Derivation of the Laplace equation

The non P-V work associated with changing (deforming) the area of the surface will be 

Substitution of Equations (6.8) and (6.9) into the left-hand side of Equation (6.4) yields 

For the often-encountered special case of spherical geometry, r =ri = r, the Laplace equation takes the simple form 

It is emphasized that the proper surface quantity to be used in Equations (6.11) and (6.12) is a, not y. It is only for the special case of fluid systems, where a = y, that the surface energy may be used in the Laplace equation and 

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The Young Equation. The principle of balancing forces used in the derivation of the Laplace equation can also be used to derive another important equation in surface thermodynamics, the Young equation. Consider a liquid droplet in equilibrium... 

Figure A.l. Displacement of a liquid surface allowing derivation of the Laplace equation.

Having defined what we mean by the surface of a liquid and its curvature, we now consider the derivation of Laplace s law for a spherical drop of liquid with radius R. In this case the radii of curvature at every point on the surface are simply equal to R. The usual derivation of the Laplace equation follows from a consideration of the net change in free energy of the liquid droplet resulting from a change in the radius R. Consider the Helmholtz free energy of a spherical drop of liquid consisting of only one type of molecule... 

One may easily verify that the above equation reduces to the Laplace equation in Eq. (5) for a spherical surface when one inserts J = IjR. When we consider the shape of a droplet in a gravitational field in the next section, the above expression is used to describe the shape. Furthermore, we present in Section II.C a detailed derivation of the Laplace equation in Eq. (7) as it pertains to the nonspherical shape of a droplet in a gravitational field. First, we need to consider the interaction of the liquid surface in contact with the solid substrate on which the droplet resides. 

An alternative derivation of Young s equation follows the same route as the derivation of the Laplace equation using a notional change of the location of the dividing surface. Consider the surface free energy of the system depicted in Fig. 4. Around the line of three-phase contact a cylinder is drawn with length L and radius R, and implicitly we assume that R and L approach... 

Several graphical curve-fitting techniques have been developed (see Padday [53] for details) that can be used in conjunction with the numerical integration of the Laplace equation by Bashforth and Adams (and by subsequent workers) to determine d and to obtain y v. Smolders [54,55] used a number of coordinate points of the profile of the drop for curve fitting. If the surface tension of the liquid is known and if 0 > 90, a perturbation solution of the Laplace equation derived by Ehrlich [56] can be used to determine the contact angle, provided the drop is not far from spherical. Input data are the maximum radius of the drop and the radius at the plane of contact of the drop with the solid surface. The accuracy of this calculation does not depend critically on the accuracy of the interfacial tension. 

This is the simplest model of an electrocatalyst system where the single energy dissipation is caused by the ohmic drop of the electrolyte, with no influence of the charge transfer in the electrochemical reaction. Thus, fast electrochemical reactions occur at current densities that are far from the limiting current density. The partial differential equation governing the potential distribution in the solution can be derived from the Laplace Equation 13.5. This equation also governs the conduction of heat in solids, steady-state diffusion, and electrostatic fields. The electric potential immediately adjacent to the electrocatalyst is modeled as a constant potential surface, and the current density is proportional to its gradient ... 

Equation (2.7) and the derived equations (2.8) and (2.9) are the weighted residual statements for all common approximating techniques of the Laplace equation. 

Derivation of the Laplace transformed state equations from the original bond graph in Fig. 4.11 yields... 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

A similar relationship may be derived for each component in a multi-component liquid using the transfer shown schematically in Figure 6.4 in which component i is transferred. In this case the free-energy change for the transfer is Vi A P and application of the Laplace equation yields a Gibbs-Thomson equation for each component in the liquid ... 

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. 

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

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. 

With the Laplace operator V. The diffusion coefficient defined in Eq. (62) has the dimension [cm /s]. (For correct derivation of the Fokker-Planck equation see [89].) If atoms are initially placed at one side of the box, they spread as ( x ) t, which follows from (62) or from (63). 

Both Poisson s and Laplace s equations describe the behavior of the potential at regular points where the first derivatives of the field exist. To characterize the behavior of the potential at the boundary of media with different densities, let us make use of Equation (1.39) according to which a component of the field along some direction / is equal to the derivative of the potential in this direction ... 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

The Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal ... 

Inputs + Sources = Outputs + Sinks + Accumulation Formulation of differential equations in general is described in Chapter 1. Usually the ODE is of the first or second order and is readily solvable directly or by aid of the Laplace Transform. For example, for the special case of initial equilibrium or dead state (All derivatives zero at time zero), the preceding equation has the transform... 

In order to stably levitate an object, the net force on it must be zero, and the forces on the body, if it is perturbed, must act to return it to its original position. The object must be at a local potential minimum that is, the second derivatives with respect to all spatial coordinates of the potential must be positive. This may seem, at first sight, to be trivial to arrange. However, any system whose potential is a solution to Laplace s equation is automatically unstable A statement in words of Laplace s equation is that the sum of the second partial derivatives of the potential is zero, and so not all can be simultaneously positive. This has long been known for electrostatic potentials, having been stated by Earnshaw(n) Millikan s scheme for suspending charged particles is thus only neutrally stable, since the fields within a Millikan capacitor provide no lateral constraint. 

The time-domain differential equation description of systems can be used instead of the Laplace-domain transfer function description. Naturally the two are related, and we will derive these relationships later in this chapter. State variables are very popular in electrical and mechanical engineering control problems which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representations are more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. 

Since it is relatively easy to transfer molecules from bulk liquid to the surface (e.g. shake or break up a droplet of water), the work done in this process can be measured and hence we can obtain the value of the surface energy of the liquid. This is, however, obviously not the case for solids (see later section). The diverse methods for measuring surface and interfacial energies of liquids generally depend on measuring either the pressure difference across a curved interface or the equilibrium (reversible) force required to extend the area of a surface, as above. The former method uses a fundamental equation for the pressure generated across any curved interface, namely the Laplace equation, which is derived in the following section. 

Can you prove why this is so ) When x, y, and z are thermodynamic quantities, such as free energy, volume, temperature, or enthalpy, the relationship between the partial differentials of M and N as described above are called Maxwell relations. Use Maxwell relations to derive the Laplace equation for a... 

The presence of surface tension has an important implication for the pressures across a curved interface and, as a consequence, for phase equilibria involving curved interphase boundaries. The equation that relates the pressure difference across an interface to the radii of curvature, known as the Laplace equation, is derived in Section 6.4, and the implications for phase equilibria are considered for some specific cases in Section 6.5. 

The Laplace equation applied specifically to spherical surfaces can be derived in a variety of ways. Example 6.1 considers an alternative derivation that points out the thermodynamic character of the result quite clearly. 

If the electrode reaction proceeds via a non-linear mechanism, a rate equation of the type of eqn. (123) or (124) serves as a boundary condition in the mathematics of the diffusion problem. Then, a rigorous analytical derivation of the eventual current—potential characteristic is not feasible because the Laplace transfrom method fails if terms like Co and c are present. The most rigorous numerical approach will be... 

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