# Kelving equation

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The Kelvin equation was published by Lord Kelvin in 1871 and describes the vapor pressure over a curved surface (also called Kelvin pressure ). ${\ displaystyle p_ {k}}$

## calculation

The following applies:

${\ displaystyle R \ cdot T \ cdot \ ln {\ frac {p_ {k}} {p_ {v}}} = \ gamma \ cdot V_ {m} \ cdot \ left ({\ frac {1} {r_ { 1}}} + {\ frac {1} {r_ {2}}} \ right)}$

With

• ${\ displaystyle p_ {v}}$the (usual) vapor pressure over a non-curved surface (index : vapor = vapor)${\ displaystyle _ {v}}$
• ${\ displaystyle R}$the gas constant
• ${\ displaystyle T}$the temperature in K
• ${\ displaystyle \ gamma}$the interfacial tension
• ${\ displaystyle V_ {m}}$the molar volume of the fluid
• ${\ displaystyle r}$the radii of curvature of the surface, on two mutually perpendicular trajectories

### drops

The best-known special case of the Kelving equation describes a drop with a radius : ${\ displaystyle r_ {1} = r_ {2} = r> 0}$

${\ displaystyle R \ cdot T \ cdot \ ln {\ frac {p_ {k}} {p_ {v}}} = {\ frac {2 \ cdot \ gamma \ cdot V_ {m}} {r}}> 0 }$

Due to the interfacial tension of a liquid , its vapor pressure increases with decreasing radius. From this follows one of the most important consequences of this equation: large drops have a smaller Kelvin pressure than smaller ones. Therefore, in a mixture of drops of different sizes, the large drops grow while the smaller ones disappear ( Ostwald ripening ). The molecules in the areas of higher pressure migrate into the areas of lower pressure. This explains why over-saturated steam in the liquid phase is condensed .

The same equation is also valid for spherical bubbles in liquids. Examples are carbon dioxide bubbles in mineral water bottles, steam bubbles when boiling water or the evaporation of precursor molecules in bubblers in CVD and CVS .

### Cylindrical pore

Another special case of the Kelving equation applies to a cylindrical pore with a radius and wetted with liquid ${\ displaystyle r_ {1} = r <0 \ left (\ mathrm {and} \; r_ {2} \ rightarrow \ infty \ Rightarrow {\ frac {1} {r_ {2}}} \ rightarrow 0 \ right) :}$

${\ displaystyle R \ cdot T \ cdot \ ln {\ frac {p_ {k}} {p_ {v}}} = {\ frac {\ gamma \ cdot V_ {m}} {r}} <0 \ quad \ Rightarrow \ ln {\ frac {p_ {k}} {p_ {v}}} <0 \ Leftrightarrow p_ {k}

## Curvature effect

The curvature effect occurs when liquid particles form on condensation nuclei . It is shown that over the curved surfaces of the resulting liquid drops of a higher saturated vapor pressure prevails than on a flat surface. This is due to the higher surface tension of the drops (see Young-Laplace equation ) against which work has to be done.

The exact relationship is reflected in the Kelvin equation named after Lord Kelvin (also Thomson equation):

${\ displaystyle \ ln {\ frac {p} {p_ {0}}} = {\ frac {2 \ cdot \ gamma} {r \ cdot \ rho \ cdot R _ {\ mathrm {S}} \ cdot T}} = {\ frac {r _ {\ mathrm {krit, e-fach}}} {r}}}$

The individual symbols stand for the following quantities:

• p - saturation vapor pressure above the drop (measurement)
• p 0 - saturation vapor pressure over a flat liquid surface ( Magnus formula )
• R S - mass-specific gas constant (e.g. for water vapor = 461.6 J / (kgK))
• ${\ displaystyle \ gamma}$- surface tension (depending on temperature)
• ρ - density of the liquid
• r - drop radius
• r crit, e-fold - critical radius at a saturation ratio , with Euler's number e ≈ 2.7182; :${\ displaystyle {{\ frac {p} {p_ {0}}} = e}}$${\ displaystyle {r _ {\ mathrm {krit, e-fold}} = {\ frac {2 \ cdot \ gamma} {\ rho \ cdot R _ {\ mathrm {S}} \ cdot T}}}}$

The smaller the drops are, the easier it is for them to evaporate again. Above all, when droplets are formed in the atmosphere through nucleation or condensation , an increase in vapor pressure or supersaturation is necessary in order to keep the droplets in equilibrium when the droplet radius is reduced.

