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Laplace,

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. [Pg.8]

Returning to equilibrium shapes, these have been determined both experimentally and by solution of the Young-Laplace equation for a variety of situations. Examples... [Pg.9]

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... [Pg.10]

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... [Pg.12]

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. [Pg.13]

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... [Pg.53]

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... [Pg.54]

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. [Pg.73]

Consider the situation illustrated in Fig. VI-9, in which two air bubbles, formed in a liquid, are pressed against each other so that a liquid film is present between them. Relate the disjoining pressure of the film to the Laplace pressure P in the air bubbles. [Pg.251]

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. [Pg.257]

If the solid in question is available only as a finely divided powder, it may be compressed into a porous plug so that the capillary pressure required to pass a nonwetting liquid can be measured [117]. If the porous plug can be regarded as a bundle of capillaries of average radius r, then from the Laplace equation (II-7) it follows that... [Pg.364]

A second ideal model for adhesion is that of a liquid wetting two plates, forming a circular meniscus, as illustrated in Fig. XII-13. Here a Laplace pressure P = 2yz.A (h ws the plates together and, for a given volume of liquid. [Pg.454]

For constant 6 and y (the contact angle was found not to be very dependent on pressure), one obtains from the Laplace equation. [Pg.578]

The solution was first obtained independently by Wertheim [32] and Thiele [33] using Laplace transfonns. Subsequently, Baxter [34] obtained the same solutions by a Wiener-Hopf factorization teclmique. This method has been generalized to charged hard spheres. [Pg.481]

The MS approximation for the RPM, i.e. charged hard spheres of the same size in a conthuium dielectric, was solved by Waisman and Lebowitz [46] using Laplace transfomis. The solutions can also be obtained [47] by an extension of Baxter s method to solve the PY approximation for hard spheres and sticky hard spheres. The method can be fiirtlier extended to solve the MS approximation for unsynnnetrical electrolytes (with hard cores of unequal size) and weak electrolytes, in which chemical bonding is municked by a delta fiinction interaction. We discuss the solution to the MS approximation for the syimnetrically charged RPM electrolyte. [Pg.492]

Surface properties enter tlirough the Yoimg-Laplace equation of state for the surface pressure ... [Pg.726]

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... [Pg.746]

Equation (A3.3.73) is referred to as the Gibbs-Thomson boundary condition, equation (A3.3.74) detemiines p on the interfaces in temis of the curvature, and between the interfaces p satisfies Laplace s equation, equation (A3.3.71). Now, since ] = -Vp, an mterface moves due to the imbalance between the current flowing into and out of it. The interface velocity is therefore given by... [Pg.748]

The solution of Laplace s equation, (A3.3.71), with these boundary conditions is, for [Pg.749]

Micelles can solubilize gases. It has been demonstrated [49] that the Laplace model gives a good description of such solubilization for the case of ionic micelles ... [Pg.2592]

The Laplace equation, which defines tire pressure difference, AP, across a curved surface of radius, r. [Pg.2761]

T. Schlick. Pursuing Laplace s vision on modern computers. In J. P. Mesirov, K. Schulten, and D. W. Sumners, editors. Mathematical Applications to Biomolecular Structure and Dynamics, volume 82 of IMA Volumes in Mathematics and Its Applications, pages 219-247, New York, New York, 1996. Springer-Verlag. [Pg.260]


See other pages where Laplace, is mentioned: [Pg.136]    [Pg.142]    [Pg.546]    [Pg.6]    [Pg.6]    [Pg.43]    [Pg.53]    [Pg.55]    [Pg.60]    [Pg.122]    [Pg.171]    [Pg.363]    [Pg.578]    [Pg.724]    [Pg.747]    [Pg.748]    [Pg.852]    [Pg.890]    [Pg.1943]    [Pg.2592]    [Pg.144]    [Pg.144]   
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An Often-Used Device for Solving Electrochemical Diffusion Problems The Laplace Transformation

Application of FFT Fast Laplace Transform (FLT)

