)12]
and [Rh4(CO)11PPh3] obtained from experimental run-1 were
selected to calculate the volume of reaction. The limiting partial
molar volume of PPh3 from the binary study was used since it
could not be observed and thus determined directly from a multi-
∞
¯
component study. A partial molar volume of CO, VCO equal to
52 cm3 mol−1 reported by Connoly and Kandalic was used.43 As
a result, the reaction volume determined by eqn (17) is equal to
−17.0 5.7 cm3 mol−1.
Discussion
Multi-component non-reactive system
Multi-component reactive system
It was shown that using three semi-batch experiments pre-designed
prior to experimental work, compositional data can be optimally
distributed in the composition space of this quaternary system of
three solutes. This experimental design methodology was crucial
to ensure that the response surface would capture the needed
system characteristics, such as sufficientinformation on the surface
curvature in the limit of infinite dilution as well as the non-
linearities that might arise due to interactions between solutes
and/or solvent. In addition, the optimal distribution will also
be cost effective, as fewer experiments are needed, and such a
distribution allows more easily discernable outliers in the data
set.44
In this multi-component organometallic study, density measure-
ments were performed based on the above experimental design in
a region of ultra-dilute concentration (ca. xi less than 140 × 10−6
mole fraction). Such dilute solutions are a very common situation
in organometallic chemistry and homogeneous catalysis due to
the limited solubility of many of the compounds involved as well
as frequent cost constraints. The very dilute concentrations used
introduce a new and serious complication in the computations,
and this arises due to the fact that the concentration range of
solvent and solutes spans many orders of magnitude.
In the most general approach (see ref. 11), total molar volume
Vm (T, P, x) is a response directly obtained from the bulk density
measurements. However, for very dilute solutions it cannot be
directly applied without modification due to the very different
range of concentrations spanned by the solvent and the solutes
i.e. xsolvent ≈ 0.99999–0.999999 and xsolute ≈ 0.00001–0.000001 that
appear on the right hand side of eqn (6). This creates a problem of
numerical ill-conditioning. In order to circumvent this problem, eqn
(6) was rearranged, to generate a new modified response surface
Y defined by eqn (11) and (12), so that only solute concentrations
are required on the right hand side.
The determination of the limiting partial molar volumes of reactive
species from a multi-component reactive solution appears possible
using the present methodological approach, at least for relatively
simple systems like those present here. However, in order to achieve
accurate parameter estimation, simultaneous spectroscopic and
density measurements are necessary. The former measurements
together with signal processing/multivariate analysis permit the
accurate quantitative determination of concentrations needed
for the proper construction of the response surface model. As
shown in the Results section, the spectroscopy analysis resulted in
the identification of two impurities/side products in the system,
namely, [Rh6(CO)16] and [Rh4(CO)10(PPh3)2]. The simultaneous
application of at least one spectroscopic tool and one bulk fluid
property measuring instrument, appear to be a pre-requisite for
the accurate determination of the physico-chemical properties of
reactive species in a multi-component reactive system.
The determined limiting partial molar volumes of the solutes
in this multi-component reactive system compared well with
those that could be determined from separate binary measure-
ments (e.g. [Rh4(CO)12]). The ability to determine the volumetric
characteristics of reacting organometallics presents a host of
new opportunities (vide infra). Moreover, the reaction volume
could also be determined and the sign was negative. Under
conditions where an observable equilibrium situation might exist,
increasing pressure will shift the equilibrium to the product side.
The uncertainty in the value of the reaction volume is actually
quite large, and this has occurred due to the accumulation of the
uncertainties of all the limiting partial molar volumes involved.
This is an area where future improvement in the methodology can
and should be made.
Corresponding solid state molar volumes
Two closely related response surface models, linear and linear–
bilinear, were developed to describe Y from the multi-component
data. Both the linear and the linear–bilinear models provided good
The volumetric behavior of a dissolved species, i.e. a metal
complex in solution at infinite dilution, should be expected to
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be different from its corresponding solid-state molar volume,
obtained from its crystallographic data. In solution, the volumetric
behavior of a solute is characterized by its limiting partial molar
volume, which can be conveniently partitioned into intrinsic and
solvation parts.45 The intrinsic part is represented by the van
der Waals radius of species and the solvation part is believed
to represent all changes associated with changes in polarity,
electrostriction, and dipole interactions.46 In this regard, the solid-
state molar volumes of the following metal carbonyl clusters
are calculated from their solid state density and respective
molar mass: [Mo(CO)6] (134.69 cm3 mol−1),47,48 [Mn2(CO)10]
(213.11 cm3 mol−1),49 and [Re2(CO)10] (224.23 cm3 mol−1),49 and
[Rh4(CO)12] (296.73 cm3 mol−1).50 As noted in this study, the
first three metal carbonyl complexes showed an increase in their
volumes when they are dissolved in toluene at infinite dilution
of ca. 31%, 25%, and 24%, respectively, while the latter carbonyl
complex only increases its volume by about 6% when it is dis-
solved in n-hexane. The results apparently indicate that there are
distinctly different volumetric aspects associated with the process
of dissolution/solvation since such dissimilar behavior occurs.
