Statistical Physics 3

1. Statistical Physics 3

2. Topics • Recap • The Molar Heat Capacity • The Einstein Model of a Solid • Summary

3. Recap • Particles come in two classes: • bosons and fermions • Fermions obey the Pauli exclusion principle. Bosons do not • The collective behavior of bosons is described by the Bose-Einstein distribution • The collective behavior of fermions is described by the Fermi-Dirac distribution

4. The Molar Heat Capacity In 1819, Dulong and Petit noted that the molar heat capacity for solids is approximately where R is the gas constant This empirically derived law is called the Dulong-Petit law

5. The Molar Heat Capacity Let’s assume that the heat supplied to a solid is transformed into the kinetic and potential energy of each atom To explain the Dulong-Petit law we need to know how the heat is divided up amongst the degrees of freedom

6. The Molar Heat Capacity Molar Energy of a Solid The Dulong-Petit law can be explained using the equipartition theorem of classical physics: the average energy of each degree of freedom is kT/2 If we assume each atom has 6 degrees of freedom, 3 translational and 3 vibrational, then where we have used R = NA k

7. The Molar Heat Capacity Heat Capacity at Constant Volume By definition, the heat capacity of a substance at constant volume is Classical physics therefore predicts A value independent of temperature

8. The Molar Heat Capacity The Dulong-Petit law, however, is valid only at high temperatures

9. The Einstein Model of a Solid In 1907, Einstein extended Planck’s ideas to matter: he proposed that the energy values of atoms are quantized and proposed the following simple model of a solid: Each atom is independent Each vibrates in 3-dimensions Each vibration has energy

10. The Einstein Model of a Solid In effect, Einstein modeled one mole of solid as an assembly of 3NAdistinguishable oscillators. He used the Boltzmann distribution to calculate the average energy of an oscillator in this model

11. The Einstein Model of a Solid To compute the average, note that it can be written as where and b = 1/kT Z is called the partition function

12. The Einstein Model of a Solid With b = 1/kT and En = ne, the partition function for the Einstein model is which, follows from the result

13. The Einstein Model of a Solid Differentiating with respect to b

14. The Einstein Model of a Solid …and multiplying by –1/Z we obtain Einstein’s result for the average energy of an oscillator. The total energy of the solid is just 3NA times that result

15. The Einstein Model of a Solid The heat capacity in Einstein’s model is given by where TE = e/k is called the Einstein temperature Exam 2: Question 4 Show that CV→ 0 as T → 0 and CV → 3NA = 3R as T → ∞

16. The Einstein Model of a Solid Einstein, Annalen der Physik 22 (4), 180 (1907)

17. The Debye Model of a Solid

18. Summary • The Dulong-Petit law is expected from the equipartition theorem of classical physics. However, it fails at temperatures low compared with the Einstein temperature • If the energy in solids is assumed to be quantized, however, models can be developed that agree with the observed behavior of the heat capacity with temperature. Einstein was the first to suggest such a model.