The Distribution of Molecules over Velocities Maxwell Distribution

Июль 31, 2022

Главная
Математика
The Distribution of Molecules over Velocities Maxwell Distribution

Содержание

2. BAR CHART Smooth CHART – distribution function x x+a ΔP x SOME MATHEMATICS: the probability distribution
3. Normal distribution The normal (or Gauss) distribution – is the smooth approximation of the Newton’s binomial
4. Gauss Distribution The normal distribution is very often found in nature. Examples: Eagle and Tails game
5. Statistical Entropy in Molecular Physics: the logarithm of the number of possible micro-realizations of a state
6. Entropy is the additive quantity. J/К Statistical Entropy in Physics. For the state of the molecular
7. Not a strict proof, but plausible considerations. ~ Statistical Entropy and the Entropy of Ideal Gas
8. ~ Statistical Entropy and the Entropy of Ideal Gas Ω~ VNTiN/2 /N!; S=k ln Ω =kNln(VTi/2/NC)=
9. The Distributions of Molecules over Velocities and Energies Maxwell and Boltzmann Distributions That will be the
10. If gas is in thermodynamic equilibrium state –the macroscopic parameters (temperature, pressure) are kept stable and
11. Each velocity vector can be presented as a point in the velocity space, As all the
12. The probability that the end of the 3-dimensional velocity vector V, will fir into the small
13. Noe some mathematics: We will calculate the derivative by dVx as Distribution of molecules over velocities
14. The only function which satisfies the equation: as well as the initial condition here α must
15. From normalization condition : we obtain: The Poisson integral: Distribution of molecules over velocities
16. THE PROPERTIES OF AVERAGES. Average of the sum of two values equals to the sum of
17. Probability Distribution and Average Values Examples: in case of even distribution of molecules over certain spherical
18. Different Kinds of Averages Y X Z V = = 3R/4 - average ( )1/2 =
19. The average of the squared velocity equals to: This integral once again can be reduced to
20. The basic assumption of thermodynamics (every degree of freedom accumulates the same energy): Distribution of molecules
21. The distribution over absolute values of velocities: , Distribution of molecules over velocities
22. Function defines the probability that velocity is within the “cubic” range: Probability to find the absolute
23. Maxwell’s Function
24. Area under the curve is always equal to 1 Maxwell’s Function
25. Stern’s experiment (1920) The outer cylinder is rotating
26. Lammert’s Experiment (1929) Two rotating discs with radial slots. One is rotating ahead of the other.
27. Most probable velocity. Most probable velocity corresponds to the maximum of the Maxwell’s function) Most probable
28. Average velocity. Average velocity by deffinition For Maxwell’s function:
29. Average squared velocity by definition Average squared velocity.
30. Most probable: Averge: Average squared: Three kinds of average velocities
31. Maxwell’s Function Vвер = (2kT/m)1/2 Vср = (8kT/πm)1/2 Vср.кв = (3kT/m)1/2 > > >
32. Example: Example: The mixture of oxygen and nitrogen (air) has the temperature T = 300 K.
33. Energy distribution function F(V)dV = F(E)dE; E = mV2/2; dV = dE/(2mE)1/2 ∫F(E)dE = 1 =
34. Energy distribution function F(V)dV = F(E)dE; E = mV2/2; dV = dE/(2mE)1/2 = 3kT/2 This is
36. Скачать презентацию

Слайд 2

BAR CHART
Smooth CHART – distribution function
x x+a
ΔP
x

SOME MATHEMATICS: the probability

distribution

Слайд 3

Normal distribution
The normal (or Gauss) distribution – is the smooth 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) =>

Слайд 4

Gauss Distribution
The normal distribution is very often found in nature.
Examples:

Eagle and Tails game
Target striking
the deviations of experimental results from the average (the dispersion of results = the experimental error)

Слайд 5

Statistical Entropy in Molecular Physics: the logarithm of the number 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.

Слайд 6

Entropy is the 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.

Слайд 7

Not a strict proof, but plausible considerations.
~
Statistical Entropy and 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)

Слайд 8

~
Statistical Entropy and the Entropy of Ideal Gas
Ω~ VNTiN/2 /N!; 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!

