Consider N gas molecules with radius r moving randomly with speed v in a cylindrical volume V. When one molecule collides with another molecule, the distance between their centers is 2r. Imagine a cylinder with a radius of 2r, with an axis parallel to the molecule's velocity. When the molecule travels for a small time interval and collides inside the cylinder, the number of collisions per unit time can be determined. Using the average relative velocity equation, the collisions for all the moving molecules per unit time can be determined. The reciprocal of the equation gives the average time between collisions, known as the mean free time. Meanwhile, the mean free path of a gas molecule is the product of the molecule's speed and the average time between collisions. By substituting the terms, the mean free path can be determined, which is inversely proportional to the number of molecules per unit volume and the cross-sectional area of the molecule. Recalling the ideal-gas equation and substituting the terms, the macroscopic properties of the gas can be obtained.