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) =>
J/K
Statistical Entropy in Physics.
5. The entropy is the measure of disorder in molecular systems.
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)
Distribution of molecules over velocities
James Clerk Maxwell
1831-1879
Distribution of molecules over velocities
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
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):
Y
X
Z
V
= 0
Calculation – on the blackboard…
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…
- Maxwell’s function
Distribution of molecules over velocities
Vвер
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