Kelvin equation
The Kelvin equation describes the change
in vapour pressure due to a curved liquid–
vapor interface, such as the surface of a
droplet. The vapor pressure at a convex
curved surface is higher than that at a flat
surface. The Kelvin equation is dependent
upon thermodynamic principles and does
not allude to special properties of
materials. It is also used for determination
of pore size distribution of a porous
medium using adsorption porosimetry.
The equation is named in honor of William
Thomson, also known as Lord Kelvin.
Formulation
The original form of the Kelvin equation,
published in 1871, is: [1]
where:
= vapor pressure at a curved
interface of radius
= vapor pressure at flat interface (
)=
= surface tension
= density of vapor
= density of liquid
,
= radii of curvature along the
principal sections of the curved
interface.
This may be written in the following form,
known as the Ostwald–Freundlich
equation:
where
is the actual vapour pressure,
is the saturated vapour pressure
when the surface is flat,
is the
liquid/vapor surface tension,
molar volume of the liquid,
universal gas constant,
the droplet, and
is the
is the
is the radius of
is temperature.
Equilibrium vapor pressure depends on
droplet size.
If the curvature is convex,
is positive,
then
If the curvature is concave,
is negative,
then
As
increases,
decreases towards
and the droplets grow into bulk liquid.
If the vapour is cooled, then
but so does
. This means
decreases,
,
increases as the liquid is cooled.
and
may be treated as approximately fixed,
which means that the critical radius
must also decrease. The further a vapour
is supercooled, the smaller the critical
radius becomes. Ultimately it can become
as small as a few molecules, and the liquid
undergoes homogeneous nucleation and
growth.
A system containing a pure homogeneous vapour and liquid in equilibrium. In a thought experiment, a non-wetting tube is
inserted into the liquid, causing the liquid in the tube to move downwards. The vapour pressure above the curved interface
is then higher than that for the planar interface. This picture provides a simple conceptual basis for the Kelvin equation.
The change in vapor pressure can be
attributed to changes in the Laplace
pressure. When the Laplace pressure rises
in a droplet, the droplet tends to evaporate
more easily.
When applying the Kelvin equation, two
cases must be distinguished: A drop of
liquid in its own vapor will result in a
convex liquid surface, and a bubble of
vapor in a liquid will result in a concave
liquid surface.
History
The form of the Kelvin equation here is not
the form in which it appeared in Lord
Kelvin's article of 1871. The derivation of
the form that appears in this article from
Kelvin's original equation was presented by
Robert von Helmholtz (son of German
physicist Hermann von Helmholtz) in his
dissertation of 1885.[2] In 2020,
researchers found that the equation was
accurate down to the 1nm scale.[3]
Derivation using the Gibbs
free energy
The formal definition of the Gibbs free
energy for a parcel of volume
, pressure
and temperature
where
is given by:
is the internal energy and
is the
entropy. The differential form of the Gibbs
free energy can be given as
where
is the chemical potential and
is
the number of moles. Suppose we have a
substance
which contains no impurities.
Let's consider the formation of a single
drop of
with radius
containing
molecules from its pure vapor. The change
in the Gibbs free energy due to this
process is
where
and
are the Gibbs energies
of the drop and vapor respectively.
Suppose we have
molecules in the
vapor phase initially. After the formation of
the drop, this number decreases to
,
where
Let
and
represent the Gibbs free
energy of a molecule in the vapor and
liquid phase respectively. The change in
the Gibbs free energy is then:
where
is the Gibbs free energy
associated with an interface with radius of
curvature
and surface tension . The
equation can be rearranged to give
Let
and
be the volume occupied by
one molecule in the liquid phase and vapor
phase respectively. If the drop is
considered to be spherical, then
The number of molecules in the drop is
then given by
The change in Gibbs energy is then
The differential form of the Gibbs free
energy of one molecule at constant
temperature and constant number of
molecules can be given by:
If we assume that
then
The vapor phase is also assumed to
behave like an ideal gas, so
where
is the Boltzmann constant. Thus,
the change in the Gibbs free energy for
one molecule is
where
pressure of
is the saturated vapor
over a flat surface and
is
the actual vapor pressure over the liquid.
Solving the integral, we have
The change in the Gibbs free energy
following the formation of the drop is then
The derivative of this equation with
respect to
is
The maximum value occurs when the
derivative equals zero. The radius
corresponding to this value is:
Rearranging this equation gives the
Ostwald–Freundlich form of the Kelvin
equation:
Apparent paradox
An equation similar to that of Kelvin can
be derived for the solubility of small
particles or droplets in a liquid, by means
of the connection between vapour
pressure and solubility, thus the Kelvin
equation also applies to solids, to slightly
soluble liquids, and their solutions if the
partial pressure
is replaced by the
solubility of the solid ( ) (or a second
liquid) at the given radius, , and
the solubility at a plane surface (
by
).
Hence small particles (like small droplets)
are more soluble than larger ones. The
equation would then be given by:
These results led to the problem of how
new phases can ever arise from old ones.
For example, if a container filled with water
vapour at slightly below the saturation
pressure is suddenly cooled, perhaps by
adiabatic expansion, as in a cloud
chamber, the vapour may become
supersaturated with respect to liquid
water. It is then in a metastable state, and
we may expect condensation to take
place. A reasonable molecular model of
condensation would seem to be that two
or three molecules of water vapour come
together to form a tiny droplet, and that
this nucleus of condensation then grows
by accretion, as additional vapour
molecules happen to hit it. The Kelvin
equation, however, indicates that a tiny
droplet like this nucleus, being only a few
ångströms in diameter, would have a
vapour pressure many times that of the
bulk liquid. As far as tiny nuclei are
concerned, the vapour would not be
supersaturated at all. Such nuclei should
immediately re-evaporate, and the
emergence of a new phase at the
equilibrium pressure, or even moderately
above it should be impossible. Hence, the
over-saturation must be several times
higher than the normal saturation value for
spontaneous nucleation to occur.