It should be noted that these results appear to be consistent
with the theory developed by Lee51 for the partial molar volume
of a solute in infinitely dilute hard-sphere binary mixtures. In this
theory, the increase of the volume in solution depends primarily
on the size of solvent molecule and the packing density of
the solvent. The greater the solute-to-solvent radius ratio, the
smaller the increased volume is achieved. However, no clear-cut
conclusions can be drawn from the present results. Too many
physico-chemical issues are associated with dissolution/solvation
such as polarity, dipole–dipole interactions,46 and issues of solute–
solvent volumetric dissimilarity.52
polynomial models), and system sampling is known to introduce
significant genuine-replicate error,55 even in systems involving
stable solutes and low vapor pressure solvent. (2) Closed systems
with on-line/in-situ measurements are the most frugal with
resources since a much larger number of measurements can be
made with a minimum amount of material. (3) Some of the
organometallics used, even in the “non-reactive” experiments were
sensitive to moisture, oxygen, light, etc., so extra and unnecessary
manipulations are not advisable. (4) Finally, the logistics of on-
line/in-situ data acquisition is particularly well suited to reactive
studies.
Classical (∂ ln Keq/∂P)T versus response surface approach
The classic (∂ ln Keq/∂P)T for the determination of reaction
volumes is well suited to reactions having an observable equi-
librium and has been used extensively to study and classify
inorganic, organic and organometallic reactions.56 Normally,
significant pressure intervals, ca. 1000 bar must be used. This
classical approach typically results in rather precise determinations
of reaction volumes (accuracy ca. 0.2 cm3 mol−1). However, as
pointed out by Kiselev et al., applied pressure affects both the free
energy of reaction and changes in the solute–solvent and solvent–
solvent interactions, and therefore, artifacts in the determined
volumes of reaction may occur.57
The present response surface approach does not require a system
with an observable equilibrium and the reaction volume can be
determined at isobaric conditions. Moreover, the present method
affords information on the individual partial molar volumes at a
particular temperature and pressure, and in principal the volumes
of interaction as well. The determination of reaction volume is
based on the differences between partial molar volumes of the
products and reactants. However, at the moment and as shown
in these examples, typical errors in the reaction volume are of the
order of 5 cm3 mol−1, due in part to the accumulation of error in
the partial molar volumes. This is an issue which requires further
consideration in future studies.
Sensitivity analysis
The present contribution appears to be the first attempt to
determine limiting partial molar volumes of ultra-dilute dissolved
organometallic compounds experimentally, in both non-reactive
and reactive multi-component systems. The limiting partial molar
volumes were determined by using two primary experimental
measurements, namely density and weight, which have accuracies
of 1 × 10−6 in g cm−3 and 10−5 in g, respectively. Assuming that
other possible errors are insignificant, and applying the standard
propagation of error method to determine the uncertainty in the
values of apparent molar volume,53,54 the estimated upper bound
limits of accuracies based on the average range of concentrations
listed in Tables 2 and 3 are ca. 2% to 3% for [Mo(CO)6],
[Mn2(CO)10], and [Re2(CO)10], ca. 1% for PPh3 and ca. 6%
for [Rh4(CO)12]. The relatively low accuracy of metal cluster
concentration is clearly due to the ultra dilute concentration
range selected for measurement. In future studies, the accuracy of
measurement can be substantially improved by selecting a higher
concentration range for study (within the solubility range).
Conclusion
The present response surface methodology provides a general
framework for determining the partial molar volumes of solutes
in both non-reactive and reactive multi-component solutions.
The methodology relies on compositional data together with
measurement of the total liquid volume in order to determine the
volumetric contributions of each constituent present. In contrast
to previous methods, the present approach does not require any
binary information what-so-ever. Accordingly, this provides the
opportunity to study reactive systems and non-isolatable species.
The potential applications of the new approach are numerous.
Issues concerning propagation of error, during the determination
of reaction volumes, and extension to systems possessing many
simultaneous solutes are areas for future study.
Sampling versus online techniques
In the present study for “non-reactive” and “reactive” systems,
a closed system with on-line density measurements and in-situ
spectroscopic measurements were used throughout. The reasons
are self evident. (1) Measurements as precise as possible were
needed in the present study (in order to have good fits to the
Acknowledgements
M. T. thanks the Singapore Millennium Foundation for an
SMF scholarship and A. D. A thanks the Graduate School of
Engineering at NUS for a research scholarship.
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The Royal Society of Chemistry 2006
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1516 | Dalton Trans., 2006, 1505–1516
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The Royal Society of Chemistry 2006
©