Application of Laplace Transform

Application of Laplace Transforms to Pharmacokinetics

Application to large amplitude methods analysis in the Laplace domain

Applying the Young-Laplace equation

Atomic Orbital Laplace-transformed MP2 Theory for Periodic Systems

Autocorrelation function Laplace transform

BEM Numerical Implementation of the 2D Laplace Equation

Bilateral Laplace transform

Boundary conditions Young-Laplace equation

Calculus Laplace transformation

Capillarity and the Young-Laplace Equation

Capillary Forces Laplace Equation (Liquid Curvature and Pressure) (Mechanical Definition)

Capillary pressure, Laplace

Complex Variables and Laplace Transforms

Conduction heat transfer Laplace equation

Continuum Poisson-Laplace equation

Correlation function Laplace inversion

Correlation functions Laplace transform

Criticisms of the inverse Laplace transform method

Curvature Young-Laplace equation

Curved Interfaces Laplaces Formula

Curved Liquid Surfaces Young-Laplace Equation

De Moivre-Laplace theorem

Derivation of the Laplace equation

Derivation of the Young-Laplace equation

Derivatives Laplace transform

Differential equations Laplace transform solution

Differential equations, Laplace transform

Differential equations, Laplace transform technique

Differential equations, solution with Laplace transforms

Diffusion Laplace transform

Diffusion equation Laplace transforms

Digital signals Laplace transform

Discrete Laplace transform

Dispersion Laplace transform

Droplets Young-Laplace equation

Electron Laplace

Electron-density distribution Laplace concentration

Emulsion Laplace pressure

Ensemble Laplace transformation

Equation Laplace

Equation Laplace transform

Equation Young-Laplace

Equation of Young and Laplace

Equation of Young-Laplace

Exponential function, Laplace transform

Fast Laplace Inversion

Fast Laplace Transform

Field correlation function Laplace transform

Final value theorem Laplace transforms

Forces Laplace

Fourier and Laplace Transformations

Fourier-Laplace transform

Fourier-Laplace transform, response

Fourier-Laplace transform, response function

Fourier-Laplace transforms analysis

Functions and a Solution of Laplaces Equation

Fundamental Solution of Poissons and Laplaces Equations

Fundamental equations Young-Laplace equation

G , Laplace transform

Gauss-Laplace equation

Generalized Laplace operator

Integral transforms Laplace transform

Integration, method Laplace transforms

Inverse Fourier-Laplace transformation

Inverse Laplace transform

Inverse Laplace transform analysis

Inverse Laplace transform techniques

Inverse Laplace transforms

Inverse of the Laplace Transformation

Inversion Laplace transforms

Inversion of Laplace Transforms by Contour Integration

Inversion of Laplace transforms

Inversion theorem, Laplace

Inverting the Laplace Transform

Kelvin-Laplace equation

LaPlace transformation equation

LaPlace transformation explained

LaPlace transformation schematized

LaPlace transforms of a constant

LaPlace transforms tabulated

LaPlace’s Law

Laplace - inverse transform function

Laplace Derivation of Dynamic Compensation

Laplace Equation (Liquid Curvature and Pressure)

Laplace Instruments

Laplace Pressure Drop

Laplace Transform Technique for Parabolic PDEs

Laplace Transform Technique for Partial Differential Equations (PDEs) in Finite Domains

Laplace Transformation and Scavenging

Laplace Transformations Building Blocks

Laplace Transforms for Processes

Laplace boundary condition

Laplace coefficients

Laplace complex variable

Laplace concentration

Laplace constant

Laplace current

Laplace distribution

Laplace domain

Laplace equation derivation

Laplace equation of capillarity

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Laplace equation, applied

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Laplace length scale

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Laplace object function

Laplace operator

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Laplace operator, Laplacian

Laplace parameter

Laplace potential

Laplace pressure

Laplace pressure difference

Laplace principle

Laplace resonance

Laplace s theory

Laplace space

Laplace summarized

Laplace tension

Laplace transform

Laplace transform INDEX

Laplace transform Kramers equation

Laplace transform Mittag-Leffler function

Laplace transform Subject

Laplace transform application

Laplace transform canonical partition function

Laplace transform case studies

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Laplace transform convolution property

Laplace transform convolution theorem

Laplace transform definition

Laplace transform dielectric relaxation

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Laplace transform electric fields

