- Temperature in modern
physics is
energy
parameter that
controls the distribution of
microscopic
particles
(i.e., atoms and
molecules)
among microscopic
energy states
(usually called energy levels).
- The image illustrates the particular case of
ideal-gas
microscopic
particles.
- The horizontal axis is particle
velocity
in meters per second (m/s).
Velocity is the conventional substitute for kinetic energy in the context of the Maxwell-Boltzmann distribution.

- The vertical axis is the number n of molecules
per meters per second (m/s).
An integral of the curves over all velocity gives the total number of molecules n = 10**6.

- In brief, the curves are the distributions of
molecules with
velocity
or, with the right conversion,
kinetic energy.
- As the image shows, the
Maxwell-Boltzmann distribution
shifts to higher velocity or
kinetic energy as
temperature increases.
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
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.

- 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 byKE = (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 isE = hc/λ = (1.239841984 ... eV)/[λ/(1 μm] .

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:

Image link: Wikimedia Commons: File:Maxwell-Boltzmann distribution 1.png.

Local file: local link: boltzmann_distribution.html.

File: Thermodynamics file: maxwell_boltzmann_distribution.html.