Download 4 – Thermodynamic and dielectric properties

”The Physics and Chemistry of Water”
4 – Thermodynamic and dielectric
properties
• Extensively studied, particularly to generate
steam tables for power generation.
• Properties characterized from 2 to 2000 K,
and up to 400000 atm.
• Water is easily supercooled (to -90 ◦C). Plenty
of data available on liquid water < 0 ◦C.
• ”Anomalies” in the thermodynamic properties can be understood in terms of H-bond
structural changes (surprise, surprise).
• H2O does not make ideal solutions, not even
H2O/D2O mixtures behave ideally.
• Comparing with other group 6 hydrides, melting points, boiling points and critical points
are over 100 K higher than expected by extrapolation.
The ice-liquid-vapour triple point
P-T diagram for H2 O. The triple point is at 0.01 ◦ C,
and 4.58 mm Hg, or 610 Pa. (From Eisenberg)
Volume per gram at the triple point
Water
1.00
Ice
1.09
Vapour
206000
The critical point
P-V diagram for H2 O near the critical point (From Eisenberg)
Critical constants (From Robinson)
Temperature, Tc Pressure, Pc Volume Vc
(K)
(MPa)
(cm3 /mol)
H2 O
647.096
22.064
55.95
D2 O
643.89
21.671
56.22
Some notes:
a) Vc is only 3 times the volume of the liquid
at the triple point, while the vapour density
increases a factor of 60000 in going from the
triple point to the critical point.
b) Tc for D2O is lower than for H2O, although
the melting point, boiling point and all triple
points of the D2O phase diagram are higher
than those for H2O.
c) The triple, melting and boiling points of D2O
are higher due to smaller zero-point energy;
D2O must absorb more thermal energy than
H2O to melt.
d) Eight triple points are known for water; four
ice-liquid-ice, and three ice-ice-ice types.
e) The critical point for water is over 250 K higher
than expected by extrapolation:
Thermal energy
Enthalpy (kcal mol−1 ), entropy (2 cal mol−1 ◦ C−1 ), free energy
(kcal mol−1 ) and heat capacity (cal mol−1 ◦ C−1 ) of H2 O at 1 atm
(From Eisenberg.)
Thermodynamic constants
for phase changes
Phase changes of water at 1 atm
Fusion Vaporization
T (K)
273.15
373.15
∆Cp (J/mol · K) 37.284
-41.928
∆H (kJ/mol)
6.0095
40.656
∆S (J/mol · K) 22.000
108.951
∆E (kJ/mol)
6.0095
37.606
3
−1
∆V (cm mol ) -1.62
3.01 × 104
Sublimation
At ice I-liquid-vapour At 0 K to the
triple point
ideal vapour
T (K)
273.15
0
∆H (kJ/mol)
51.01
47.40
∆S (J/mol · K)
187.05
0
∆E (kJ/mol)
48.82
47.31
• Smaller changes in ∆H, ∆S and ∆E for fusion
than for vaporization reflect H-bond changes...!
• The enthalpy of sublimation at 0 K is a direct
measure of the intermolecular energy in ice.
• The heat of vaporization is more than twice
that of e.g. H2S, since there is considerable
hydrogen bonding in H2O at 100 ◦C (∼ 75%),
and almost all these have to be broken. Increased hydrogen bonding causes higher heat
of vaporization at lower temperatures (10% increase at 0 ◦C).
The heat capacity of liquid water is
about twice that of ice or steam
(From Ben-Naim)
Origin of the heat capacity of ice
• Intramolecular vibrations are hardly excited
at room temperature, and only intermolecular
vibrations, hindered translations and hindered
rotations (librations) contribute.
• As ice is heated from 0 K, the translations
(∼ 200 cm−1) are excited before the rotations
(∼ 500-800 cm−1). Below 80 K virtually only
hindered translations contribute.
Contributions to the heat capacity
of liquid water
When the structure of a phase changes with T ,
a configurational contribution to thermal properties has to be considered. Upon heating, water changes structure, i.e. the average relative
positions are changing, and thus also the potential energy associated with the intermolecular interactions. In ice only the average displacements
within the same structure increases.
O–O pair correlations for water (Adapted from Narten).
Thus, we may separate the configurational contribution from that originating in mechanical degrees of freedom;
Cv (observed) = Cv (conf ig.) + Cv (vibrational)
Separation of the experimental heat capacity of H2 O into
vibrational and configuration contributions (From Eisenberg).
In liquid water, the excitation of vibrations accounts for about half of the measured heat capacity, and the remaining configurational contribution is associated with the distortion and breaking of hydrogen bonds. The energy absorbed in
these processes is not available to increase the kinetic energy of water, and relatively much heat
is required to raise the temperature.
In steam, virtually all intermolecular contributions are lost, reducing the heat capacity to below
that of ice.
Anomalies in P–V–T relations
Such as...
• The minimum in Cp at 36 ◦C.
• The minimum in isothermal compressibility.
(From Eisenberg)
• The low thermal expansivity
• The density maximum
...can generally be understood in terms of two
competing effects as water is heated:
(1) The open tetrahedral structure breaks down
(configurational contribution)
(2) Intermolecular vibrations increase in amplitude, enlarging the volume (vibrational contribution)
These effects can also be described very elaborately (and quantitatively) in terms of large Hbonded clusters, see Chaplins website for details!
http://www.lsbu.ac.uk/water/clusters.html
The density maximum
(From Eisenberg)
Note that the decrease in volume between 0 and
4 ◦C is only 0.013 % of the volume at 4 ◦C, and
0.13 % of the increase from 4 to 100 ◦C.