Слайд 9

The Distributions of Molecules
over Velocities and Energies
Maxwell and Boltzmann Distributions
That

will be the Focus of the next lecture!

MEPhI General Physics

Слайд 10

If gas is in thermodynamic equilibrium state –the macroscopic parameters (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

Слайд 11

Each velocity vector can be presented as a point in 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

Слайд 12

The probability that the end of the 3-dimensional velocity vector 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

Слайд 13

Noe some mathematics:
We will calculate the derivative by dVx
as
Distribution of molecules

over velocities

Слайд 14

The only function which satisfies the equation:
as well as the

initial condition

here α must be negative! α <0

is:

Distribution of molecules over velocities

Слайд 15

From normalization condition : we obtain:
The Poisson integral:
Distribution of molecules over

velocities

Слайд 16

THE PROPERTIES OF AVERAGES.
Average of the sum of two 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):

Слайд 17

Probability Distribution and Average Values
Examples: in case of even distribution of

molecules over certain spherical volume V (balloon with radius R):

= 0

= R2/5 >0
= < x2 +y2 +z2> = 3R2/5 > 2
= <(x2 +y2 +z2)1/2> = 3R/4
Calculation – on the blackboard…

= 0;b = 0

Слайд 18

Different Kinds of Averages
Y
X
Z
V
= <(x2 +y2 +z2)1/2> = 3R/4 -

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…

Слайд 19

The average of the squared velocity equals to:
This integral once again

can be reduced to the Poisson integral:

Distribution of molecules over velocities

Слайд 20

The basic assumption of thermodynamics (every degree of freedom accumulates the

same energy):

Distribution of molecules over velocities

Слайд 21

The distribution over absolute values of velocities:
,
Distribution of molecules over velocities

Слайд 22

Function
defines the probability that velocity is within the “cubic” range:
Probability

to find the absolute value of the velocity between V and V+dV

- Maxwell’s function

Distribution of molecules over velocities

Слайд 23

Maxwell’s Function

Слайд 24

Area under the curve is always equal to 1
Maxwell’s Function

Слайд 25

Stern’s experiment (1920)
The outer cylinder is rotating

Слайд 26

Lammert’s Experiment (1929)
Two rotating discs with radial slots. One is rotating

ahead of the other. The angle distance is

Слайд 27

Most probable velocity.
Most probable velocity corresponds to the maximum of the

Maxwell’s function)
Most probable velocity
The value of the Maxwell’s function maximum:

Vвер

Слайд 28

Average velocity.
Average velocity by deffinition
For Maxwell’s function:

Слайд 29

Average squared velocity by definition
Average squared velocity.

Слайд 30

Most probable:
Averge:
Average squared:
Three kinds of average velocities

Слайд 31

Maxwell’s Function
Vвер = (2kT/m)1/2
Vср = (8kT/πm)1/2
Vср.кв =

(3kT/m)1/2

Слайд 32

Example:
Example: The mixture of oxygen and nitrogen (air) has the

temperature T = 300 K. What are the average velocities of two types of molecules:

Слайд 33

Energy distribution function
F(V)dV = F(E)dE;
E = mV2/2;
dV = dE/(2mE)1/2

∫F(E)dE = 1
= ∫E F(E)dE = 3kT/2

Слайд 34

Energy distribution function
F(V)dV = F(E)dE;
E = mV2/2;
dV = 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..

The Distribution of Molecules over Velocities Maxwell Distribution

Содержание

BAR CHARTSmooth CHART – distribution functionx x+aΔPx SOME MATHEMATICS: the probability

Normal distributionThe normal (or Gauss) distribution – is the smooth approximation

Gauss DistributionThe normal distribution is very often found in nature. Examples:

Statistical Entropy in Molecular Physics: the logarithm of the number of

Entropy is the additive quantity.J/КStatistical Entropy in Physics. For

Not a strict proof, but plausible considerations. ~Statistical Entropy and the

~Statistical Entropy and the Entropy of Ideal GasΩ~ VNTiN/2 /N!; S=k

The Distributions of Molecules over Velocities and EnergiesMaxwell and Boltzmann DistributionsThat

