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Lecture 2 – The First Law (Ch. 1) Wednesday January 9 th

Lecture 2 – The First Law (Ch. 1) Wednesday January 9 th. Statistical mechanics What will we cover (cont...) Chapter 1 Equilibrium The zeroth law Temperature and equilibrium Temperature scales and thermometers. Reading: All of chapter 1 (pages 1 - 23)

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Lecture 2 – The First Law (Ch. 1) Wednesday January 9 th

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  1. Lecture 2 – The First Law (Ch. 1) Wednesday January 9th • Statistical mechanics • What will we cover (cont...) • Chapter 1 • Equilibrium • The zeroth law • Temperature and equilibrium • Temperature scales and thermometers Reading: All of chapter 1 (pages 1 - 23) 1st homework set due next Friday (18th). Homework assignment available on web page. Assigned problems: 2, 6, 8, 10, 12

  2. Statistical Mechanics What will we cover?

  3. Probability and Statistics PHY 3513 (Fall 2006)

  4. Probability and Statistics Probability distribution function Gaussian statistics: Input parameters: Quality of teacher and level of difficulty Abilities and study habits of the students

  5. Probability and Statistics Probability distribution function Gaussian statistics: Input parameters: Quality of teacher and level of difficulty Abilities and study habits of the students

  6. The connection to thermodynamics Maxwell-Boltzmann speed distribution function Input parameters: Temperature and mass (T/m) Equation of state:

  7. Probability and Entropy Suppose you toss 4 coins. There are 16 (24) possible outcomes. Each one is equally probably, i.e. probability of each result is 1/16. Let W be the number of configurations, i.e. 16 in this case, then: Boltzmann’s hypothesis concerning entropy: where kB = 1.38 × 10-23 J/K is Boltzmann’s constant.

  8. The bridge to thermodynamics through Z js represent different configurations

  9. Quantum statistics and identical particles Indistinguishable events Heisenberg uncertainty principle The indistinguishability of identical particles has a profound effect on statistics. Furthermore, there are two fundamentally different types of particle in nature: bosons and fermions. The statistical rules for each type of particle differ!

  10. The connection to thermodynamics Maxwell-Boltzmann speed distribution function Input parameters: Temperature and mass (T/m) Consider T 0

  11. Bose particles (bosons) Internal energy = 0 Entropy = 0 # of bosons 11 10 9 8 7 6 5 4 3 2 1 0 Energy

  12. Fermi-Dirac particles (fermions) Internal energy ≠ 0 Free energy = 0 Entropy = 0 # of fermions Pauli exclusion principle 1 EF 0 Energy Particles are indistinguishable

  13. Applications Specific heats: Insulating solid Diatomic molecular gas Fermi and Bose gases

  14. The zeroth & first Laws Chapter 1

  15. Thermal equilibrium System 2 Heat System 1 Pi, Vi Pe, Ve If Pi = Pe and Vi = Ve, then system 1 and systems 2 are already in thermal equilibrium.

  16. Different aspects of equilibrium Mechanical equilibrium: Piston 1 kg 1 kg Already in thermal equilibrium Pe, Ve gas When Peand Ve no longer change (static)  mechanical equilibrium

  17. Different aspects of equilibrium Chemical equilibrium: nl↔ nv nl + nv = const. P, nv, Vv Already in thermal and mechanical equilibrium vapor P, nl, Vl liquid When nl, nv, Vl & Vv no longer change (static)  chemical equilibrium

  18. Different aspects of equilibrium Chemical reaction: A + B↔ AB# molecules ≠ const. Already in thermal and mechanical equilibrium A, B & AB When nA, nB& nAB no longer change (static)  chemical equilibrium

  19. Different aspects of equilibrium If all three conditions are met: • Thermal • Mechanical • Chemical Then we talk about a system being thermodynamic equilibrium. Question: How do we characterize the equilibrium state of a system? In particular, thermal equilibrium.....

  20. The Zeroth Law a) b) A C B C VA, PA VC, PC VB, PB VC, PC “If two systems are separately in thermal equilibrium with a third system, they are in equilibrium with each other.” c) A B VA, PA VB, PB

  21. The Zeroth Law a) b) A C B C VA, PA VC, PC VB, PB VC, PC “If two systems are separately in thermal equilibrium with a third system, they are in equilibrium with each other.” • This leads to an equation of state, q =f(P,V), where the parameter, q (temperature), characterizes the equilibrium. • Even more useful is the fact that this same value of q also characterizes any other system which is in thermal equilibrium with the first system, regardless of its state.

  22. More on thermal equilibrium • Continuum of different mechanical equilibria (P,V) for each thermal equilibrium, q. • Experimental fact: for an ideal, non-interacting gas, PV = constant (Boyle’s law). • Why not have PV proportional to q; Kelvin scale. q characterizes (is a measure of) the equilibrium. Each equilibrium is unique. Erases all information on history.

  23. Equations of state • Defines a 2D surface in P-V-qstate space. • Each point on this surface represents a unique equilibriumstate of the system. q f(P,V,q) = 0 Equilibrium state • An equation of stateis a mathematical relation between state variables, e.g. P, V & q. • This reduces the number of independent variables to two. General form: f(P,V,q) = 0 or q = f(P,V) Example: PV = nRq (ideal gas law)

  24. Temperature Scales

  25. Gas Pressure Thermometer Ice point Steam point LN2 Celsius scale P = a[T(oC) + 273.15]

  26. An experiment that I did in PHY3513 P T 17.7 79 13.8 0 3.63 -195.97

  27. The ‘absolute’ kelvin scale T(K) = T(oC) + 273.15 Triple point of water: 273.16 K

  28. Other Types of Thermometer • Thermocouple: E = aT + bT2 • Metal resistor: R = aT + b • Semiconductor: logR = a-blogT Low Temperature Thermometry How stuff works

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