The effect of pressure on viscosity
Viscosity of water from
2 to 100 ◦ C. The initial decrease at low T
is caused by a weakening
of the H-bond structure,
which makes flow easier.
As T increases, this effect is compensated by
the reduction in the available volume per molecule
(From Bett & Cappi, Nature 207, 620 (1965))).
O–O pair correlation functions for pressures up to 7.7 kbar (left),
and pressure dependence on nearest-neighbour separations (right),
indicating a structural change with pressure, and also that the
nearest-neighbour separation changes very little (From Okhulkov,
J. Chem. Phys. 100, 1578 (1994))).
Dielectric constants of ice and water
Ice
T (K) P(kbar)
²
²∞
Ih
250
0
97.5 3.1
II
243
2.3
3.66 3.66
III
243
2.3
117 4.1
V
243
5
144 4.6
VI
243
8
193 5.1
IX
173
2.3
3.74 3.74
Liquid 273
0
87.7 1.77
Remember: Ice III and ice IX have the same
structure, but IX is proton ordered, as is ice II.
The large dielectric constants for disordered ices
suggests that water molecules in these polymorphs
constantly change their orientation due to thermal agitation.
(From Israelachvili)
Dipolar orientation
Approximate dipole moments (D)
Ice
2.6
Water 2.4
Vapour 1.85
The tetrahedral structure leads to...
• ...large dipole moments in each molecule. Compared to a free molecule, each molecule is further polarized by the fields from its strongly
polar neighbours.
• ...a strong angular correlation between molecular dipole moments, meaning that when one
molecule is aligned with an external field, its
neighbours tend to be aligned also.
The inreasing permittivity of ices at higher pressures is caused by the greater density, resulting in
more dipoles per unit volume, and larger values
of the molecular dipole moments. The angular
correlation is thought to change little with the
density.
Stillinger (J. Chem. Phys., 58, 2532 (1973)): Interaction
between neighbours alone in ice would seem to favour antiparallel arrangement of water molecular dipoles over parallel...?
Dipolar orientation - quantitatively
In a first approximation, Kirkwood theory can be implemented (in the absence of external fields);
² = 2πN
m̄ · m̄∗
kT
where N is the number density of molecules, m̄(= µ̄ + αF̄ )
the average dipole moment of a molecule surrounded by its
neighbours, m∗ the vector sum of the dipole moment of a
molecule and its neighbours. The angular correlation can
be described by the Kirkwood correlation parameter
g=
m̄ · m̄∗
|m̄|2
=1+
X
Ni hcos γi i
Here Ni is the number of molecules in the ith coordination
shell, and hcos γi i the average cosine of angles formed by
the dipole moments of molecules in the ith shell with the
dipole moment of the central molecule. Combining these,
we obtain
|m̄|2 g
² = 2πN
kT
This means that ² depends not only on the permanent
dipole, but also on the dipole density and their tendency
to align each other. So, in ice, increasing pressure increases
m̄ while g remains largely unchanged.
Frequency-dependent permittivity ²(ω)
(From Israelachvili)
²(ω) of water can be represented as a Debye relaxation in
the microwave region, added to damped harmonic oscillators in the IR and UV, e.g.:
ε(iξ) = 1 +
d
1 + ξτ
+
11
X
fj
2
2
j=1 ωj + ξ + gj ξ
(1)
For the Debye relaxation d = 74.8 and 1/τ = 6.5 × 10−5
eV. Parameters for five IR and six UV terms are listed
below. This representation is simple, and agrees with calculations using spectral data.
ωj
(eV)
2.07×10−2
6.9×10−2
9.2×10−2
2.0×10−1
4.2×10−1
fj
(eV2 )
Infrared
6.25×10−4
3.50×10−3
1.28×10−3
5.54×10−4
1.35×10−2
gj
(eV)
1.5×10−2
3.8×10−2
2.8×10−2
2.5×10−2
5.6×10−2
ωj
fj
gj
2
(eV) (eV ) (eV)
Ultraviolet
8.21 3.26 0.63
10.0 3.87 0.84
11.4 12.0 2.05
13.6 63.6 3.90
17.8 114.0 7.33
25.2 24.3 5.43
(From Ederth, Langmuir 17, 3329 (2001))
Summary
Compared to other substances, liquid water seems
in many ways unusual and mysterious. However,
the unusual features can be understood in terms
of the properties of water at the molecular level,
in particular the strong association. Thus, water obeys well known laws for molecular interactions and is not anomalous, but perhaps rather
unusual.
General references
• G. Wilse Robinson et al., Water in biology, chemistry
and physics, Singapore: World Scientific 1996.
• D. Eisenberg and W. Kauzmann, The structure and
properties of water, Oxford: OUP 1969.
• A. Ben-Naim, Water and aqueous solutions, New York:
Plenum 1974.
• F. Franks, Water: a matrix of life, 2nd ed., Cambridge:
Royal Society of Chemistry 2000.
• Website by M. Chaplin, London South Bank University, http://www.lsbu.ac.uk/water/index.html. An extensive site explaining many properties of water.
• J. N. Israelachvili, Intermolecular and surface forces,
London: Academic Press 1992.