PHY203: Thermal Physics

Topic 4: Second Law of Thermodynamics

• Heat Engines

• Statements of the Second Law (Kelvin, Clausius)

• Carnot Cycle

• Efficiency of a Carnot engine

• Carnot’s theorem

• Introduction to idea of entropy

• Entropy as a function of state

• Entropy form of second law

• Example calculations

Heat Engines

•Heat engines operate in a cycle, converting heat to work then returning to original state at end of cycle.

•A gun (for example) converts heat to work but isn’t a heat engine because it doesn’t operate in a cycle.

•In each cycle the engine takes in heat Q1 from a “hot reservoir”, converts some of it into work W, then dumps the remaining heat (Q2) into a “cold reservoir”

HOT

COLD

Engine

Q1

Q2

W

HOT

COLD

Engine

Q1

Q2

W

Efficiency of a heat engine

Definition:

cycleper input heat cycleper done work

efficiency

1QW

•Because engine returns to original state at the end of each cycle, U(cycle) = 0, so W = Q1 - Q2

•Thus:12

11

21QQ

QQQ

Efficiency of a heat engine

•According to the first law of thermodynamics (energy conservation) you can (in principle) make a 100% efficient heat engine.

BUT………….

•The second law of thermodynamics says you can’t:

Kelvin Statement of Second Law:

“No process is possible whose SOLE RESULT is the complete conversion of heat into work”

 William Thomson, Lord Kelvin  (1824-1907)

HOTCOLD

HOTTERCOLDER

WARMWARM

Q

HOTCOLDQ

Both processes opposite are perfectly OK according to First Law (energy conservation)

But we know only one of them would really happen – Second Law

Heat flow

Clausius Statement of Second Law:

“No process is possible whose SOLE RESULT is the net transfer of heat from an object at temperature T1 to another object at temperature T2, if T2 > T1”

Rudolf Clausius (1822-1888)

How to design a “perfect” heat engine

1)Don’t waste any workSo make sure engine operates reversibly (always equilibrium conditions, and no friction).

2)Don’t waste any heatSo make sure no heat is used up changing the temperature of the engine or working substance, ie ensure heat input/output takes place isothermally

 Sadi Carnot (1796-1832)

HotSource

T1 Q1

Piston

Gas

a

b

Working substance (gas) expands isothermally at temperature T1, absorbing heat Q1 from hot source.

The Carnot Cycle (I): isothermal expansion

T1

P

V

T1T1

a

b

Q1Q1

The Carnot Cycle (II): adiabatic expansion

Gas isolated from hot source, expands adiabatically and temperature falls from T1 to T2.

Piston

Gas

b

c

Gas isolated from hot source, expands adiabatically, and temperature falls from T1 to T2

P

V

T1T1

a

b

T2T2c

The Carnot Cycle (III): isothermal compression

3)Gas is compressed isothermally at temperature T2 expelling heatQ2 to cold sink.

Piston

Gas

d

c

Q2

ColdSinkT2

Gas is compressed isothermally at temperature T2 expelling heat Q2 to cold sink.

T2

V

P

T1

a

b

T2cd

P

T1T1

a

b

T2T2cd

Q2Q2

The Carnot Cycle (IV): adiabatic compression

Gas is compressed adiabatically, temperature rises from T2 to T1 and the piston is returned to its original position. The work done per cycle is the shaded area.

Piston

Gas

a

d

Gas is compressed adiabatically, temperature rises from T2 to T1 and the piston is returned to its original position. Work done is the shaded area.

V

P

T1

a

b

T2cd

W

Efficiency of ideal gas Carnot engine

12

11

21QQ

QQQ

•We can calculate the efficiency using our knowledge of the properties of ideal gases

V

P

T1

a

b

T2cd

W

Q2

Q1

V

P

T1

a

b

T2cd

W

V

P

T1T1

a

b

T2T2cd

WW

Q2Q2

Q1Q1

a

bab V

VnRTWQ ln11

Isothermal expansion (ideal gas)

