Entropy

1. Physics 202 Professor Lee Carkner Lecture 15 Entropy

2. PAL #14 Internal Energy • 3 moles of gas, temperature raised from 300 to 400 K • He gas, isochorically • Q = nCVDT, CV = (f/2)R = (3/2) R • Q = (3)(3/2)R(100) = 3740 J • He gas, isobarically • Q = nCPDT, CP = CV + R = (5/2) R • Q = (3)(5/2)R(100) = 6333 J • H2 gas, isochorically • Q = nCVDT, CV = (5/2) R, f = 5 for diatomic • Q = (3)(5/2)R(100) = 6333 J • H2 gas, isobarically • Q = nCPDT, CP = CV + R = (7/2) R • Q = (3)(7/2)R(100) = 8725 J

3. PAL #14 Internal Energy • 4 moles of N2 gas isobaric expansion from 0.45 m3 to 0.78 m3 and 457 K • pressure = p =nRT/V = (4)(8.31)(457)/(0.78) = 19475 Pa • initial temp = T = pV/nR = (19475)(0.45)/(4)(8.31) = 263.7 K • W=pDV = (19475)(0.78-0.45) = 6427 J • Q=nCpDT = (4)(7/2)(8.31)(457-263.7) =22489 J • adiabatic process starts at the same point, ends where V= 0.78 m3. • piVig=pfVfg • pf= piVig/Vfg= (19475)(0.45)1.4/(0.78)1.4 = 9017 Pa

4. Randomness • Classical thermodynamics is deterministic • Every time! • But the real world is probabilistic • It is possible that you could add heat to a system and the temperature could go down • The universe only seems deterministic because the number of molecules is so large that the chance of an improbable event happening is absurdly low

5. Reversible • Why? • The smashing plate is an example of an irreversible process, one that only happens in one direction • Examples: • Perfume diffuses throughout a room • Heat transfer

7. Entropy • What do irreversible processes have in common? • The degree of randomness of system is called entropy • In any thermodynamic process that proceeds from an initial to a final point, the change in entropy depends on the heat and temperature, specifically: DS = Sf –Si = ∫ (dQ/T)

8. Isothermal Entropy DS = (1/T) ∫ dQ DS = Q/T • Like heating something up by 1 degree

9. Heat Reservoir • Something that is too big to change temperature • A heat reservoir can gain or lose heat without changing temperature • Since Q = mcDT, if m is very large, DT can be very small

10. Second Law of Thermodynamics (Entropy) • Consider objects A and B that exchange heat Q with each other isothermally: • We always find that the positive term is always a larger than the negative term, so: DS>0 • Entropy always increases

11. Entropy Problems Using Q/T • Need to find heat • Sign of DS is sign of Q (positive in and negative out) • T constant for phase change or heat reservoir • For total entropy, must add all sources and sinks of heat

12. General Entropy • From the first law and the ideal gas law, we get DS = nRln(Vf/Vi) + nCVln(Tf/Ti) • Note that we only need to know the initial and final conditions, not the path

13. Statistical Mechanics • We will use statistical mechanics to explore the reason why gas diffuses throughout a container • The box contains 4 indistinguishable molecules

14. 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 • If all microstates are equally probable than the configuration with equal distribution is the most probable

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

16. 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

17. Irreversibility • Irreversible processes move from a low probability state to a high probability one • All real processes are irreversible, so entropy will always increases • The universe is stochastic

18. Arrows of Time • Three arrows of time: • Direction in which entropy increases • Direction that you do not remember • Direction of increasing expansion of the universe

19. Entropy and Memory • Memory requires energy dissipation as heat • Psychological arrow of time is related to the thermodynamic

20. Synchronized Arrows • Why do all the arrows go in the same direction? • Can life exist with a backwards arrow of time? • Does life only exist because we have a universe with a forward thermodynamic arrow? (anthropic principle)

21. Fate of the Universe • Head towards the Big Crunch • Will the others reverse as well? • Expand forever

22. Heat Death • Everything in the universe trying to be same temperature • Universe gets more and more disordered • Left with white dwarfs, neutron stars and radiation • Can live off of compact objects, but eventually will convert them all to heat

23. Next Time • Read: 20.5-20.7

24. Suppose it is 0 F outside today. What would the temperature need to be outside tomorrow (in F) to be twice as hot? • -34 • 0 • 100 • 458 • 510

25. How much heat does it take to change the temperature of one mole of a monatomic ideal gas 1 degree K in a constant volume process? How much heat does it take to change the temperature of one mole of a monatomic ideal gas 1 degree K in a constant pressure process? • 1 J : 1 J • 1 J : 12.5 J • 12.5 J : 12.5 J • 12.5 J : 20.8 J • 8.3 J : 16.6 J

26. What is the change in internal energy for an ideal monatomic gas whose temperature increases 1 degree K in a constant volume process? What is the change in internal energy for an ideal monatomic gas whose temperature increases 1 degree K in a constant pressure process? • 1 J : 1 J • 1 J : 12.5 J • 12.5 J : 12.5 J • 12.5 J : 20.8 J • 8.3 J : 16.6 J