Laplace-Kelvin equation

Upon capillary condensation of water in PEMs, the relative humidity, f /P, determines capillary pressure, P , and capillary radius, via the Kelvin-Laplace equation ... 

According to the thermodynamics of capillary condensation, the Kelvin-Laplace equation determines the free energy of... 

Another conceptual difficulty is the question of adequacy of macroscale thermodynamic parameters in nanometer size pores, where the molecular-scale effects prevail. Much work has been done to check the limits of the validity of the Kelvin-Laplace equation [32]. Predictions of this equation have been verified in experiments and MDs simulations in pores with capillary diameters as small as 2 nm [89, 90]. 

Using Equation 2.26 for the capillary pressure and employing the Kelvin-Laplace equation... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

What is the Laplace equation What is the Kelvin equation What are the differences between the two ... 

Laplace equation A thermodynamic derivation Determining surface tension from the Kelvin equation Heat of immersion from surface tension and contact angle Surface tension and the height of a meniscus at a wall Interfacial tensions from the Girifalco-Good-Fowkes equation... 

The beginning of the study of surface effects is old. Equation (12.48) was first reported in 1806 by P. S. Laplace in his celebrated Mecanique celeste, while W. Thomson (Lord Kelvin) reported equation (12.49) in Proc. Roy. Soc. (London), 9, 255 (1858). 

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

The Young-Laplace equation (3.4/3.5) shows that, pA>pB, the pressure inside a bubble or drop exceeds that outside. For a sphere, Ap=pA - Pb = 2y/R, so that Ap varies with the radius, R. Thus the vapour pressure of a drop should be higher, the smaller the drop. This is shown by a related equation, the Kelvin equation [13,26], which is described here. 

The above equation is intuitively understandable because it is the basis for the Kelvin equation If the pressure difference is equated with the Young-Laplace equation it yields the Kelvin one. It should be noted here that Eq.(5) does not suffer from the incorrectness of the Kelvin equation for nanopores because it does not include any pore-size related factor. 

Unlike the Laplace equation, which can only be applied to fluid surfaces, the Kelvin equation is valid for any phase boundary, also solid-gas, solid-liquid, and even solid-solid. As for the Laplace equation, Eq. (10.9) can also be modified to accommodate nonspherical curved surfaces then 2/(l/i + l/i 2) should be used instead of r. The values for the solubility should give the thermodynamic activity of the substance involved, for the equation to be generally valid for ideal or ideally dilute solutions, concentrations can be used. Most gases in water show ideal behavior. It should further be noted that Eq. (10.9) applies for one component if a particle contains several components, it should be applied to each of them separately. Finally, there are some conditions that may interfere with the... 

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 aim of this book is to show that there is a great deal of science in ice cream, and in particular to demonstrate the link between the microscopic structure and the macroscopic properties. It is naturally biased towards physics, physical chemistry and materials science as these are the areas in which I trained. The book is aimed at schools and universities, and a scientific background is required to understand the more technical sections. I have attempted to make it readable by 16-18-year-olds, and many sections are suitable for adaptation by GCSE science teachers. I have unashamedly made reference to the giants of chemistry and physics such as Newton, Einstein, Boyle, Gibbs, Kelvin, Laplace and Young where the laws and equations that bear their names are relevant. I hope that as a result teachers reading this book will find in ice cream useful illustrations of a number of scientific principles. Some... 

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 most important application of the Young-Laplace equation is possibly the derivation of the Kelvin equation. The Kelvin equation gives the vapour pressure of a curved surface (droplet, bubble), P, compared to that of a flat surface, P °. The vapour pressure (P) is higher than that of a flat surface for droplets but lower above a liquid in a capillary. The Kelvin equation is discussed next. 

The Kelvin equation is derived from the Young-Laplace equation and the principles of phase equilibria. It gives the vapour pressure, P, of a droplet (curved surface) over the ordinary vapour pressure (P ) for a flat surface (see Appendix 4.2 for the derivation) ... 

Young-Laplace equation (pressure Applied in the derivation of the Kelvin... 

We provide here a derivation of the Kelvin equation based on the Young-Laplace equation and phase equilibrium principles. 

When, however, due consideration is given to the full space dependence, the transverse Kelvin-Laplace wave motion is described by the following set of nonlinear partial differential equations... 

After contact (Figure 5.12d) upon retraction the liquid will form a bridging meniscus (Figure 5.12e). The meniscus breaks at a distance where the condition set by the Kelvin equation cannot be fulfilled anymore [565, 567] (Figure 5.12f). Even for constant volume, the meniscus becomes unstable at a certain distance [527] because no solutions of the Laplace equation exist anymore [515, 568, 569). The meniscus also breaks if the retraction velocity is so high and the liquid is so viscous that it becomes kinetically unstable [520, 570]. For layers of nonvolatile liquids, for example, for lubrication layers on magnetic hard disks, capillary condensation and evaporation are not an option. The liquid in the bridging meniscus has to be provided by direct flow from the adsorbed layer. 

With the preceding introduction to the handling of surface excess quantities, we now proceed to the derivation of the third fundamental equation of surface chemistry (the Laplace and Kelvin equations, Eqs. II-7 and III-18, are the other two), known as the Gibbs equation. 

The consequence of Laplace pressure is very important in many different processes. One example is that, when a small drop comes into contact with a large drop, the former will merge into the latter. Another aspect is that vapor pressure over a curved liquid surface, pcur, will be larger than on a flat surface, pf,at. A relation between pressure over curved and flat liquid surfaces was derived (Kelvin equation) ... 

The cause for this change in vapor pressure is the Laplace pressure. The raised Laplace pressure in a drop causes the molecules to evaporate more easily. In the liquid, which surrounds a bubble, the pressure with respect to the inner part of the bubble is reduced. This makes it more difficult for molecules to evaporate. Quantitatively the change of vapor pressure for curved liquid surfaces is described by the Kelvin equation ... 

A. u = Tc( + ) M 111P. Laplace-Kelvin equation. Difference in fluid pressure A.11 across two-fluid interface. Related to surface tension Tc and the curvature radii r and r2... 

The surface stress of some solids in a liquid might be determined by measuring solubility changes of small particles [97,98]. As small liquid drops have an increased vapor pressure in gas, small crystals show a higher solubility than larger ones. The reason is that, due to the curvature of the particles surface, the Laplace pressure increases the chemical potential of the molecules inside the particle. This is described by the Kelvin equation, which can be written (Ref. 3, p. 380)... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

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. 

---