P

V

T1T1

a

b

Q1Q1

Va Vb

Isothermal compression (ideal gas)

d

ccd V

VnRTWQ ln22

V

P

T1

a

b

T2cd

Q2

V

P

T1

a

b

T2cd

Q2

V

P

T1

a

b

T2cd

P

T1T1

a

b

T2T2cd

Q2Q2

a

b

d

c

VV

nRT

VV

nRT

QQ

ln

ln11

1

2

1

2

(2)

(1) 1

21

1

12

11

da

cb

VTVT

VTVT

Adiabatic processes

)2()1( d

c

a

b

VV

VV

2

2

1

1

1

21

TQ

TQ

TT

Efficiency of ideal gas Carnot engine

V

P

T1

a

b

T2cd

W

Q2

Q1

V

P

T1

a

b

T2cd

W

V

P

T1T1

a

b

T2T2cd

WW

Q2Q2

Q1Q1

Can you do better than a Carnot?

HOT

COLD

Carnot

Q1

Q2

W“SuperCarnot”

Q3

Q4

W

1QW

c 3Q

Wsc

31 QQcsc

HOT

COLD

Engine

Q1

Q2

W

HOT

COLD

Engine

Q1

Q2

W

A Carnot engine is reversible……..

…..so you can drive it backwards

HOT

COLD

Carnot

Q1

Q2

W “SuperCarnot”

Q3

Q4

•Drive Carnot backwards with work output from “Super Carnot”

•Heat leaving hot reservoir =Q3 – Q1, which is negative

•So, net heat enters the hot reservoir

•Since the composite engine is an isolated system, this heat can only have come from the cold reservoir

•Net result, transfer of heat from cold body to hot body, FORBIDDEN BY CLAUSIUS STATEMENT OF SECOND LAW

HOT

COLD

Carnot

Q1

Q2

W “SuperCarnot”

Q3

Q4

•Carnot’s Theorem:

“No heat engine operating between a hot (Th) and a cold (Tc) reservoir can be more efficient than a Carnot engine operating between reservoirs at the same temperatures”

•It follows, by exactly the same argument, that ALL Carnot engines operating between reservoirs at same Tc, Th, are equally efficient (ie independent of “working substance”) – our ideal gas result holds for all Carnot Cycles

So……..

h

c

TT

1

0h

h

c

c

h

h

c

c

TQ

TQ

TQ

TQ Conservation

of “Q/T”

For all Carnot Cycles, the following results hold:

What about more general cases?????

0h

h

c

c

h

h

c

c

TQ

TQ

TQ

TQ

Was derived from expressions for efficiency, where only the magnitude of the heat input/output matters. If we now adopt the convention that heat input is positive, and heat output is negative we have:

0h

h

c

c

TQ

TQ

The expression

Arbitrary reversible cycle can be built up from tiny Carnot cycles (CCs)

Q1

Q3

Q2

Q4

For 2 CCs shown:

04

4

3

3

2

2

1

1

TQ

TQ

TQ

TQ

For whole reversible cycle:

0n n

n

TQ

In infinitesimal limit, Q→dQ, → ,fit to cycle becomes exact:

0

cycle

TdQ For any reversible cycle:

Arbitrary reversible cycle

Entropy

To emphasise the fact that the relationship we have just derived is true for reversible processes only, we write:

0cycle

rev

TdQ

We now introduce a new quantity, called ENTROPY (S)

TdQ

dS rev

Entropy is conserved for a reversible cycle

Is entropy a function of state?

P

V

A

B

path1

path2

For whole cycle:

0 cycle

rev

TdQ

S

2 path1 path

2 path1 path

0

B

A

revB

A

rev

A

B

revB

A

rev

TdQ

TdQ

TdQ

TdQ

ABBABA SSSS 2) (path1) (path

Entropy change is path independent → entropy is a thermodynamic function of state

Reversible cycle

Example calculation:

Calculate the entropy change of a 10g ice cube at an initial temperature of -10°C, when it is reversibly heated to completely form liquid water at 0°C……..

