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

Thermo & Stat Mech - Spring 2006 Class 62 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 63 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 64 Carnot Cycle is Best A Carnot cycle is the most efficient possible, operating between two reservoirs at temperatures T 1 and T 2. 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 65 Carnot Cycle is Best

Thermo & Stat Mech - Spring 2006 Class 66 Carnot Cycle is Best Then, |Q 2  < |Q 2 |. Also, |W| = |Q 2  – |Q 1  = |Q 2 | – |Q 1 |. So, |Q 2 | – |Q 2  = |Q 1 | – |Q 1 , and |Q 1  < |Q 1 | 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 67 For a Carnot Engine or

Thermo & Stat Mech - Spring 2006 Class 68 For infinitesimal cycles Any cycle can be represented as a sum of infinitesimal Carnot cycles. Then,

Thermo & Stat Mech - Spring 2006 Class 69 Carnot Cycles

Thermo & Stat Mech - Spring 2006 Class 610 Entropy For reversible processes. Entropy is a state variable.

Thermo & Stat Mech - Spring 2006 Class 611 Carnot Cycle

Thermo & Stat Mech - Spring 2006 Class 612 Carnot Cycle on T-S Plot

Thermo & Stat Mech - Spring 2006 Class 613 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 = TdS for any reversible process.) These two quantities are equal.

Thermo & Stat Mech - Spring 2006 Class 614 Irreversible “Carnot” Cycle

Thermo & Stat Mech - Spring 2006 Class 615 Clausius Inequality Irreversible cycle In general

Thermo & Stat Mech - Spring 2006 Class 616 Entropy Change The equal sign applies for reversible processes.

Thermo & Stat Mech - Spring 2006 Class 617 Free Expansion of a Gas

Thermo & Stat Mech - Spring 2006 Class 618 Free Expansion

Thermo & Stat Mech - Spring 2006 Class 619 Isothermal Expansion

Thermo & Stat Mech - Spring 2006 Class 620 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 621 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,  S = 0 For the free expansion, there was no reservoir.

Thermo & Stat Mech - Spring 2006 Class 622 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.  S  0

Thermo & Stat Mech - Spring 2006 Class 623 First and Second Laws First Law:dU = đQ – đW First law, combined with the second law: (for reversible processes) dU = TdS – PdV