Features Extended:

  1. The probability density function for the Maxwell-Boltzmann distribution is

        f(v) = [m/(2πkT)]**(3/2)*(4πv**2)*exp[-(1/2)mv**2/(kT)] ,

    where for the microscopic particles v is velocity, m is the mass, T is Kelvin temperature, Boltzmann's contant k = 1.380649*10**(-23) J/K = (8.617333262 ... )*10**(-5) eV/K (exact) ≅ 10**(-4) eV/K ≅ 10**(-10) MeV/K, and (1/2)*m*v**2 is kinetic energy.

    The v**2 in f(v) initially causes it to grow with v, but the exponential function exp[-(1/2)*m*v**2/(kT)] eventually causes f(v) to decrease to Zorro---er, zero---as v goes to infinity.

    The energy parameter kT is the e-folding energy. An increase in kinetic energy by kT causes a decrease in exp[(1/2)*m*v**2/(kT)] by a factor exp(-1).

    The energy parameter kT is essentially temperature in energy units.

  2. For the Maxwell-Boltzmann distribution, the maximizing velocity (i.e., the most probable velocity), the mean velocity, and the root-mean-square (rms) velocity are, respectively, 
      v_max = sqrt(2kT/m) = (390.3153 ... m/s )*sqrt(T/293 K)*sqrt(31.998/A) ,

  v_mean = sqrt[8kT/(πm)] = (440.4237 ... m/s)*sqrt(T/293 K)*sqrt(31.998/A) , and

  v_rms = sqrt(3kT/m) = (478.0367 ... m/s)*sqrt(T/293 K)*sqrt(31.998/A)  ,
    where note v_max < v_mean < v_rms, Boltzmann's contant k = 1.380649*10**(-23) J/K = (8.617333262 ... )*10**(-5) eV/K (exact) ≅ 10**(-4) eV/K ≅ 10**(-10) MeV/K, m = A*u (i.e., atomic mass times the atomic mass unit (u) = (1/12) C-12 = 1.660 539 066 60(50)*10**(-27) kg), fiducial value 293 K = 20 C is the temperature of the green curve, and fiducial value 31.998 is the atomic mass of molecular oxygen (O_2) (see Wikipedia: Maxwell-Boltzmann distribution: Typical speeds). Note that the mean kinetic energy KE=(1/2)mv**2 is given by 
      KE = (3/2)KT = (3.7892568 ... )*10**(-2) eV]*(T /293 K)  ,
    where the electron-volt (eV) = 1.602176634*10**(-19) J (exact) (HyperPhysics: Average Molecular Kinetic Energy). The electron-volt is the natural unit for microscopic scale energies. For comparison, the energy of photon from the de Broglie relation E=hc/λ is 
      E = hc/λ =  (1.239841984 ... eV)/[λ/(1 μm]  .

File: Thermodynamics file: maxwell_boltzmann_distribution_1bb.html.