Maxwell/Boltzmann and an Ideal Gas

G.R.Dixon, 2/12/2006

Imagine that a hollow metal sphere has an internal volume of 22.4 liters = 2.24E-2 cubic meters. The spherical radius is R = .175 meters. Inside the sphere is 1 mole = N_{o} molecules of an ideal gas, where N_{o} = 6.025E23 molecules per mole is Avogadro’s number. (The volume of an ideal gas molecule is zero. There are no internal vibrations or spinning around a center of mass. Should two molecules collide, they simply exchange velocities.) Let the molecular mass equal 2.016 (the same as H_{2}). The mass of each molecule is then 2.016/N_{o} = 3.346E-27 kg.

Classically we could (in principle) know the position and velocity of each molecule at some initial moment. Each set of positions and velocities defines an initial state. There are an infinite number of possible initial states. And each initial state presumably has an equal probability.

Experience seems to confirm that (a) all initial states exert the same pressure on the container wall, and (b) the pressure does not vary with the passage of time (provided R and T, the gas temperature, do not vary). This being the case, we can choose any of the possible initial states for analysis. We shall choose the state where (a) the molecules are equally distributed within the spherical volume and (b) each molecule moves back and forth along a line through the sphere’s center. That is, each molecule’s velocity is normal to two diametrically opposed points on the spherical container’s inner surface.

** The Maxwell/Boltzmann distribution** specifies a formula for N(v)dv, the number of molecules with speeds in the range v to v+dv. Each of these molecules imparts an outward-pointing momentum of magnitude 2mv whenever it elastically rebounds from the container wall. In 1 second any one of these molecules rebounds v sec/2R times. Thus in 1 second the molecules with speeds in the range v to v+dv rebound from the wall vN(v)dv sec/2R times, and collectively impart momentum of mv

. (1)

According to Maxwell/Boltzmann, at an absolute temperature of T the number of molecules with speeds in the range v to v+dv is

, (2)

where k = 1.38E-23 joule/molecule is Boltzmann’s constant. Substituting in Eq. 1:

. (3)

Mathematically,

. (4)

Thus

, (5)

where V = 4pR^{3}/3 is the spherical volume. Rearranging Eq. 5 produces

, (6)

which is the combined laws of Boyle/Mariotte and Gay-Lussac/Charles. Note that N_{o}k is usually called the absolute gas constant and is denoted as "R" (not to be confused with our spherical volume’s radius).

It is noteworthy that m, the molecular mass, does not enter into Eq. 6. That is, Eq. 6 applies to __all__ ideal gases. If a different molecule (say a more massive one) is substituted, the pressure is the same! True enough the more massive molecules, with speeds in the range v to v+dv, impart a greater momentum each time they rebound from the container wall. However, it is clear in Eq. 2 that their numbers in each range of speeds will differ from those of the lighter molecule.

In an ideal gas E, a given molecule’s energy, is purely kinetic: E = mv^{2}/2. And the number of molecules with energies in the range E to E+dE is the same as N(v)dv. Thus m<v^{2}>/2, the mean energy, can be calculated from Maxwell/Boltzmann:

. (6)

The absolute temperature is therefore proportional to the mean kinetic energy:

. (7)

The average molecular speed is

. (8)

Mathematically,

. (9)

Thus

. (10)

For our "hydrogenesque" ideal gas molecule of mass m = 3.346E-27kg: at a temperature of 273K, <v> = 1693 m/sec = 1 mile/sec … about Mach 5!