Equation of Young and Laplace

The equation of Young and Laplace describes one of the fundamental laws in interface science If an interface between two fluids is curved, there is a pressure difference across it provided the system is in equilibrium. The Young-Laplace equation relates the pressure difference between the two phases AP and the curvature of the interface. In the absence of gravitation, or if the objects are so small that gravitation is negligible, the Young-Laplace equation is 

ri and ri are the two principal radii of curvature. AP is also called Laplace pressure. Equation (5.1) is also referred to as the Laplace equation. 

1) Thomas Young, 1773-1829. English physidan and physidst, professor in Cambridge. 

2) Pierre-Simon Laplace, Marquis de Laplace, 1749-1827. French natural sdentist. 

How high is the pressure in a spherical bubble with a radius p, of 1 mm and a bubble of 10 nm radius in pure water, compared to the pressure outside For a bubble, the two radii of curvature are identical to that of a sphere ri = t2 = ry,. Therefore, 

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To derive the equation of Young and Laplace we consider a small part of a liquid surface. First, we pick a point X and draw a line around it which is characterized by the fact that all points on that line are the same distance d away from X (Fig. 2.6). If the liquid surface is planar, this would be a flat circle. On this line we take two cuts that are perpendicular to each other (AXB and CXD). Consider in B a small segment on the line of length dl. The surface tension pulls with a force 7 dl. The vertical force on that segment is 7 dl sin a. For small surface areas (and small a) we have sin a d/R where R is the radius of curvature along AXB. The vertical force component is... 

When applying the equation of Young and Laplace to simple geometries it is usually obvious at which side the pressure is higher. For example, both inside a bubble and inside a drop, the pressure is higher than outside (Fig. 2.7). In other cases this is not so obvious because the curvature can have an opposite sign. One example is a drop hanging between the planar ends of two cylinders (Fig. 2.7). Then the two principal curvatures, defined by... 

The shape of a liquid surface is determined by the Young-Laplace equation. In large structures we have to consider also the hydrostatic pressure. Then the equation of Young and Laplace becomes... 

To describe wettability in a porous reservoir rock requires inclusion of both the fluid surface interaction and curvature of pore walls. Both are responsible for the capillary rise seen in porous media. The fundamental equation of capillarity is given by the equation of Young and Laplace [2]... 

In ceramics, mercury porosimetry is more widely used than gas adsorption because pores sizes in the macropore range are more common. The technique is based on the phenomenon of capillary rise (Fig. 3.11). A liquid that wets the walls of a narrow capillary (contact angle, 0 < 90°) will climb up the walls of the capillary. However, the level of a liquid that does not wet the walls of a capillary (0 > 90°) will be depressed. For a nonwetting liquid, a pressure must be applied to force the liquid to flow up the capillary to the level of the reservoir. For a capillary with principal radii of curvature and r2 in two orthogonal directions, this pressure is given by the equation of Young and Laplace ... 

Binder removal during thermal degradation has features that are similar to those encountered in the drying of a moist granular material. Let us consider a model in which interconnected pores of two different radii are present (Fig. 6.57a). Even though the pores have different radii (r and r ), initially liquid evaporates from them at the same rate so that the radii of the menisci (r ) are equal. The capillary tension in the liquid is given by the equation of Young and Laplace ... 

Mercury porosimetry is based on the capillary rise phenomenon whereby an excess pressure is required to cause a non-wetting liquid to climb up a narrow capillary. The pressure difference across the interface is given by the equation of Young and Laplace [3 sic] and its sign is such that the pressure is less in the liquid than in the gas (or... 

At the interface of water and air, there are unbalanced forces owing to the surface tension of the fluid. In the unsaturated zone, this transition zone is known as the capillary fringe or tension-saturated zone and these unbalanced forces result in capillary rise of groundwater and dissolved constituents. Assuming a constant hemispherical fluid meniscus and pore radius, the equation of Young and LaPlace can be combined with an equation for hydrostatic pressure to produce a relation for computing the capillary rise of fluid ... 

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