Thermodynamics and Statistical Mechanics

1. Thermodynamics and Statistical Mechanics Entropy and the Second Law of Thermodynamics Thermo & Stat Mech - Spring 2006 Class 6

2. Second Law • Kelvin-Planck statement: • No series of processes is possible whose sole result is the absorption of heat from a thermal reservoir and the complete conversion of this energy to work. • There are no perfect engines! Thermo & Stat Mech - Spring 2006 Class 6

3. Second Law • Clausius statement: • No series of processes is possible whose sole result is the transfer of heat from a reservoir at a given temperature to a reservoir at a higher temperature. • There are no perfect refrigerators! Thermo & Stat Mech - Spring 2006 Class 6

4. Carnot Cycle is Best A Carnot cycle is the most efficient possible, operating between two reservoirs at temperatures T1 and T2. Proof: Assume there is a more efficient engine. Let it produce work, and use that work to run a Carnot refrigerator between the same two reservoirs. Thermo & Stat Mech - Spring 2006 Class 6

5. Carnot Cycle is Best Thermo & Stat Mech - Spring 2006 Class 6

6. Carnot Cycle is Best Then, |Q2¢| < |Q2|. Also, |W| = |Q2¢| – |Q1¢| = |Q2| – |Q1|. So, |Q2| – |Q2¢|=|Q1| – |Q1¢|, and |Q1¢| < |Q1| also. Heat has been taken out of the low temperature reservoir and put into the high temperature reservoir with no expenditure of work! Not possible. Thermo & Stat Mech - Spring 2006 Class 6

7. For a Carnot Engine or Thermo & Stat Mech - Spring 2006 Class 6

8. For infinitesimal cycles Any cycle can be represented as a sum of infinitesimal Carnot cycles. Then, Thermo & Stat Mech - Spring 2006 Class 6

9. Carnot Cycles Thermo & Stat Mech - Spring 2006 Class 6

10. Entropy For reversible processes. Entropy is a state variable. Thermo & Stat Mech - Spring 2006 Class 6

11. Carnot Cycle Thermo & Stat Mech - Spring 2006 Class 6

12. Carnot Cycle on T-S Plot Thermo & Stat Mech - Spring 2006 Class 6

13. Carnot Cycle The area enclosed by the cycle on a P-V plot is the net work done per cycle. (đW = PdV) The area enclosed by the cycle on a T-S plot is the net heat added per cycle. (đQ = TdSfor any reversible process.) These two quantities are equal. Thermo & Stat Mech - Spring 2006 Class 6

14. Irreversible “Carnot” Cycle Thermo & Stat Mech - Spring 2006 Class 6

15. Clausius Inequality Irreversible cycle In general Thermo & Stat Mech - Spring 2006 Class 6

16. Entropy Change The equal sign applies for reversible processes. Thermo & Stat Mech - Spring 2006 Class 6

17. Free Expansion of a Gas Thermo & Stat Mech - Spring 2006 Class 6

18. Free Expansion Thermo & Stat Mech - Spring 2006 Class 6

19. Isothermal Expansion Thermo & Stat Mech - Spring 2006 Class 6

20. Isothermal Expansion Reversible route between same states. đQ = đW + dU Since T is constant, dU = 0. Then, đQ = đW. Thermo & Stat Mech - Spring 2006 Class 6

21. Entropy Change The entropy of the gas increased. For the isothermal expansion, the entropy of the Reservoir decreased by the same amount. So for the system plus reservoir, DS = 0 For the free expansion, there was no reservoir. Thermo & Stat Mech - Spring 2006 Class 6

22. Second Law of Thermodynamics • The entropy of an isolated system increases in any irreversible process and is unaltered in any reversible process. This is the principle of increasing entropy. • DS³ 0 Thermo & Stat Mech - Spring 2006 Class 6

23. First and Second Laws • First Law: dU = đQ – đW • First law, combined with the second law: • (for reversible processes) • dU = TdS – PdV Thermo & Stat Mech - Spring 2006 Class 6