There are two ways of resolving this
paradox. In the first place, we know the
statistical basis of the second law of
thermodynamics. In any system at
equilibrium, there are always fluctuations
around the equilibrium condition, and if the
system contains few molecules, these
fluctuations may be relatively large. There
is always a chance that an appropriate
fluctuation may lead to the formation of a
nucleus of a new phase, even though the
tiny nucleus could be called
thermodynamically unstable. The chance
of a fluctuation is e−ΔS/k, where ΔS is the
deviation of the entropy from the
equilibrium value.[4]
It is unlikely, however, that new phases
often arise by this fluctuation mechanism
and the resultant spontaneous nucleation.
Calculations show that the chance, e−ΔS/k,
is usually too small. It is more likely that
tiny dust particles act as nuclei in
supersaturated vapours or solutions. In
the cloud chamber, it is the clusters of ions
caused by a passing high-energy particle
that acts as nucleation centers. Actually,
vapours seem to be much less finicky than
solutions about the sort of nuclei required.
This is because a liquid will condense on
almost any surface, but crystallization
requires the presence of crystal faces of
the proper kind.
For a sessile drop residing on a solid
surface, the Kelvin equation is modified
near the contact line, due to intermolecular
interactions between the liquid drop and
the solid surface. This extended Kelvin
equation is given by[5]
where
is the disjoining pressure that
accounts for the intermolecular
interactions between the sessile drop and
the solid and
is the Laplace
pressure, accounting for the curvatureinduced pressure inside the liquid drop.
When the interactions are attractive in
nature, the disjoining pressure,
is
negative. Near the contact line, the
disjoining pressure dominates over the
Laplace pressure, implying that the
solubility,
is less than
. This implies
that a new phase can spontaneously grow
on a solid surface, even under saturation
conditions.[6]
See also
Condensation
Gibbs–Thomson equation
Ostwald–Freundlich equation
References
1. Sir William Thomson (1871) "On the
equilibrium of vapour at a curved surface of
liquid," (https://books.google.com/books?id
=ZeYXAAAAYAAJ&pg=PA448#v=onepage&
q&f=false) Philosophical Magazine, series
4, 42 (282) : 448-452. See equation (2) on
page 450.
2. Robert von Helmholtz (1886)
"Untersuchungen über Dämpfe und Nebel,
besonders über solche von Lösungen" (http
s://books.google.com/books?id=9xVbAAA
AYAAJ&pg=PA508#v=onepage&q&f=false)
(Investigations of vapors and mists,
especially of such things from solutions),
Annalen der Physik, 263 (4): 508–543. On
pages 523–525, Robert von Helmholtz
converts Kelvin's equation to the form that
appears here (which is actually the
Ostwald–Freundlich equation).
3. Ouellette, Jennifer (2020-12-09). "Physicists
solve 150-year-old mystery of equation
governing sandcastle physics" (https://arst
echnica.com/science/2020/12/physicists-s
olve-150-year-old-mystery-of-equation-gove
rning-sandcastle-physics/) . Ars Technica.
Retrieved 2021-01-25.
4. 1. Kramers, H. A. Brownian motion in a field
of force and the diffusion model of
chemical reactions. Physica 7, 284–304
(1940).
5. Sharma, Ashutosh (1 August 1998).
"Equilibrium and Dynamics of Evaporating
or Condensing Thin Fluid Domains: Thin
Film Stability and Heterogeneous
Nucleation" (https://pubs.acs.org/doi/abs/
10.1021/la971389f#) . Langmuir. 14 (17):
4918. doi:10.1021/la971389f (https://doi.or
g/10.1021%2Fla971389f) . Retrieved
15 October 2021.
6. Borkar, Suraj; Ramachandran, Arun (30
September 2021). "Substrate colonization
by an emulsion drop prior to spreading" (htt
ps://www.ncbi.nlm.nih.gov/pmc/articles/P
MC8484436) . Nature Communications. 12
(5734): 3. Bibcode:2021NatCo..12.5734B (h
ttps://ui.adsabs.harvard.edu/abs/2021Nat
Co..12.5734B) . doi:10.1038/s41467-02126015-2 (https://doi.org/10.1038%2Fs4146
7-021-26015-2) . ISSN 2041-1723 (https://
www.worldcat.org/issn/2041-1723) .
PMC 8484436 (https://www.ncbi.nlm.nih.go
v/pmc/articles/PMC8484436) .
PMID 34593803 (https://pubmed.ncbi.nlm.
nih.gov/34593803) .
Further reading
W. J. Moore, Physical Chemistry, 4th ed.,
Prentice Hall, Englewood Cliffs, N. J.,
(1962) p. 734–736.
S. J. Gregg and K. S. W. Sing, Adsorption,
Surface Area and Porosity, 2nd edition,
Academic Press, New York, (1982)
p. 121.
Arthur W. Adamson and Alice P. Gast,
Physical Chemistry of Surfaces, 6th
edition, Wiley-Blackwell (1997) p. 54.
Butt, Hans-Jürgen, Karlheinz Graf, and
Michael Kappl. "The Kelvin Equation".
Physics and Chemistry of Interfaces.
Weinheim: Wiley-VCH, 2006. 16–19.
Print.
Anton A. Valeev,"Simple Kelvin Equation
Applicable in the Critical Point Vicinity"
(https://web.archive.org/web/20160303
042442/http://www.world-science.ru/eu
ro/pdf/2014/5/4.pdf) ,European Journal
of Natural History, (2014), Issue 5, p. 1314.
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