The Kelvin equation describes the change in
vapour pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
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
A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usu ...
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:
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
The Ostwald–Freundlich equation governs boundaries between two phases; specifically, it relates the surface tension of the boundary to its curvature, the ambient temperature, and the vapor pressure or chemical potential in the two phases.
The ...
:
where
is the actual vapour pressure,
is the
saturated vapour pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
when the surface is flat,
is the liquid/vapor
surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to ...
,
is the
molar volume of the liquid,
is the
universal gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment pe ...
,
is the radius of the droplet, and
is
temperature.
Equilibrium vapor pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
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
decreases, but so does
. This means
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.
The change in vapor pressure can be attributed to changes in the
Laplace pressure
The Laplace pressure is the pressure difference between the inside and the outside of a curved surface that forms the boundary between two fluid regions. The pressure difference is caused by the surface tension of the interface between liquid and ...
. 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
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
) in his dissertation of 1885. In 2020, researchers found that the equation was accurate down to the 1nm scale.
Derivation using the Gibbs free energy
The formal definition of the
Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and p ...
for a parcel of volume
, pressure
and temperature
is given by:
:
where
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
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
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
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
. Thus, the change in the Gibbs free energy for one molecule is
:
where
is the
saturated vapor pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases ...
of
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
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and p ...
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
by the solubility at a plane surface (
). 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
A cloud chamber, also known as a Wilson cloud chamber, is a particle detector used for visualizing the passage of ionizing radiation.
A cloud chamber consists of a sealed environment containing a supersaturated vapour of water or alcohol. An ...
, 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
ångström
The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...
s 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''
−Δ''S''/''k'', where Δ''S'' is the deviation of the entropy from the equilibrium value.
It is unlikely, however, that new phases often arise by this fluctuation mechanism and the resultant spontaneous nucleation. Calculations show that the chance, ''e''
−Δ''S''/''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
:
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 curvature-induced 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.
See also
*
Condensation
Condensation is the change of the state of matter from the gas phase into the liquid phase, and is the reverse of vaporization. The word most often refers to the water cycle. It can also be defined as the change in the state of water vapor to ...
*
Gibbs–Thomson equation The Gibbs–Thomson effect, in common physics usage, refers to variations in vapor pressure or chemical potential across a curved surface or interface. The existence of a positive interfacial energy will increase the energy required to form small pa ...
*
Ostwald–Freundlich equation
The Ostwald–Freundlich equation governs boundaries between two phases; specifically, it relates the surface tension of the boundary to its curvature, the ambient temperature, and the vapor pressure or chemical potential in the two phases.
The ...
References
{{Reflist
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. Valee
"Simple Kelvin Equation Applicable in the Critical Point Vicinity"''European Journal of Natural History'', (2014), Issue 5, p. 13-14.
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