1. Too many particles… can’t keep track! Use pressure (p) and volume (V) instead. Temperature (T) measures the tendency of an object to spontaneously give up/absorb energy to/from its surroundings. (p and T will turn out to be related to the too many particles mentioned above) Thermal Physics

2. P, V, T Pressure, Volume, Temperature F/A L3L3 Something to do with heat

3. Equations of state An equation of state is a mathematical relation between state variables, e.g. p, V & T. This reduces the number of independent variables to two. General form: f (p,V,T) = 0 Example:pV – nRT = 0(ideal gas law) Defines a 2D surface in p-V-T state space. Each point on this surface represents an unique state of the system. f (p,V,T) = 0 Equilibrium state

4. Ideal gas equation of state Robert Boyle (1627 – 1691) Boyle’s law p  1/V Jacques Charles (1746 – 1823) Charles’ law V  T Joseph Louis Gay-Lussac (1778 - 1850) Gay-Lussac’ law p  T pV = Nk B T k B = 1.38  10 -23 J/K

5. Universe (system + surroundings) System Surroundings Heat Heat is energy in transit

6. Temperature is what you measure with a thermometer Temperature is the thing that’s the same for two objects, after they’ve been in contact long enough. Long enough so that the two objects are in thermal equilibrium. Time required to reach thermal equilibrium is the relaxation time. What is temperature?

7. AC BC Diathermal wall Zeroth law of thermodynamics If two systems are separately in thermal equilibrium with a third system, they are in thermal equilibrium with each other. C can be considered the thermometer. If C is at a certain temperature then A and B are also at the same temperature.

8. How can we define temperature using the microscopic properties of a system?

9. Most likely macrostate the system will find itself in is the one with the maximum number of microstates. Macrostate Number of Microstates (  )

10. 1.Each microstate is equally likely 2.The microstate of a system is continually changing 3.Given enough time, the system will explore all possible microstates and spend equal time in each of them (ergodic hypothesis).

12. Most likely macrostate the system will find itself in is the one with the maximum number of microstates. E 1  1 (E 1 ) E 2  2 (E 2 ) E  (E)

13. Most likely macrostate the system will find itself in is the one with the maximum number of microstates. E 1  1 (E 1 ) E 2  2 (E 2 ) E 1  1 (E 1 ) E 2  2 (E 2 )

14. Most likely macrostate the system will find itself in is the one with the maximum number of microstates. E 1  1 (E 1 ) E 2  2 (E 2 )

15. Using this definition of temperature we need to describe real systems

16. E  (E) Microcanonical ensemble: An ensemble of snapshots of a system with the same N, V, and E

17. Canonical ensemble: An ensemble of snapshots of a system with the same N, V, and T (red box with energy  << E. E-   (E-  ) I()I() Red box is small only in terms of energy, its volume could still be large

19. Boltzmann Factor (canonical ensemble)

20. Canonical ensemble Reservoir

22. The red ball is the particle from the canonical ensemble in thermal equilibrium with the reservoir. It occupies the same volume as the reservoir which in this case are the rest of particles in an ideal gas.

23. Spherical coordinates

24. Monatomic ideal gas

26. First try to find the probability that the red particle has a certain velocity

30. Maxwell-Boltzmann speed distribution

31. T = 10 T = 100 T = 1000

33. In velocity space: Or since its velocity space

34. In velocity space: Or since its velocity space

35. In velocity space: Or since its velocity space

36. Once again:

37. Remember all this is happening in velocity space

38. This is what happens in real space  vdt A

39.  A

42. The pressure on the wall due to all the particles in the gas is: