Question

Why maxwellian speeds have gaussian surface because?

Answer 1

The kinetic molecular theory is used to determine the motion of a molecule of an ideal gas under a certain set of conditions. However, when looking at a mole of ideal gas, it is impossible to measure the velocity of each molecule at every instant of time. Therefore, the Maxwell-Boltzmann distribution is used to determine how many molecules are moving between velocities v and v + dv. Assuming that the one-dimensional distributions are independent of one another, that the velocity in the y and z directions does not affect the x velocity, for example, the Maxwell-Boltzmann distribution is given by

dNN=(m2πkBT)1/2e−mv22kBTdv(1)

where

dN/N is the fraction of molecules moving at velocity v to v + dv,

m is the mass of the molecule,

kb is the Boltzmann constant, and

T is the absolute temperature.1

Additionally, the function can be written in terms of the scalar quantity speed c instead of the vector quantity velocity. This form of the function defines the distribution of the gas molecules moving at different speeds, between  c1  and  c2 , thus

f(c)=4πc2(m2πkBT)3/2e−mc22kBT(2)

Finally, the Maxwell-Boltzmann distribution can be used to determine the distribution of the kinetic energy of for a set of molecules. The distribution of the kinetic energy is identical to the distribution of the speeds for a certain gas at any temperature.