Ek = nNAmv2/2 = nMv2/2
In this expression m is the mass of a single molecule, so the molar mass M is the product mNA. Since the kinetic energy is also 3nRT/2, the square root of the square of the mean velocity, known as the root-mean-square velocity v(rms), of the molecules of the gas is proportional to the square root of its molar mass. The root-mean-square velocity, like the actual distribution of velocities embodied in the Maxwell law, is a function only of the absolute temperature.
v(rms) = (the square root of)3RT/M
v(rms) = (the square root of)3(8.3143)(298.15)/(0.0319998) = 482.1 m/s
A speed of 482.1 m/s is 1726 km/h, much faster than a jetliner can fly and faster than most rifle bullets.
The distance any single molecule travels between collisions will vary from very short to very long distances, but the average distance that a molecule travels between collisions in a gas can be calculated. This distance is called the mean free path l of the gas molecules. If the root-mean-square velocity is divided by the mean free path of the gas molecules, the result will be the number of collisions one molecule undergoes per second. This number is called the collision frequency Z1 of the gas molecules.
The postulates of the kinetic-molecular theory of gases permit the calculation of the mean free path of gas molecules. The gas molecules are visualized as small hard spheres. A sphere of diameter d sweeps through a cylinder of cross-sectional area (pi)d2 and length v(rms) each second, colliding with all molecules in the cylinder, as shown in the Figure below.
Z1 = (pi)d2v(rms)(the square root of)2
This total number of collisions must now be divided by the number of molecules which are present per unit volume. The number of gas molecules present per unit volume is found by rearrangement of the ideal gas law to n/V = p/RT and use of Avogadro's number, n = N/NA; thus N/V = pNA/RT. This gives the mean free path of the gas molecules, l, as
(v(rms)/Z1)/(N/V) = l = RT/(pi)d2pNA(the square root of)2
According to this expression, the mean free path of the molecules should get longer as the temperature increases; as the pressure decreases; and as the size of the molecules decreases.
l = (8.3143 kg m2/s2K mol)(298.15 K)/3.14159(370 x 10-12 m)2(101325 kg/m s2) (6.0225 x 10+23 mol-1)((the square root of)2),
so l = 6.7 x 10-8 m = 67 nm. The utility of SI units and of the quantity calculus in this example should be obvious.
Z1 = (pi)d2pNA(the square root of)2/RT = v(rms)/l
For oxygen at room temperature, each gas molecule collides with another every 0.13 nanoseconds (one nanosecond is 1.0 x 10-9 s), since the collision frequency is 7.2 x 10+9 collisions per second per molecule.
For an ideal gas, the number of molecules per unit volume is given using pV = nRT and n = N/NA as
N/V = NAp/RT
which for oxygen at 25oC would be (6.0225 x 10+23 mol-1)(101325 kg/m s2)/(8.3143 kg m2/s2 K mol)(298.15 K) or 2.46 x 10+25 molecules/m3. The number of collisions between two molecules in a volume, Z11, would then be the product of the number of collisions each molecule makes times the number of molecules there are, Z1N/V, except that this would count each collision twice (since two molecules are involved in each one collision). The correct equation must be
Z11 = (pi)d2p2NA2v(rms) (the square root of)2/2R2T2
If the molecules present in the gas had different masses they would also have different speeds, so an average value of v(rms) would be using a weighted average of the molar masses; the partial pressures of the different gases in the mixture would also be required. Although such calculations involve no new principles, they are beyond our scope. However, the number of collisions which occur per second in gases and in liquids are extremely important in chemical kinetics, so we shall return to this topic in other sections.
v(rms)1/v(rms)2 = (the square root of)(M2/M1) = (the square root of)(d2/d1)
This equation gives the velocity ratio in terms of either the molar mass ratio or the ratio of densities d. The ratio of root-mean-square velocities is also the ratio of the rates of effusion, the process by which gases escape from containers through small holes, and the ratio of the rates of diffusion of gases.
This equation is called Graham's law of diffusion and effusion because it was observed by Thomas Graham (1805-1869) well before the kinetic-molecular theory of gases was developed. As an empirical law, it simply stated that the rates of diffusion and of effusion of gases varied as the square root of the densities of the gases. Graham's law is the basis of many separations of gases. The most significant is the separation of the isotopes of uranium as the gases 238UF6 and 235UF6. Fluorine has only one isotope, so the separation on the basis of molar mass is really a separation on the basis of isotopic mass.