1. Statistical Mechanics Physics 202 Professor Lee Carkner Lecture 19

2. PAL # 18 Engines  Engine #1 W = 10, Q H = 45   =W/Q H = 0.22   Engine #2 Q L = 25, Q H = 30   = 1 – Q L /Q H = 0.17   Engine #3 T H = 450 K, T L = 350 K   C = 1 – T L /T H = 0.22   Engine #4 W = 20, Q H = 30, T H = 500, T L = 400   = 0.66 >  C = 0.2   Engine #5 W = 20, Q H = 15   = 1.33 > 1 

3. Engines and Refrigerators   Heat from the hot reservoir is transformed into work (+ heat to cold reservoir)   By an application of work, heat is moved from the cold to the hot reservoir

4. A Refrigerator  A refrigerator depends on 2 physical principles:   Boiling liquids absorb heat, condensing liquids give off heat (heat of vaporization)  Heat can be moved from a cold region to a hot region by adjusting the pressure so that the circulating fluid boils in the cold region and condenses in the hot   n.b., the refrigerator is not the cold region (where we keep our groceries), it is the machine on the back that moves the heat

5. Refrigerator Cycle Liquid Gas Compressor (work =W) Expansion Valve Heat removed from inside cold region by evaporation Heat added to room by condensation High Pressure Low Pressure QLQL QHQH

6. Refrigerator Diagram

7. Refrigerator as a Thermodynamic System  K = Q L /W  K is called the coefficient of performance  Q H = Q L + W W = Q H - Q L  This is the work needed to move Q L out of the cold area

8. Refrigerators and Entropy  We can rewrite K as:  From the 2nd law (for a reversible, isothermal process):  So K becomes: K C = T L /(T H -T L )   Refrigerators are most efficient if they are not kept very cold and if the difference in temperature between the room and the refrigerator is small

9. Perfect Refrigerator

10. Perfect Systems  A perfect engine converts Q H directly into W with Q L = 0 (no waste heat)   Perfect refrigerators are impossible (heat won’t flow from cold to hot)  But why?   Violates the second law:  If T L does not equal T H then Q L cannot equal Q H  Perfect systems are impossible

11. Entropy   Entropy always increases for irreversible systems   Entropy always increases for any real, closed system (2nd law)  Why?   The 2nd law is based on statistics

12. Statistical Mechanics  Statistical mechanics uses microscopic properties to explain macroscopic properties   Consider a box with a right and left half of equal area 

13. Molecules in a Box  There are 16 ways that the molecules can be distributed in the box   Since the molecules are indistinguishable there are only 5 configurations  Example:  If all microstates are equally probable than the configuration with equal distribution is the most probable

14. Configurations and Microstates Configuration I 1 microstate Probability = (1/16) Configuration II 4 microstates Probability = (4/16)

15. Probability  There are more microstates for the configurations with roughly equal distributions   Gas diffuses throughout a room because the probability of a configuration where all of the molecules bunch up is low

16. Multiplicity  The multiplicity of a configuration is the number of microstates it has and is represented by: W = N! /(n L ! n R !)  n! = n(n-1)(n-2)(n-3) … (1)   For large N (N>100) the probability of the equal distribution configurations is enormous

17. Microstate Probabilities

18. Entropy and Multiplicity  The more random configurations are most probable   We can express the entropy with Boltzmann’s entropy equation as:  Where k is the Boltzmann constant (1.38 X 10 -23 J/K)  ln N! = N (ln N) - N

19. Irreversibility  Irreversible processes move from a low probability state to a high probability one   Increase of entropy based on statistics  Why doesn’t the universe seem random?  

20. Arrows of Time  Three arrows of time:  Thermodynamic   Psychological   Cosmological 

21. Entropy and Memory  When we remember things, order is increased   A brain or a computer cannot store information without the output of heat 

22. Fate of the Universe  The universe is expanding, and there does not seem to be enough mass in the universe to stop the expansion   Entropy keeps increasing   Stars burn out   Can live off of compact objects, but eventually will convert them all to heat  