Surface tension gases. The pressure inside an ideal (one surface) soap bubble can be derived from thermodynamic free energy considerations. At constant temperature and particle number, , the differential Helmholtz energy is given by where is the difference in pressure inside and outside of the bubble, and is the surface tension. In equilibrium, , and so, For a spherical bubble, the volume and surface area are given simply by and Substituting these relations into the previous expression, we find which is equivalent to the Young–Laplace equation when . For real soap bubbles, the pressure is doubled due to the presence of two
Surface area a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: "the area of the whole is the sum of the areas of the parts". More rigorously, if a surface "S" is a union of finitely many pieces "S", …, "S" which do not overlap except at their boundaries, then Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface,