Caption: The ideal-gas Maxwell-Boltzmann distribution for velocity for 10**6 oxygen O_2 molecules for temperatures Celsius temperatures -100 C (red curve), 20 C (green curve), 600 C (blue curve): i.e., Kelvin temperatures 173.15 K, 293.15 K, 873.15 K. The Maxwell-Boltzmann distribution is a result in statistical mechanics.
Features:
Velocity is the conventional substitute for kinetic energy in the context of the Maxwell-Boltzmann distribution.
An integral of the curves over all velocity gives the total number of molecules n = 10**6.
In general for distributions of microscopic particles, increasing temperature shifts the distribution of microscopic particles to higher energy levels.
The formula for the distribution depends on the physical system.
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.
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] .