Droplets Kelvin equation

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

While Eq. III-18 has been verified for small droplets, attempts to do so for liquids in capillaries (where Rm is negative and there should be a pressure reduction) have led to startling discrepancies. Potential problems include the presence of impurities leached from the capillary walls and allowance for the film of adsorbed vapor that should be present (see Chapter X). There is room for another real effect arising from structural peiturbations in the liquid induced by the vicinity of the solid capillary wall (see Chapter VI). Fisher and Israelachvili [19] review much of the literature on the verification of the Kelvin equation and report confirmatory measurements for liquid bridges between crossed mica cylinders. The situation is similar to that of the meniscus in a capillary since Rm is negative some of their results are shown in Fig. III-3. Studies in capillaries have been reviewed by Melrose [20] who concludes that the Kelvin equation is obeyed for radii at least down to 1 fim. 

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

This Kelvin equation says that the vapor pressure over a droplet depends exponentially on the inverse of the droplet radius. Thus, as the radius decreases, the vapor pressure over the droplet increases compared to that over the bulk liquid. This equation also holds for water coating an insoluble sphere (Twomey, 1977). 

The vapor pressure over a pure liquid droplet at equilibrium ps depends on its radius of curvature. The Kelvin equation gives this relationship as ... 

The two extremes of insoluble nuclei are nuclei which are easily wetted and those which are not. Nuclei which are easily wetted rapidly take on the appearance of a droplet and subsequently behave as one. To predict droplet growth or evaporation, these particles with easily wettable surfaces can be considered to be pure drop nuclei, and the Kelvin equation can be used directly (but with a lower limit on nucleus size). 

Langmuir s equation indicates that a droplet will not evaporate when S = pip, a 1.0. But according to the Kelvin equation, curvature effects will cause small droplets to evaporate, even when S exceeds 1.0. How can this apparent contradiction be resolved One way is to replace the p, - pt term in Eq. 15.14 with an equivalent term from Kelvin s equation which takes curvature into account. Recalling Kelvin s equation... 

Jjet us consider as an example the case of a saturated vapor which has been suddenly and adiabatically compressed to a vapor pressure P which is in excess of its equilibrium vapor pressure Po at the final temperature T. In order for liquid to form, it must grow by the growth of small droplets. If, however, we consider a very small droplet of the liquid phase present in the vapor, it will have an excess free energy, compared to bulk liquid, that is due to its extra surface. The magnitude of the excess surface energy is 4irrV, where surface tension and r is the radius of the drop. In order for the drop and vapor to be in equilibrium, the vapor pressure P must exceed the saturation vapor pressure Po by an amount which can be calculated from the Gibbs-Kelvin equation... 

The origin of the metastability of sujjersaturated vapors can be understood in an elementary way using ideas from continuum thermodynamics. It is clear that small aggregates of vapor molecules are intermediates in the transformation from gas-phase monomers to condensate. As a first approximation we can view these small cluster intermediates as microscopic continuum droplets, which by virtue of the Kelvin equation are known to have elevated vapor pressures. This elevation in vapor pressure is given by... 

Comparison of the molecular dynamics calculations with the predictions of classical thermodynamics indicates that the Laplace formula is accurate for droplet diameters of 20 tT j (about 3400 molecules) or larger and predicts a Ap value within 3% of the molecular dynamic.s calculations for droplet diameters of 15 o-y (about 1400 molecules). Interestingly, vapor pres-sures calculated from the molecular dynamics simulations suggested that the Kelvin equation is not consistent with the Laplace formula for small droplets. Possible explanations are the additional assumptions on which the Kelvin relation is ba.sed including ideal vapor, incompressible liquid, and bulk-like liquid phase in the droplet. 

CrystalHsation, which is by far the most serious instability problem with suspoemulsions (particularly with many agrochemicals), arises from the partial solubihty of the suspension particles into the oil droplets. The process is accelerated at higher temperatures and also on temperature cycling. The smaller particles will have a higher solubility than their larger counterparts, due to the fact that the higher the curvature the higher the solubihty, as described by the Kelvin equation [11],... 

Kelvin Equation An expression for the vapor pressure of a droplet of liquid, RT In(p/pQ) = 2 iVfr where R is the gas constant, T is the absolute temperature, p is the vapor pressure of the liquid in bulk, pQ is that of the droplet, 7 is the surface tension, V is the molar volume, and r is the radius of the liquid. See also Young—Laplace Equation. 

The Kelvin equation tells us that the vapor pressure over a curved interface always exceeds that of the same substance over a flat surface. A rough physical interpretation of this so-called Kelvin effect is depicted in Figure 10.11. The vapor pressure of a liquid is determined by the energy necessary to separate a molecule from the attractive forces exerted by its neighbors and bring it to the gas phase. When a curved interface exists, as in a small droplet, there are fewer molecules immediately adjacent to a molecule on the surface than when the surface is flat. Consequently, it is easier for the molecules on the surface of a small drop to escape into the vapor phase and the vapor pressure over a curved interface is always greater than that over a plane surface. 

In Chapter 10 we showed that the vapor pressure over a curved interface always exceeds that of the same substance over a flat surface. The dependence of the water vapor pressure on the droplet diameter is given by the Kelvin equation (10.86) as... 

The difference in thermodynamic equilibrium between crystals of different size is defined by the Gibbs-Thomson or Ostwald-Freundlich equation (Mullin 1993). The original Kelvin equation was written to describe droplets of liquid in a vapor in terms of vapor pressures around droplets of different size. However, in crystallization studies, the vapor pressure may be related, through the solution activity, to the equilibrium concentration in solution for crystals of different size, faking the assumption that concentration can be used directly fpr vapor pressure gives... 

