Standard Maxwell-Boltzmann Distribution.pptx

A standard Maxwell-Boltzmann distribution (B) is defined by analogy to the concept of the standard Gaussian distribution. The most important statistical properties of B, as well as a simple method for generating random numbers from the standard Maxwell-Boltzmann distribution are presented. Given that the properties of B are already known, it is advantageous to describe any arbitrary Maxwell-Boltzmann distribution as a function of the standard Maxwell-Boltzmann distribution B. By using this approach, it is possible to demonstrate that the temperature of a material is a function only of the fluctuating component of the average molecular kinetic energy, and that it is independent of its macroscopic kinetic energy.

1. Standard Maxwell- Boltzmann Distribution : Definition , Properties and Applications Medellín , 20/11/2017 www.forschem.org 1 Hugo Hernández, Dr.rer.nat . [email protected]

2. Introduction 2

3. Background Statistics Random variable : Probability density function : Expected value : Variance :   3        

4. Maxwell- Boltzmann Distribution 4   Equation of molecular motion ( relative to the system ):     Net force acting on molecule in - direction : Normal distribution (Central Limit Theorem )   Therefore , the molecular velocity in a single direction is also normally distributed !   Standard normal random variable   Unit direction vector

5. Maxwell- Boltzmann Distribution 5   Overall relative molecular speed: The molecular speed is a random variable with the following probability density:     Maxwell-Boltzmann distribution! Distribution of the norm of a 3-D vector of zero -mean normal random variables!

6. Maxwell- Boltzmann Distribution 6 Molecular speed distributions for potassium gas obtained from the Maxwell-Boltzmann distribution at 300 K, 544 K and 1000 K

7. Maxwell- Boltzmann Distribution 7 Predicted (solid line) vs. experimental (data points) distributions of molecular speeds for potassium gas in a vacuum oven at 544 K [1] Miller , R. C., & Kusch, P. (1955). Velocity distributions in potassium and thallium atomic beams. Physical Review, 99(4), 1314.

8. Generalized Maxwell- Boltzmann Distribution in Multicomponent Systems 8    

9. Definition of the Standard maxwell- boltzmann distribution 9

10. Standard Maxwell- Boltzmann Distribution B 10      

11. Standard Maxwell- Boltzmann Distribution B 11 Probability density function for the standard Maxwell-Boltzmann random variable    

12. Standard Maxwell- Boltzmann Distribution B 12   The molecular speed distribution for species j simplifies into: where the average molecular speed is:     The molecular velocity distribution for species j is:     Unit direction random vector

13. Standard Maxwell- Boltzmann Distribution : Random numbers generation 13   Histogram of 10.000 random numbers generated compared to the corresponding standard Maxwell-Boltzmann probability density function   Uniform random number [0, 1]

14. Some Properties of the Standard maxwell- boltzmann distribution 14

15. Standard Maxwell- Boltzmann Distribution : Properties 15    

16. Standard Maxwell- Boltzmann Distribution : Properties 16

17. Standard Maxwell- Boltzmann Distribution : Properties 17      

18. Standard Maxwell- Boltzmann Distribution : Properties 18  

19. Standard Maxwell- Boltzmann Distribution : Properties 19  

20. applications of the Standard maxwell- boltzmann distribution 20

21. Standard Maxwell- Boltzmann Distribution : Applications Thermodynamics of an ideal gas: 21       Average temperature  

22. Standard Maxwell- Boltzmann Distribution : Applications Thermodynamics of an ideal gas: 22        

23. Standard Maxwell- Boltzmann Distribution : Applications Thermodynamics of an ideal gas: 23   Virial equation    

24. Standard Maxwell- Boltzmann Distribution : Applications Thermodynamics of an ideal gas: 24        

25. Standard Maxwell- Boltzmann Distribution : Applications Kinetic theory of gases: 25   Relative molecular speed between two molecules:     Time between consecutive collisions:  

26. Standard Maxwell- Boltzmann Distribution : Applications Kinetic theory of gases: 26 Time between collisions for Maxwell-Boltzmann molecular speed distributions:    

27. Standard Maxwell- Boltzmann Distribution : Applications Kinetic theory of gases: 27     Standard Collision Probability Density Function:      

28. Standard Maxwell- Boltzmann Distribution : Applications Kinetic theory of gases: 28   Conventional exponential :       27% error 247 % error 61% error

29. Standard Maxwell- Boltzmann Distribution : Applications Molecular Diffusion : 29        

30. Standard Maxwell- Boltzmann Distribution : Applications Molecular Diffusion : 30      

31. Additional applications ? 31

32. Further Reading Hernandez, H. (2016). Modelling the effect of fluctuation in nonlinear systems using variance algebra - Application to light scattering of ideal gases, ForsChem Research Reports, 2016-1, doi : 10.13140/RG.2.2.36501.52969 . Hernandez , H. (2017). Standard Maxwell-Boltzmann distribution: Definition and properties. ForsChem Research Reports 2017-2. doi : 10.13140/RG.2.2.29888.74244 . Hernandez, H. (2017). On the generalized validity of the Maxwell-Boltzmann distribution and the zeroth Law of Thermodynamics. ForsChem Research Reports 2017-4. doi : 10.13140/RG.2.2.26937.16480 . Hernandez, H. (2017). Molecular Free Path Statistical Distribution of Multicomponent Systems. ForsChem Research Reports 2017-6. doi : 10.13140/RG.2.2.15605.58088 . Hernandez , H. (2017). Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution. ForsChem Research Reports 2017-7. doi : 10.13140/RG.2.2.26218.31689 . Hernandez , H. (2017). Multicomponent Molecular Collision Kinetics: Collision Rate and the Collision Frequency Paradox. ForsChem Research Reports 2017-8. doi : 10.13140/RG.2.2.32983.27048 . Hernandez , H. (2017). Multicomponent Molecular Diffusion: A Mathematical Framework. ForsChem Research Reports 2017-9. doi : 10.13140/RG.2.2.14828.46724 . Hernandez, H. (2017). Multivariate Probability Theory: Determination of Probability Density Functions. ForsChem Research Reports 2017-13. doi : 10.13140/RG.2.2.28214.60481 . 32