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
Platonic solid by where "θ" is the dihedral angle. The midradius "ρ" is given by where "h" is the quantity used above in the definition of the dihedral angle ("h" = 4, 6, 6, 10, or 10). Note that the ratio of the circumradius to the inradius is symmetric in "p" and "q": The surface area, "A", of a Platonic solid {"p", "q"} is easily computed as area of a regular "p"-gon times the number of faces "F". This is: The volume is computed as "F" times the volume of the pyramid whose base is a regular "p"-gon and whose height is