Lecture 09 The Distribution of Molecules over Velocities Maxwell

probability distribution

approximation of the Newton’s binomial formula
Parameters of Gauss distribution:
x –

some random value
μ — the most probable (or expected) value of the random (the maximum of the distribution function)
σ — dispersion of the random value.

In case of the Eagles and Tails game: μ = N/2, σ = (N/2)1/2 << N

Pk=(2/πN)1/2exp(-2(k-N/2)2/N) =>

Examples:
Eagle and Tails game
Target striking

the deviations of experimental results from the average (the dispersion of results = the experimental error)

of possible micro-realizations of a state with certain macro-parameters, multiplied

by the Boltzmann constant.


In the state of thermodynamic equilibrium, the entropy of a closed system has the maximum possible value (for a given energy).
If the system (with the help of external influence)) is derived from the equilibrium state - its entropy can become smaller. BUT…
If a nonequilibrium system is left to itself - it relaxes into an equilibrium state and its entropy increases
The entropy of an isolated system for any processes does not decrease, i.e. ΔS > 0 , as the spontaneous transitions from more probable (less ordered) to less probable (more ordered) states in molecular systems have negligibly low probability

J/K

Statistical Entropy in Physics.

5. The entropy is the measure of disorder in molecular systems.

additive quantity.

J/К
Statistical Entropy in Physics.
For the state of the

molecular system with certain macroscopic parameters we may introduce the definition of Statistical Entropy as the logarithm of the number of possible micro-realizations (the statistical weight of a state Ώ) - of a this state, multiplied by the Boltzmann constant.

the Entropy of Ideal Gas
the number of variants

of realization of a state (the statistical weight of a state) shall be higher, if the so called phase volume, available for each molecule (atom), is higher: Phase volume Ω1 ~Vp3~VE3/2 ~VT3/2

As molecules are completely identical, their permutations do not change neither the macrostate, nor the microstates of the system. Thus we have to reduce the statistical weight of the state by the factor ~ N! (the number of permulations for N molecules)

the phase volume for N molecules shall be raised to the power N: Ω ~ VNT3N/2.
For multy-atomic molecules, taking into account the possibilities of rotational and oscillational motion, we shall substitute 3 by i : Ω ~ VNT iN/2

Ω~ VNTiN/2 /N!; S=k ln Ω =kNln(VTi/2/NC)=
= v(Rln(V/v) + cVlnT +s0)

S=k ln Ω =kNln(VTi/2/NC)=
= v(Rln(V/v) + cVlnT +s0)
The statistical entropy proves

to be the same physical quantity, as was earlier defined in thermodynamics without even referring to the molecular structure of matter and heat!

Distributions

That will be the Focus of the next lecture!



MEPhI General

Physics

(temperature, pressure) are kept stable and the distribution of molecules

over velocities and energies remains also stable in time and space.
This distribution was first derived in 1859 by J.C.Maxwell.

Distribution of molecules over velocities

James Clerk Maxwell
1831-1879

the velocity space,
As all the directions are equal –

the distribution function can not depend on the direction, but only on the modulus of velocity f(V)

Distribution of molecules over velocities

V, will fir into the small cube

nearby the velocity V can be calculated by multiplying probabilities:

The probability to have the x-component of the velocity within the range between Vx and Vx+dVx

From the other hand, as all the directions are equal, this probability may depend only on the modulus of velocity

dP(V)

Distribution of molecules over velocities

molecules over velocities

the initial condition
here α must be negative! α

molecules over velocities

of molecules over velocities

values equals to the sum of their averages

= +
Average of the product of two values equals to the product of their averages ONLY in case if those two values DO NOT depend on each other
= only if x and y are independent variables

Probability Distribution and Average Values

Knowing the distribution of a random value x we may calculate its average value :

Moreover, we may calculate the average for any function ψ(x):

of molecules over certain spherical volume V (balloon with radius

R):

Y

X

Z

V

= 0

= R2/5 >0
= < x2 +y2 +z2> = 3R2/5 > 2
= <(x2 +y2 +z2)1/2> = 3R/4

Calculation – on the blackboard…

= 0;b = 0

- average
()1/2 = 0,61/2R > - squared average
=

0; 1/2 = R/51/2 >0

Median average rmed ; the quantity of molecules with rrmed

rmed = R/21/3 = 0,7937R > ()1/2 =0,7756R > = 0,75R

This all is about even distribution of molecules over space in spherical balloon.
What about the distribution of molecules over velocities and energies?
It can be spherically symmetric, but it can not be even as formally there is no upper limit of velocity…

again can be reduced to the Poisson integral:
Distribution of molecules

over velocities

the same energy):
Distribution of molecules over velocities

velocities

Probability to find the absolute value of the velocity between

V and V+dV

- Maxwell’s function

Distribution of molecules over velocities

rotating ahead of the other. The angle distance is

the Maxwell’s function)



Most probable velocity

The value of the Maxwell’s

function maximum:

Vвер

Vср.кв = (3kT/m)1/2
>
>
>

the temperature T = 300 K. What are the average

velocities of two types of molecules:

dE/(2mE)1/2
∫F(E)dE = 1
= ∫E F(E)dE = 3kT/2

dE/(2mE)1/2
= 3kT/2
This is the distribution of molecules over

kinetic energies. he question is: how the distribution will look like, if to take into account also the potential energy (gravity)?
That will be the topic of the next lecture..