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Thermodynamics

Boltzmann (Gibbs) Distribution

Maxwell-Boltzmann Distribution

Lana Sheridan

De Anza College

May 8, 2020

Last time

• (^) heat capacities for ideal gases
• (^) adiabatic processes

Reminder: Adiabatic Process in Ideal Gases

For an adiabatic process (Q = 0):

PV γ^ = const.

and:

TV γ−^1 = const.

(Given the first one is true, the second follows immediately from the ideal gas equation, P = nRTV .)

Weather and Adiabatic Process in a Gas

On the eastern side of the Rocky Mountains there is a phenomenon called chinooks.

These eastward moving wind patterns cause distinctive cloud patterns (chinook arches) and sudden increases in temperature.

Temperature and the Distribution of Particles’

Energies

In a gas at temperature T , we know the average translational KE of the molecules.

However, not all of the molecules have the same energy, that’s just the average.

How is the total energy of the gas distributed amongst the molecules?

Temperature and the Distribution of Particles’

Energies

Ludwig Boltzmann first found the distribution of the number of particles at a given energy given a thermodynamic system at a fixed temperature.

Assuming that energy takes continuous values we can say that the number of molecules per unit volume with energies in the range E to E + dE is:

N[E ,E +dE] =

∫ (^) E +dE

E

nV (E ) dE

Where nV (E ) = n 0 e−E^ /kB^ T

and n 0 is a constant setting the scale: when E = 0, nV (E ) = n 0.

The Boltzmann Distribution

This particular frequency distribution:

nV (E ) ∝ e−E^ /kB^ T

is called the Boltzmann distribution or sometimes the Gibbs distribution (after Josiah Willard Gibbs, who studied the behavior of this distribution in-depth).

This distribution is even easier to understand for discrete energy levels.

The probability for a given particle to be found in a state with energy Ei drawn from a sample at temperature T :

p(Ei ) =

Z

e−Ei^ /kB^ T

where Z is simply a normalization constant to allow the total probability to be 1. (The partition function.)

The Boltzmann Distribution

p(Ei ) =

Z

e−Ei^ /kB^ T

If we know the energies of two states E 1 and E 2 , E 2 > E 1 , we can find the ratio of the number of particles in each:

nV (E 2 ) nV (E 1 )

= e−(E^2 −E^1 )/kB^ T

States with lower energies have more particles occupying them.

(Somewhat Contrived) Example

Suppose a type of atom has only 2 energy states, separated in energy by 12.0 eV.^1 For a very large sample of these atoms, at what temperature would 1% of the atoms in the sample be in the excited (higher energy) state?

∆E = E 2 − E 1 = 12 eV

(Somewhat Contrived) Example

Suppose a type of atom has only 2 energy states, separated in energy by 12.0 eV.^1 For a very large sample of these atoms, at what temperature would 1% of the atoms in the sample be in the excited (higher energy) state?

nV (E 2 ) nV (E 1 )

(^1) This does not describe any real atom.

(Somewhat Contrived) Example

Suppose a type of atom has only 2 energy states, separated in energy by 12.0 eV.^2 For a very large sample of these atoms,

∆E = E 2 − E 1 = 12 eV

At what temperature would the number of atoms in each state be equal?

(^2) This does not describe any real atom.

(Somewhat Contrived) Example

Suppose a type of atom has only 2 energy states, separated in energy by 12.0 eV.^2 For a very large sample of these atoms,

∆E = E 2 − E 1 = 12 eV

At what temperature would the number of atoms in each state be equal?

nV (E 2 ) nV (E 1 )

(^2) This does not describe any real atom.

Aside: Lasers

Lasers emit coherent light. One photon interacts with an atom and causes another to be emitted with the same phase.

This starts a cascade.

Inside a laser cavity there are atoms that are in a very strange state: a higher energy level is more populated than a lower one. This is called a “population inversion”.

Aside: Lasers

This is necessary for the photon cascade. Since:

nV (E 2 ) nV (E 1 )

= e−(E^2 −E^1 )/kB^ T^ , E 2 > E 1

we can associate a “negative temperature”, T , to these two energy states in the atoms.