If gas is in thermodynamic equilibrium state –the macroscopic parameters (temperature,

Each velocity vector can be presented as a point in the

The probability that the end of the 3-dimensional velocity vector V,

Noe some mathematics: We will calculate the derivative by dVx asDistribution of molecules

The only function which satisfies the equation: as well as the

From normalization condition : we obtain: The Poisson integral:Distribution of molecules over

THE PROPERTIES OF AVERAGES. Average of the sum of two values

Probability Distribution and Average ValuesExamples: in case of even distribution of

Different Kinds of AveragesYXZV = <(x2 +y2 +z2)1/2> = 3R/4 -

The average of the squared velocity equals to: This integral once again

The basic assumption of thermodynamics (every degree of freedom accumulates the

The distribution over absolute values of velocities:,Distribution of molecules over velocities

Function defines the probability that velocity is within the “cubic” range: Probability

Maxwell’s Function

Area under the curve is always equal to 1Maxwell’s Function

Stern’s experiment (1920)The outer cylinder is rotating

Lammert’s Experiment (1929)Two rotating discs with radial slots. One is rotating

Most probable velocity.Most probable velocity corresponds to the maximum of the

Average velocity.Average velocity by deffinitionFor Maxwell’s function:

Average squared velocity by definitionAverage squared velocity.

Most probable:Averge:Average squared:Three kinds of average velocities

Maxwell’s FunctionVвер = (2kT/m)1/2 Vср = (8kT/πm)1/2 Vср.кв =

Example: Example: The mixture of oxygen and nitrogen (air) has the

Energy distribution functionF(V)dV = F(E)dE; E = mV2/2; dV = dE/(2mE)1/2

Energy distribution functionF(V)dV = F(E)dE; E = mV2/2; dV = dE/(2mE)1/2

Похожие презентации

BAR CHART
Smooth CHART – distribution function
x x+a
ΔP
x

SOME MATHEMATICS: the probability

Normal distribution
The normal (or Gauss) distribution – is the smooth approximation

Gauss Distribution
The normal distribution is very often found in nature.
Examples:

Entropy is the additive quantity.
J/К
Statistical Entropy in Physics.
For

Not a strict proof, but plausible considerations.
~
Statistical Entropy and the

~
Statistical Entropy and the Entropy of Ideal Gas
Ω~ VNTiN/2 /N!; S=k

The Distributions of Molecules
over Velocities and Energies
Maxwell and Boltzmann Distributions
That

Noe some mathematics:
We will calculate the derivative by dVx
as
Distribution of molecules

The only function which satisfies the equation:
as well as the

From normalization condition : we obtain:
The Poisson integral:
Distribution of molecules over

THE PROPERTIES OF AVERAGES.
Average of the sum of two values

Probability Distribution and Average Values
Examples: in case of even distribution of

Different Kinds of Averages
Y
X
Z
V
= <(x2 +y2 +z2)1/2> = 3R/4 -

The average of the squared velocity equals to:
This integral once again

The distribution over absolute values of velocities:
,
Distribution of molecules over velocities

Function
defines the probability that velocity is within the “cubic” range:
Probability

Area under the curve is always equal to 1
Maxwell’s Function

Stern’s experiment (1920)
The outer cylinder is rotating

Lammert’s Experiment (1929)
Two rotating discs with radial slots. One is rotating

Most probable velocity.
Most probable velocity corresponds to the maximum of the

Average velocity.
Average velocity by deffinition
For Maxwell’s function:

Average squared velocity by definition
Average squared velocity.

Most probable:
Averge:
Average squared:
Three kinds of average velocities

Maxwell’s Function
Vвер = (2kT/m)1/2
Vср = (8kT/πm)1/2
Vср.кв =

Example:
Example: The mixture of oxygen and nitrogen (air) has the

Energy distribution function
F(V)dV = F(E)dE;
E = mV2/2;
dV = dE/(2mE)1/2

Energy distribution function
F(V)dV = F(E)dE;
E = mV2/2;
dV = dE/(2mE)1/2