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 6

1

Thermodynamics and Statistical Mechanics

Entropy and the Second Law of Thermodynamics


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!


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!


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.


5



6


and '

' where,' 22 Q

W

Q

W

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.


7

For a Carnot Engine

TQTT

Q

Q

1

2

1

2

1

2

1

2

TT

QQ or 0

2

2

1

1 TQ

TQ


8

For infinitesimal cycles

02

2

1

1 TQd

TQd

0i i

i

TQd

Any cycle can be represented as a sum of infinitesimalCarnot cycles. Then,


9

Carnot Cycles


10

Entropy

0 TQd

TQd

i i

i

dSTQd

For reversible processes.Entropy is a state variable.


11

Carnot Cycle


12

Carnot Cycle on T-S Plot


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 = TdS for any reversible process.)

These two quantities are equal.


14

Irreversible “Carnot” Cycle

0

or Then

11 and

2

2

1

1

2

2

1

1

2

1

2

1

2

1

2

1

2

1

T

Q

T

Q

T

Q

T

Q

T

T

Q

Q

Q

Q

Q

Q

Q

Q


15

Clausius Inequality

0 so 02

2

1

1

T

Qd

T

Qd

T

Qd

Irreversible cycle

In general 0T

Qd


16

Entropy Change

T

QddS

The equal sign applies forreversible processes.


17

Free Expansion of a Gas


18

Free Expansion


19

Isothermal Expansion


20

Isothermal Expansion

Reversible route between same states.

f

iTQd

SđQ = đW + dU Since T is constant, dU = 0. Then, đQ = đW.

dVV

nRTPdVWd

VdV

nRVdV

TnRT

TWd

TQd

dS


21

Entropy Change

2ln2

ln2

nRVV

nRVdV

nRSV

V

The entropy of the gas increased.For the isothermal expansion, the entropy of theReservoir decreased by the same amount. So for the system plus reservoir, S = 0

For the free expansion, there was no reservoir.


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.

S 0


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 1 Thermodynamics and Statistical Mechanics Entropy and the Second Law of Thermodynamics.

Documents

thermo stat mech spring

best slide

gas slide

general slide

isothermal expansion

free expansion slide

ts plot slide

carnot engine