# The kelvin equation

The Kelvin equation relates the vapour pressure of a liquid to the effect of surface curvature i.e. a meniscus with radius. The equation illustrates that due to the surface tension and the surface being larger than the volume, there is much higher evaporation rate for smaller droplet liquids when compared to that of larger droplets. (Gordon 1988)

ln(P/P0 ) = 2γVm /rRT

(Hiemenz, CP 1997)

The Kelvin equation also shows that smaller particles exhibits higher solubility than larger particles. This occurs because of much greater pressure inside the droplet than outside and therefore gas diffusion can occur across the plateau borders of the droplet, weakening the walls (from high to low pressure). (Farlex 2010)

### KELVIN EQUATION PARAMETERS

P: Actual vapour pressure

P0: Saturated vapour pressure

γ: Surface tension of liquid

Mv: Molar volume

ρ: Density

R: Universal gas constant (8.314 J K-1 mol-1)

T: Temperature in Kelvin

The Kelvin equation has a variety of significant effectsas it provides clarifications for such phenomena including supersaturation of vapours, superheating, Ostwald ripening and capillary condensation. This effect is known as Kelvin effect and also Gibbs-Kelvin effect or Gibbs-Thomson effect. (Barnes 2005)

### Supersaturation

The effect on the Kelvin equation could be explained in terms of supersaturation of vapour. The term of supersaturation refers to the state in which a solution has overly dissolved material than that of a saturated solution. (Economic expert 2010) Supersaturation could also be referred to the compound's vapour having greater partial pressure than that of the vapour pressure of the compound. (NationMaster encyclopedia 2005)

In a polydisperse suspension the material will dissolve until it has reached its saturated level at that temperature. When the sample is stored, the temperature will fluctuate slightly regardless of storage location. For example, if the sample is stored in the fridge, the temperature could vary and so the temperature of suspension could increase leading to a rise in saturation level allowing smaller particles to further dissolve until a new saturation level is reached. When the temperature drops, the suspension is affected in terms of consisting more than usual of the dissolved species. The new concentration is described as supersaturation. The remaining dissolved species would precipitate back on all the particles, the smaller particles preferentially dissolve and so larger particles having a coat will enlarge. Hence, supersaturated solutions are a result because of alteration of the saturated system, i.e. temperature, volume or pressure.

An example of supersaturation of vapour is in the cloud formation. In forming clouds, the rise of warm humid air allows water from a surface of a liquid i.e. ocean to evaporate and rise into the atmosphere. As it rises, the atmosphere becomes cooler and so the water vapour would condense. Small group of dispersed molecules are gathered to form water drops. As because of the small droplets have a high vapour pressure, they tend to evaporate and disappear. Hence, once the droplets are formed from condensation of water vapour, they would evaporate again. One aspect of the Kelvin equation is that cloud formation is not seen because clouds are made of water droplets having a small radius which consequently disappear. Air is said to be supersaturated with water as the quantity of tiny condensed water droplets or ice crystals are suspended in the atmosphere. (Perlman 2010)

### Rock candy

Rock candy is a type of confectionery mineral composed of relatively large sugar crystals. The candy is formed by allowing a supersaturated solution of sugar and water to crystallize onto a surface suitable for crystal nucleation, such as a string or stick...

### Superheating

In the phenomenon of superheating, the homogeneous substance without boiling is heated at a temperature in which is greater than the liquids normal boiling point. In order for a vapour bubble to expand and consequently burst at the surface (to boil), the temperature heating must be highly sufficient (raised above the boiling point) that the vapour pressure will go beyond the atmospheric pressure. (Patrick 2006) An insufficient temperature that is below will cause the vapour bubble to decrease and so disappear. (The Engineering Tool Box 2005)

However, in terms of superheating, the liquid is seen that it does not boil despite having the vapour pressure exceeding the atmospheric pressure. As a result, the surface tension restrains the growth of the bubbles. In superheating, spontaneous nucleation of liquid vapours at an instant occurs creating a large bubble at the bottom of the beaker with high internal pressure and forces the liquid above it to expel. Boiling chips which are also known as boiling stones or anti-bumping granules are purposely added to the liquid to ensure boiling is easier without the liquid becoming superheated. They are small and insoluble porous stones composing of calcium carbonate or silicon carbide. Inside the stones, they have pores that allow cavities to trap air and supply spaces in which the solvent bubbles can develop (act as nucleation sites i.e. for the formation of the vapour bubbles in on cavities). (UBC Chemistry department 2002)

