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
Solid angle of a planar triangle is equal to "", for the sum of the four internal solid angles of a tetrahedron as follows: where ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC. A useful formula for calculating the solid angle subtended by the triangular surface ABC where formula_17 are the vector positions of the vertices A, B and C has been given by Oosterom and Strackee (although the result was known earlier by Euler and Lagrange): where denotes the scalar triple product of the three vectors; When