To derive Laplace's equation using this 'local' approach, the authors consider a thin film of fluid L that contains a portion of the LV surface (see figure 5). #Poisson relations thermodynamics calculator freeThis equilibrium condition in the local approach is that the free energy of any volume element changes when surfaces move (in contrast with the free energy of the entire system, which remains constant. In equilibrium, any displacement of the surfaces is reversible and, the work done by external forces on a volume element (\delta w) will increase the Helmholtz free energy (F) (at constant temperature). Gibb's approach involves looking at the energy of a system from a global perspective, whereas the approach proposed by the authors involves analyzing local energy variations of a volume element. The authors derive Young's Equation and Laplace's equation from a different approach using the condition of energy balance during a reversible displacement of a volume element located at an interface. Typically, thermodynamically, the two equations above are derived using the condition of minimum energy (Gibbs' approach). The mechanical equilibrium of an element of volume, located along the contact line between the liquid-vapor (LV) surface and a solid S can be explained by the action of two surface tensions of the fluid-solid surfaces and, which equilibrate Young derived the following equation which explains the equilibrium contact angle \Theta: Basically, the larger the curvature (smaller radii), the larger the difference in pressure between the liquid and vapor. This equation relates the pressure discontinuity at the surface with the surface curvature. Laplace Derived the second general equation for the equilibrium of fluids in 1806: The surface forces arise from a surface tension Figure 1 shows that the pressure on a thin fluid element must be balanced by surface forces acting on its contour. However, additional if we are dealing with the surface of a fluid (as opposed to the bulk) additional surface forces need to be considered for the system to be in mechanical equilibrium. Where p=density of the fluid and the force of gravity is exerted downward along the z axis The relationship between the hydrostatic pressure (P) and the acceleration due to gravity (g) at any point in the bulk of a fluid at rest is: Overview of Laplace's and Young's Equations These advantages are: the derivations are simpler, they allow for the analysis of nonequilibrium situations, and they allow a natural identification of the surface energy with the surface tension of the liquid-vapor interface. They argue that their derivations have several advantages compared to the more traditional 'global' approaches which minimize the energy at constant entropy or the Helmholtz free energy at constant total volume. The authors also derive Young's and Laplace's equations using energy balance on a 'local' volume element located a the surface. The authors of this paper revisit Gibbs' derivation and simplify it to make it more accessible to the undergraduate level student. Subsequently, in 1880, Gibbs showed, thermodynamically, that these laws were necessary conditions for the equilibrium of heterogeneous systems. At the time of their conception, the laws were supported from a purely mechanical approach. Laplace's Law and Young's equation were established in 18 respectively. The fundamental laws governing the mechanical equilibrium of solid-fluid systems are Laplace's Law (which describes the pressure drop across an interface) and Young's equation for the contact angle. Laplace's Law, Young's equation, surface energy, surface tension Introduction Title: "Thermodynamic derivations of the mechanical equilibrium conditions for fluid surfaces: Young's and Laplace's equations"
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