When there is an addition of boiling chips to a solvent and is heated, it would release very small bubbles. These tiny bubbles prevent bumping, loss of solution and hence allows liquid to boil easier without being superheated. Without the addition of boiling chips in a solvent, the heated liquid could be defined as superheated, an unstable state and therefore would be able to release a quantity of bubble of gas all of a sudden or perhaps violently. As a consequence of the sudden expel of vapour from the container of the liquid, it could lead to from ruining of the experiment to causing harsh skin burns. (RDG 2005)

### Ostwald ripening

In regards to droplets that are spherical and being in contact with their vapour, the two radii situated in the liquid phase are equal and so maintain positive signs. Therefore in reference to the Kelvin equation, the droplet's vapour pressure will be higher than the same liquid with a flat surface. An example is that for a radius of water droplet of 10nm, the increase in vapour pressure is approximately 10%. The smaller the droplets radius, the greater the vapour pressure. (Barnes 2005)

For instance, in the case a polydispersion system (droplets with different sizes), the smaller particles has the tendency to dissolve (i.e. have greater solubility) or evaporate attaching themselves onto the surface of the larger ones, leading the larger particles to continue to grow with time but the smaller particles disappearing. This spontaneous process known as Ostwald ripening was first recognised in 1896 by Wilhelm Ostwald. (Chemie.De 2010)

A real life effect of the Kelvin equation occurs in clouds in which the larger droplets grow expecting to be heavy enough to fall as rain. Similarly, this process also exists for crystals contained in solutions. The larger crystals will tend to grow rather than the smaller ones resulting in a process of Ostwald ripening. This process is an important ageing effect occurring in any polydisperse systems. It could be observed in foams, emulsions as well as aqueous sols. (Adamson 1990)

In terms of foam instability, the reason of instability could be due to Ostwald ripening. The Ostwald ripening's driving force arises because of the variation of Laplace pressure between bubbles of unequal size that is the smaller and larger foam bubbles. (Barnes 2005)

The smaller bubbles having a greater gas pressure than the larger bubbles causes an effect on the foam stability. For the period of foam storage, the bubbles rise in the gravity field to the top of the liquid due to the large density existing between the liquid and the gas phases. From here, they shape a loosely closed-packed or honeycomb like structure from which the liquid drains forming foam that turns out to be dryer. The thin films created can collapse due to gravity and causing coalescence of bubbles. When increased of gas solubility with pressure happens, gas between bubbles are able to diffuse from the smaller to the larger bubbles (lower internal pressure). A large bubble in the foam that is surrounded by smaller bubbles could ultimately cause a big hole in the structure of the foam. (Hubbard2002)

Ostwald ripening is the cause of foam instability when a larger pressure difference of foam bubbles that are spherical cam be seen. In the case of polyhedral foam (with planar liquid lamella), the foam bubbles pressure difference is not great and so foam instability is not because of the Ostwald process. (Tadros 2005)

Since the surface of the bubble behaves like viscoelastic behaviour, the process of Ostwald ripening in foam bubbles could be reduced by addition of a gas that is only soluble in the liquid. As a result of this process, it also has an effect of decreasing of specific surface area in time due to the highly soluble particles. (Lyklema 2005) The Ostwald ripening is important as it is used to increase the efficiency in both industrial and analytical filtration. (Barnes 2005)

### Capillary Condensation

The Kelvin effect can also be observed in terms of capillary condensation due to the existence of a curvature, a meniscus. Capillary condensation arises because of the vapour's adsorption in a capillary which creates a liquid surface with tiny radius of curvature. (Fisher 2002)

The liquids vapour pressure being lower than that of the same liquid with a flat surface is a consequence of the radii being positioned in the vapour phase. If the actual vapour pressure is greater than the vapour pressure calculated from using the Kelvin equation for the curved surface than condensation would likely to occur. The actual vapour pressure for a flat surface may possibly be lower than its saturation vapour pressure. (Patrick 2006)

As the capillary is filling by capillary condensation, immediately a meniscus is generated (which depends on the liquids surface tension) at the liquid-vapour interface enabling the equilibrium to be under the Psat (saturation vapour pressure). At low vapour pressure, the liquid condenses in smaller radius of the capillary. As the vapour pressure increases, further condensation of the liquid at the larger radius capillary is filled causing the radius of the meniscus to increase (capillary becoming full). The Kelvin equation implies that the Pv/Psat rises inside a capillary and the meniscus radius will further expand causing a creation of more flat surface. Hence, the vapour condensation is able to occur smoothly over the range of vapour pressures. For any liquid-vapour interface involving a meniscus, the Kelvin equation therefore relates the change between the equilibrium vapour pressure and the saturation vapour pressure.

Capillary condensation as an important factor is significant by means for occurring porous structures that is both synthetic and naturally. These structures allow scientists to use capillary condensation for determination of the distribution of pore sizes as well as surface area through adsorption isotherms. (Barnes, 2005)

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