The other way around, the vapor pressure behaves with concave surfaces, e.g. B. in capillaries . Here the vapor pressure is reduced as the capillary diameter decreases :

${\ displaystyle r <0 \ Rightarrow \ ln {\ frac {p} {p_ {0}}} <0 \ Leftrightarrow p

This explains, for example, the hygroscopic effect of fine-pored, wettable materials.

## Critical radius

critical radius and number of molecules of
pure water droplets
at 0 ° C (75.65 mN / m, 18.0153 g / mol, 0.99984 g / cm 3 )
Saturation ratio ${\ displaystyle S = {\ frac {p} {p_ {0}}}}$ critical radius in nm Molecular number N
1 ∞ (flat surface)
1.01 120 2.46 · 10 8
1.1 12.6 2.80 · 10 5
1.5 2.96 3.63 · 10 3
2 1.73 727
e ≈ 2.71828 ... 1.20 242
3 1.09 183
4th 0.87 91
5 0.75 58
10 0.52 20th

A critical radius follows from the Kelving equation for a given temperature and a given saturation ratio . If the radius of a liquid droplet is smaller than the critical radius, it will evaporate, if it is larger, the droplet will grow. Only particles with the critical radius are in equilibrium with the given supersaturated vapor phase and do not change. In the adjacent table the critical radii and number of molecules of pure water droplets are calculated. For the spontaneous formation of a droplet from only twenty molecules, the air would have to be e.g. B. 1000% oversaturated, which rarely occurs in nature. This proves the need for condensation nuclei . ${\ displaystyle {S = {\ frac {p} {p_ {0}}}}}$${\ displaystyle {r _ {\ mathrm {krit}}}}$

The range in which the saturation ratio is greater than one, but not (with a practical probability) the critical radius is spontaneously reached, determines the Ostwald-Miers range .

## Derivation

The Kelvin equation can be derived from the equations of state of equilibrium thermodynamics using various approximations ; in particular, the liquid phase is treated as an incompressible liquid and the gaseous phase as an ideal gas . It is also assumed that the difference in pressure inside and outside the drop, determined by surface tension and curvature, is much greater than the difference between Kelvin pressure and vapor pressure ( ). Detailed derivations can be found in ${\ displaystyle 2 \ gamma / r \ gg p_ {K} -p_ {v}}$

## literature

• Walter J. Moore: Fundamentals of physical chemistry. de Gruyter, Berlin 1990, ISBN 3-11-009941-1 , p. 459 f.
• E. Zmarsly, W. Kuttler, H. Pethe: Meteorological-climatological basic knowledge. An introduction with exercises, tasks and solutions. Ulmer Verlag, Stuttgart 2002, ISBN 3-8252-2281-0 .
• SJ Gregg, KSW Sing: Adsorption, Surface Area and Porosity. 2nd Edition. Academic Press, New York 1982, p. 121
• Arthur W. Adamson, Alice P. Guest: Physical Chemistry of Surfaces. 6th edition. Wiley, New York 1997, ISBN 0-471-14873-3 , p. 54.

## Individual evidence

1. (Sir) William Thomson (Baron Kelvin): LX. On the Equilibrium of Vapor at a Curved Surface of Liquid . In: (London, Edinburgh and Dublin) Philosophical Magazine (and Journal of Science) Series 4 . tape 42 , no. 282 , December 1871, p. 448–452 (English, online on the pages of the Thuringian University and State Library Jena ).
2. a b J. G. Powles: On the validity of the Kelvin equation . In: Journal of Physics A: Mathematical and General . tape 18 , 1985, pp. 1551–1553 , doi : 10.1088 / 0305-4470 / 18/9/034 .
3. Walter J. Moore: Fundamentals of physical chemistry. de Gruyter, Berlin 1990, ISBN 3-11-009941-1 , p. 459 f.
4. KP Galvin: A conceptually simple derivation of the Kelvin equation . In: Chemical Engineering Science . tape 60 , 2005, pp. 4659-4660 , doi : 10.1016 / j.ces.2005.03.030 .