Laplace formula

This equation can be deduced from the general Laplace formula that expresses the force exerted on a conductor of length dl, through which a current / passes, in a magnetic field of intensity B. The orientation of the Lorentz force (F = I dl AB) can be found by different approaches such as the right-handed three-finger rule or using the orientations of a direct trihedron. 

The interface between the droplet and the gas is not discontinuous the average molecular density decreases over a narrow region from the liquid side to the vapor. When the size of the droplet becomes sufhctently small compared with the thickness of the transition layer, the use of classical thermodynamics and the bulk surface tension become inaccurate the Kelvin relation and Laplace formula no longer apply. This effect has been studied by molecular dynamics calculations of the behavior of liquid droplets composed of 41 to 2(X)4 molecules that interact through a Lennard-Jones (LI) intermolecular potential (Thomp.son et al., 1984). The results of this analysis are shown in Fig. 9.5, in which the nondimensional pressure difference between the drop interior and the surrounding vapor (Pd — p)rr / ij is... 

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. 

Because the right-hand side is always positive, the effect of using fine particles instead of bulk material is to increase the equilibrium constant—that is, to shift the equilibrium to the right. The smaller the particles, the larger will be the effect. This phenomenon occurs because the high internal pressure in the particles which can be calculated from the Laplace formula leads to an increased thermodynamic activity of the particle substance. 

The factors that determine the crystal structure of particles formed in aerosol reactors have not been studied systematically. In this section, we identify key theoretical concepts and review relevant experimental observations. Consideration is limited to single-component systems. Panicle crystal structure depends on a combination of thermodynamic (equilibrium) factors and rate processes. The equilibrium shape of a particle is detennined by the surface energies of its crystal face.s according to the Wulff construction (Chapter 8). Another factor that inay enter into the process is the excess pressure inside small particles according to the Laplace formula (Chapter 9). Thus the equilibrium form may vary with panicle size depending on the phase diagram,... 

Examples.—(1) Assuming the Newton-Laplace formula that the square of the velocity of propagation, V, of a compression wave (e.g., of sound) in a gas varies directly as the adiabatic elasticity of the gas, E, and inversely as the density, p, or V2 cc E /p show that F2 oc yRT. Hints Since the compression wave travels so rapidly, the changes of pressure and volume may be supposed to take place without gain or loss of heat. Therefore, instead of using Boyle s law, pv = constant, we must employ pvy = constant. Hence deduce yp = v. dp/dv = Eq. Note that the volume varies inversely as the density of the gas. Hence, if... 

Young-Laplace formula 7 cos 9 = gs — %s- This can be seen by using the relation pts = d /dh to evaluate the integral explicitly, and observing that 7(/io) equals to the gas-solid surface tension 7, while 7(00) = 7 + 7gs is the sum of the gas-liquid and liquid-solid surface tensions. 

The history of the barometric formula is described in the literature [8]. Famous scientists laid the foundations to the barometric formula, among ihtmGalileC Torricelli Pascal, and Boyle. The explicit formula goes back to Laplace. For this reason, the barometric formula is sometimes called Laplace formula. 

Then the expression for the energy contribution changes to (check that both expressions are identical after using the Laplace formula)... 

Transport of liquid in the continuous network of capillaries, as well as in the microcracks, is forced by the hydrauhc pressure or by the capillary action. The latter one is the effect of surface tension of water causing the pressure in capillaries, which can be expressed by Laplace formula ... 

The energy of the vapor-solid interface 7 is expressed in the same way, with pt replaced by pv These expressions can be used in the Young-Laplace formula that derives the contact angle 6 from the balance condition for a three-phase contact line ... 

The value of the disjoining pressure for a liquid layer situated between an air bubble and a solid substrate may be calculated from the Laplace formula ... 

Changes of interface characteristics enable a certain nominalization of the contact energy, which is in the centre of thermodynamic descriptions of any heterogeneous systems putting on display the phase separating interfaces. A direct consequence is the famous Laplace formulae, = (1// 1//" ),... 

The computation of the incomplete Gamma functions is a vital part of evaluating of the ERIs for all integral methods except the Rys-Gauss quadrature. In the evaluation of Fm(T ) two formulae can be used depending on the value of the argument, T. Firstly, there is the asymptotic formula in which the Laplace formula (see equation 13) is utilized. 

It is interesting to note that independent, direct calculations of the PMC transients by Ramakrishna and Rangarajan (the time-dependent generation term considered in the transport equation and solved by Laplace transformation) have yielded an analogous inverse root dependence of the PMC transient lifetime on the electrode potential.37 This shows that our simple derivation from stationary equations is sufficiently reliable. It is interesting that these authors do not discuss a lifetime maximum for their formula, such as that observed near the onset of photocurrents (Fig. 22). Their complicated formula may still contain this information for certain parameter constellations, but it is applicable only for moderate flash intensities. 

Thus, on any irregular pattern the Laplace operator is approximated to first order by the difference operator A specified by formula (13). 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

An interesting approach has been employed in paper [74] to find the distribution f(li, l2) of copolymer chains for numbers l and h of monomeric units Mi and M2. This distribution is evidently equivalent to the SCD, because the pair of numbers k and I2 unambiguously characterizes chemical size (l = h + l2) and composition ( 1 = l] //, 2 = h/l) of a macromolecule. The essence of this approach consists of invoking the Superposition Principle [81] that enables the problem of finding the Laplace transform G(pi,p2) of distribution f(li,k) to be reduced to the solution of two subsidiary problems. The first implies the derivation of the expression for the generating function [/(z1",z 2n ZjX,z ) of distribution P(ti, M2 mt, m2), and the second is concerned with finding the Laplace transforms g (pi,p2) and (pi,p2) of distributions (Eq. 91). With these two problems solved, it is possible to obtain the characteristic function G(pi,p2) of distribution f(li,h) using the Superposition Principle formula... 

In order to make the meaning of this theorem more precise, let us consider the Laplace inverse of (Q (z) ky, which is defined by a formula similar to (65) ... 

Progress has been recently made in constructing an iterative inverse Laplace transform method which is not exponentially sensitive to noise. This Short Time Inverse Laplace Transform (STILT) method is based on rewriting the Bromwich inversion formula as ... 

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