Lecture 18Second law of

thermodynamics. Carnot cycle. Absolute zero.

ACT: Leaving the fridge open

If you leave the door of your fridge open, you will get a heart-stopping electricity bill, but you will also:

A. Freeze the kitchen

B. Warm up the kitchen

H C C Q Q W Q

Now the fridge and the kitchen are one system. We are taking QC out of this system and dumping QH into it. Overall, heat is added!

Second law of thermodynamics

It is impossible for any process to have as its sole result the transfer of heat from a cooler to a hotter body (“refrigerator” or Clausius statement)

i.e.,

It is impossible to build a 100%-efficient heat engine (e = 1)

It is impossible for any system to undergo a process in which it absorbs heat form a reservoir at a single temperature and convert the heat completely into mechanical work, with the system ending in the same state as it began (“engine” or Kelvin-Plank statement)

Or

i.e.,

It is impossible to build a workless refrigerator (K ∞)

Clausius Kelvin-Plank

100%-efficient engineWorkless refrigerator

Kelvin-Plank Clausius

100%-efficient engine

Workless refrigerator

The Carnot cycle

Cycle made with reversible isothermal and adiabatic processes.

A

B

D

C

ACT: Carnot cycle

Where in the cycle is heat absorbed? 1

2

4

3

A. At point 1

B. Between 1 and 2

C. Between 4 and 1

To absorb heat, we need a process between two states. 1 is a state.

No heat transfer between 4 and 1 (adiabatic)

Between 1 and 2, temperature is alwaysTH

1 2

1 2 1 2

0

0

U

Q W

Heat exchange in Carnot’s cycle

1 2

21 2 1 2 H

1

0

ln 0

U

VQ W nRT

V

1

2

4

3

3 4

43 4 3 4 C

3

0

ln 0

U

VQ W nRT

V

1 1H 1 C 4T V T V 1 1

H 4 3

C 1 2

T V V

T V V

4 3

1 2

V V

V V1 1

H 2 C 3T V T V

2H

H 1 2 1

C 3 4 4C

3

ln

ln

VnRT

Q Q V

Q Q VnRT

V

32

1 4

VV

V V

H

C

T

T H H

C C

Q T

Q T

Carnot efficiency

H

We

Q C

H

1T

T C H

H

Q Q

Q

C

H

1Q

Q C

CarnotH

1T

eT

Carnot’s cycle is completely reversible. Run backwards, it is a Carnot refrigerator

CC C

H CH C

QQ TK

T TW Q Q

C

CarnotH C

TK

T T

In-class example: Carnot engine

A Carnot engine can operate between different sets of reservoirs.

1: TH = 80°C, TC = 10°C

2: TH = 10°C, TC = −50°C

3: TH = − 60°C, TC = −100°C

Rank their efficiencies (largest to smallest).

A. 1,2,3

B. 3,2,1

C. 3,1,2

D. 2,1,3

E. 2,3,1

1: TH = 353K, TC = 283K

2: TH = 283K, TC = 223K

3: TH = 213K, TC = 173KC

1

2831 0.198

353e C

CarnotH

1T

eT

2

2231 0.212

283e

3

1731 0.188

213e

Carnot is the ideal cycle

The Carnot cycle is completely reversible. So let’s use it as a refrigerator, and couple it to a hypothetical superengine with eSE > eCarnot.

Carnot Super engine

QH

QC

QH + Δ

WΔ

QC

TC

TH

Δ

Δ

TC

TH

Superengine produces more work than Carnot for these temperatures

Required to balance energy in SE

This is equivalent to this!

Impossi

ble

No engine can be more efficient than a Carnot engine operating between the same two temperatures.

Absolute zero

The colder you try to go, the less efficient the refrigerator gets:

Since heat leaks will not disappear as the object is cooled, you need more cooling power the colder it gets.The integral of the power required diverges as T 0.

Therefore you cannot cool a system to absolute zero

The best refrigerator you can get (Carnot) has performance

CCarnot

H C

TK

T T

C

CCarnot 0

H C

0T

TK

T T