Because the free energy of formation of a surface is always positive, a particle that consists only of surfaces (that is, platelets or droplets of atomic dimensions) would be thermodynamically unstable. This is also apparent from the Kelvin equation [Eq. 3.70], which states that a particle that falls below a certain size will have an increased vapor pressure and will therefore evaporate. There must be a stabilizing influence, however, that allows small particles of atomic dimensions to form and grow a common occurrence in nature. This influence is given by the free energy of formation of the bulk condensed phase. In this process, n moles of vapor are transferred to the liquid phase under isothermal conditions. This work of isothermal compression is given by... 

The steeply rising portion of the Kohler curves represents a region where solute effects dominate. As the droplet diameter increases, the relative importance of the Kelvin effect over the solute effect increases, and finally beyond the critical diameter the domination of the Kelvin effect is evident. In this range all Kohler curves approach the Kelvin equation, represented by the equilibrium of a pure water droplet. Physically, the solute concentration is so small in this range (recall that each Kohler curve refers to fixed solute amount) that the droplet becomes similar to pure water. 

As the size of a particle decreases, the contribution of the surface to the total flee energy becomes increasingly significant. This trend is evident in the Kelvin equation, a classic thermodynamic equation relating the vapor pressure of a droplet to its physical size. This equation is often applied in the determinalion of the pore size distribution through adsorption porosimetry. For this appHcalion, the equation can be written ... 

Types IVand V Capillary condensation in mesopores. Initially, a multilayer is adsorbed on the capilla walls. When the pressure is further increased, liquids droplets form preferentially at sites where the curvature fulfills the Kelvin equation. When two opposing droplets touch, the pore is filled. On desorption, pores whose radius is smaller than the Kelvin radius are emptied. The adsorption branch indicates the extent ofthe pores, and the desorption branch the size ofthe pore openings [Evertt 1976]. 

Capillary condensation is the reverse effect from the pressure increase of small droplets. As shown by Kelvin from thermodynamic arguments, small droplets with a strongly curved surface have a larger equilibrium vapor pressure than planar surfaces. On the other hand, the Kelvin equation (12.30) predicts that the equilibrium vapor pressure of a surface with a negative radius, this is a hole, is still lower than the equilibrium vapor pressure of a planar surface. This is the reason for capillary condensation. The vapor pressure of a spherical droplet is... 

We emphasize still another example from physical chemistry. In the derivation of the Kelvin equation, i.e., the vapor pressure of small droplets, we immediately and naturally accept that the shape of the droplets is spherical. Whoever spills mercury knows this. It is obviously not at all necessary to justify spherical droplets. Spherical droplets are in our common sense. But an extremal principle is behind the spherical droplets. A body with a given volume seeks to reach a minimum boundary surface, if no disturbance is aside. Particles that are bigger than a certain size exhibit an obvious deviation from the spherical shape under the effect of gravity. 

We can now transform the Kelvin equation (4.223), valid for pure water, by consideration of Raoulfs equation for a droplet with dissolved matter. The total change in vapor pressure consists of the vapor change above the curved surface and the vapor change of the solution ... 

We consider a suspension of spherical droplets. The droplets are at rest relative to the ambient vapor/gas mixture. The droplet concentration is sufficiently small to rule out droplet interactions. According to the droplet concentration a spherical volume is assigned to every droplet as vapor reservoir. The temperature T and the partial pressures pv (vapor) and pg (gas) at the outer edge of the volume represent the thermodynamic state. The droplet temperature Td is taken to be constant throughout the droplet however may vary during the growth process. The partial pressure of the vapor at the surface pd is set equal to the equilibrium vapor pressure over a spherical surface as given by the Kelvin equation... 

There is a remarkable consequence of Eq. (3.47). If there are no droplets in the vapor phase, condensation would not be possible, as a droplet with d = 0 ends up with an infinite saturation pressure, resulting in immediate evaporation. Only due to statistical fluctuations or seed particles (dust, microorganisms) [44, 45] droplet formation can happen at all. Once a droplet has been formed, a small growth in size makes the saturation pressure fall rapidly, leading to an extremely fast growth of the droplet. For evaporation, the situation is vice versa, so that a small decrease of the droplet size leads to rapid evaporation. Thus, the Kelvin equation describes an unstable equilibrium, where small disturbances lead to disappearance or growth of the droplets. 

The Kelvin equation describes the dependence of the vapor pressure from the droplet size. Its derivation is based on the equilibrium of forces on a droplet, which is visualized in Figure C.l. 

The same result can be obtained by nebuUzation of an analyte solution into small droplets that are then flash heated by exposure to hot gas, as in thermospray and APCI sources. Thus, it is conamon to find that a compound that will not 5deld a meaningful mass spectrum using conventional Cl (solid sample heated in an insertion probe) will do so when subjected to flash heating in APCI. Anotber physical phenomenon that contributes to this effect is the increase of vapor pressure of an analyte as the sample size becomes very small, as in a nebulized solution. This increase is described by the Kelvin equation (Thomson 1871, Moore 1972) ... 

Equation (3.6) is called Young-Laplace equation, in which R is the harmonic mean of the principal radii of curvature. The capillary pressure promotes the release of atoms or molecules from the particle surface. This leads to a decrease of the equilibrium vapour pressure with increasing droplet size Kelvin equation) ... 

The Kelvin equation describes the increase in vapor pressure as a function of the increasing curvature of liquid droplets in a gas phase. This is equivalent for the dissolution pressure of solid particles in a liquid phase it increases with decreasing particle size below approx. 1 The parameters necessary for calculating the vapor pressure are more easily accessible in case of liquids. Therefore a model calculation was performed for liquids with high, medium and very low vapor pressure (ether, water and oleic... 

---