Laplace transform examples

Laplace transform factorization

Laplace transform final value theorem

Laplace transform friction

Laplace transform function

Laplace transform grand partition function

Laplace transform input signals

Laplace transform integral

Laplace transform inverse transforms

Laplace transform inversion

Laplace transform inversion, Tables

Laplace transform limit functions

Laplace transform lumped parameter equivalent with

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Laplace transform periodic potentials

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Laplace transform technique for partial

Laplace transform technique for partial differential equations

Laplace transform technique ordinary differential equations

Laplace transform technique partial differential equations

Laplace transform time-dependent friction

Laplace transform times

Laplace transform transfer functions

Laplace transform using partial fractions

Laplace transform wave function

Laplace transform, harmonic oscillators

Laplace transform, identifiability

Laplace transform, linear viscoelasticity

Laplace transform, linear viscoelasticity elastic-viscoelastic correspondence

Laplace transform, process transfer

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Laplace transformation

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Laplace transforms unit impulse

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Laplace waves

Laplace, Pierre

Laplace, Pierre Simon

Laplace, Pierre Simon, Marquis

Laplace, Simon

Laplace-Domain Analysis of Conventional Feedback Control Systems

Laplace-Fourier space

Laplace-Kelvin laws

Laplace-Runge-Lenz vector

Laplace-Schwarzschild radius

Laplace-Transformation Fundamentals

Laplace-Young formula

Laplaces Equation in Spherical Coordinates

Laplace’s equation

Laplace’s formula

Laplace’s transformation

Lavoisier and Laplace

Law of LaPlace

Linearization and Laplace Transformation

Liquid surfaces and the Laplace-Young equation

Marquis de Laplace

Mass transport Laplace transform

Master equations Laplace transform

Mathematical methods Laplace transform technique

Mathematical methods Laplace transforms

Matrix exponential by the Laplace

Matrix exponential by the Laplace transform method

Method of Laplace Transforms

Modified Laplace transform

Moivre-Laplace approximation

Moivre-Laplace theorem

Molecular forces, Laplace

Moments from Laplace transform

Moments from Laplace transforms

Numerical Fourier/Laplace transform

Numerical Procedures for Solving the Laplace Equation

One-sided Laplace transform

Partial differential equations Laplace transform

Partial differential equations standard Laplace transforms

Partial differential equations, Laplace

Pharmacokinetics Laplace transformation

Phase equilibrium Laplace equation

Pressure Young-Laplace equation

Pressure difference Young-Laplace equation

Properties of Laplace Transforms

Properties of the Laplace Transformation

Properties of the Laplace transform

Pulse function Laplace transform

Reverse Laplace transform

Shift of Integral Path Laplace Transform

Solids Laplace transform technique

Solution of Differential Equations with Laplace Transforms

Solution of the Laplace and Poisson Equations

Solving First-Order Differential Equations Using Laplace Transforms

Some Useful Properties of Laplace Transforms

Spherical polar coordinates Laplace

Standard Laplace transforms

Step function Laplace transform

Stochastic differential equations Laplace transforms

Surface energy Laplace equation

Surfaces Laplace equation

Table of Laplace Operations

Table of Laplace Transformations

Table of Laplace Transforms

Techniques that use the Laplace equation to measure surface energy

The Laplace Transform in Kinetic Calculations

The Laplace Transform of a Constant

The Laplace Transformation

The Laplace equation

The Laplace transform

The Method of Laplace Transforms

The Young-Laplace Equation

The specific rate function k(E) as an inverse Laplace transform

Three-dimensional Laplace equation

Time-pi (t) profiles using Laplace transform

Total solutions, Laplace transform

Transforms Laplace transform

Transition probability, Laplace

Transition probability, Laplace transform

Trigonometric functions Laplace transforms

Two-compartment mammillary model for intravenous administration using Laplace transform

Two-dimensional Laplace equation

Use of Laplace transformation

Useful Properties of Laplace Transform limit functions

Vapour pressure Young-Laplace equation

Young and Laplace equation

Young-Laplace

Young-Laplace equation definition

Young-Laplace equation for the pressure difference across a curved surface

Young-Laplace equation from Newton mechanics

Young-Laplace equation from curvature

Young-Laplace equation from plane geometry

Young-Laplace law

Young-Laplace method

Young-Laplace relationship

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