Specific heat capacity of ice = 2090 J kg-1K-1

Specific latent heat of fusion for water = 3.3105 J kg-1

Irreversible processes

Carnot Engine

h

c

revh

cc T

TQ

Q

11

Irreversible Engine

h

c

irrevh

c

TT

Q

Q

11irrev

For irreversible case:

h

c

irrevh

c

h

c

irrevh

c

TT

Q

Q

TT

Q

Q

0

irrevc

c

irrevh

h

TQ

TQ

h

h

irrevc

c

TQ

T

Q

Irreversible processes

Following similar argument to that for arbitrary reversible cycle:

0cycle

irrev

TdQ

P

V

A

B

Path 1(irreversible)

Path 2(reversible)

For irreversible cycle

0)(

)(

)(

)(

rev

rev

irrev

irrev

A

B

rev

B

A

irrev

TdQ

TdQ

)(

)(

)(

)(

rev

rev

irrev

irrev

B

A

rev

B

A

irrev

TdQ

TdQ

Irreversible cycle

Irreversible processes

)(

)(

)(

)(

rev

rev

irrev

irrev

B

A

rev

B

A

irrev

TdQ

TdQ

)(

)(

)(

)(

rev

rev

irrev

irrev

B

A

B

A

irrev dST

dQ

BAAB

B

A

SSSdSrev

rev

)(

)(T

dQdS irrev

T

dQdS

Equality holds for reversible change, inequality holds for irreversible change

General Case

“Entropy statement” of Second Law

We have shown that:

T

dQdS

For a thermally isolated (or completely isolated) system, dQ = 0

0dS

“The entropy of an isolated system cannot decrease”

What is an “isolated system”

The Universe itself is the ultimate “isolated system”, so you sometimes see the second law written:

“The entropy of the Universe cannot decrease”

(but it can, in principle, stay the same (for a reversible process))

It’s usually a sufficiently good enough approximation to assume that a given system, together with its immediate surroundings constitute our “isolated system” (or universe)………

Entropy changes: a summary For a reversible cycle:S(system) = S(surroundings) = 0S(universe) = S(system) + S(surroundings) = 0

For a reversible change of state (A→B):S(system) = -S(surroundings) = not necessarily 0S(universe) = S(system) + S(surroundings) = 0

For an irreversible cycle S(system) = 0; S(surroundings) > 0

For a irreversible change of state (A→B):S(system) ≠ - S(surroundings) S(universe) = S(system) + S(surroundings) > 0

Example: entropy changes in a Carnot Cycle

Entropy calculations for irreversible processes

•Suppose we add or remove heat in an irreversible way to our system, changing state from A to B

•At first sight we might think it’s a problem to use the relation:

)(

)(

rev

rev

B

A

revBA T

dQS

•However, because S is a function of state, S for the change of state must be the same for all paths, reversible or not.

•So, we can “pretend” the heat required to produce the given change of state was added or removed reversibly and use the formula anyway!

•BUT… the entropy change of the surroundings must be different for the reversible (S(universe)=0) and irreversible (S(universe)>0) cases.

Entropy calculations for adiabatic processes

•Here, we have no “dQ” term at all, so where do we start with the calculation………..

•To calculate the entropy change of the system we can “invent” a non-adiabatic process that takes the system between the same 2 states as our actual, adiabatic process and use that “dQ” to do the calculation.

•Again we rely on the path independence of S

Example: Joule expansion of an ideal gas: (irreversible, adiabatic change of state)

Before After

Rigid, adiabatic wall

GASVi

GASVf

Before After

Rigid, adiabatic wall

GASVi

GASVf

Joule expansion of an ideal gas

•For this process, U(gas)=0

•For ideal gas, U is a function of T only, so T = 0

•So, our “model” process is a reversible, isothermal expansion from Vi to Vf (S(gas) = nRln(Vf/Vi), see Carnot cycle calculation)

•But, for Joule expansion, S(universe) = S(gas): gas is an isolated system

•For reversible isothermal expansion, S(surroundings) =-nRln(Vf/Vi), S(universe) = 0

Some past (part) exam questions….

1)A copper block of heat capacity 150J/K, at an initial temperature of 60°C is placed in a lake at a temperature of 10°C. Calculate the entropy change of the universe as a result of this process

2)A thermally insulated 20 resistor carries a current of 10A for 1 second. The initial temperature of the resistor is 10C, its mass is 5x10-3kg, and its heat capacity is 850 Jkg-1K-1. Calculate the entropy change in a) the resistor and b) the universe.