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
Cylinder . The area of the side is known as the "", . An "open cylinder" does not include either top or bottom elements, and therefore has surface area (lateral area) The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore, where is the diameter of the circular top or bottom. For a given volume, the right circular cylinder with the smallest surface area has . Equivalently, for a given surface area, the right circular cylinder with the largest volume has , that is,