Statistical Physics 1

1. Statistical Physics 1

2. Topics • Introduction • The Boltzmann Distribution • The Maxwell Distribution • Summary

3. Introduction We believe we now have the basic laws that, in principle, can be used to predict the detailed behavior of an arbitrarily large assembly of atoms and molecules But even a tiny piece of matter consists of millions of atoms In practice, the complexity of the calculation is far beyond the capability of any conceivable computer and we need a different approach

4. Introduction Happily, for most applications we are not interested in the precise behavior of each atom, but only the collective behavior of the assembly, which can be described with only a few variables, such as temperature, pressure and volume Statistical physics is the study of the collective behavior of large assemblies of atoms and molecules using probabilistic reasoning

5. The Boltzmann Distribution The Austrian physicist Boltzmann asked the following question: in an assembly of atoms, what is the probability that an atom has total energy between E and E+dE? His answer: where Ludwig Boltzmann 1844 - 1906

6. The Boltzmann Distribution is called the Boltzmann distribution, e-E/kT is the Boltzmann factor and k = 8.617 x 10-5 eV/K is the Boltzmann constant The Boltzmann distribution applies to identical, but distinguishable particles Ludwig Boltzmann 1844 - 1906

7. The Boltzmann Distribution The number of particles with energy E is given by where g(E) is the statistical weight, i.e., the number of states with energy E. However, in classical physics the energy is continuous so we must replace g(E) by g(E)dE, which is the number of states with energy between E and E + dE. g(E) is then referred to as the density of states.

8. The Boltzmann Distribution Example 8-1The Law of Atmospheres Classically, the total energy of a gas molecule of mass m, near the Earth’s surface, is where z is the vertical distance above the ground z Wanted: the fraction of particles between z and z+dz So we can write

9. The Boltzmann Distribution Example 8-1The Law of Atmospheres A basic rule of probability is: sum, or integrate, over quantities whose values are either unknown or not of interest. We are interested only in z. After integrating the Boltzmann distribution with respect to p we get z

10. The Boltzmann Distribution Example 8-1The Law of Atmospheres The fraction of molecules between z and z + dz is z At T = 300K, the ratio of the fraction at z = 1000 m to that at z = 0 m is just fB(1000) / fB(0) = 0.893

11. The Boltzmann Distribution Example 8-2H Atoms in First Excited State At temperature T, the atoms of a gas will occupy different energy levels. For hydrogen, the energy difference E2 - E1 between the 1st excited state and the ground state is 10.2 eV. What is the ratio of the number of atoms in the 1st excited state to the number in the ground state at T= 5800 K (the temperature of the Sun’s “surface”)?

12. The Boltzmann Distribution Example 8-2 • Number of atoms in state E • Ratio of number of atoms in E2 and E1

13. The Boltzmann Distribution Example 8-2 • Ratio of statistical weights. The degeneracy for each orbital quantum number l is given by 2l+1. For the ground state of hydrogen, l = 0, which gives 1. For the 1st excited state l = 0 and l = 1, which gives 4. But for each of these states the electron has 2 spin states. So we have g1 = 2 and g2 = 8. So

14. The Boltzmann Distribution Example 8-2 4. For T = 5800 K (kT ≈ 0.5 and DE = E2-E1 = 10.2 eV) we have Even at the Sun’s surface there are relatively few atoms in the 1st excited state. This is because the energy gap DE >> kT

15. The Maxwell Distribution An important application of the Boltzmann distribution is the distribution of molecular speeds v in a gas of N molecules: This distribution, in fact, was derived by James Clerk Maxwell before Boltzmann’s work. But it is an important (and famous) special case.

16. The Maxwell Distribution Most probable Average RMS Different summaries of the molecular speed computed from the Maxwell distribution

17. The Maxwell Distribution Example The average speed of a nitrogen molecule at T = 300 K is given by With k = 1.39 x 10-23 J/K and m = 4.68 x 10-26 kg one gets <v> = 475 m/s = 1700 km/h

18. Summary • Statistical physics is the study of the collective behavior of large assemblies of particles • Ludwig Boltzmann derived the following energy distribution for identical, but distinguishable, particles • The Maxwell distribution of molecular speeds is a famous application of Boltzmann’s general formula