Steps of the Carnot Cycle animationhttp://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/carnot.htm

Details on the Carnot Cycle

The isothermal expansion (ab) and compression (cd):

0isothermalU (T is constant and U(T) is a function of T only for an Ideal Gas.)

ln bH ab H

a

VQ W nRT

V

(ab : isothermal expansion)

ln ln ( )dC cd C C c d

c

c

d

V VQ W nRT nRT V V

V V

(cd : isothermal compression)

Details on the Carnot Cycle

(by definition)

The adiabatic expansion (bc) and compression (da):

0adiabaticQ

From Section 19.8, we learned that 1 for adiabatic processes.TV const

(bc: adiabatic expansion)1 1H b C cT V T V

(da: adiabatic compression)1 1H a C dT V T V

Dividing these two equations gives,

b c

a d

V V

V V

Efficiency of the Carnot Cycle

From definition, we have 1 C

H

Qe

Q

Using our results for QC and QH from the isothermal processes,

ln ln1 1

ln lnC c d C c d

H b a H b a

nRT V V T V Ve

nRT V V T V V

Then, from the adiabatic processes, we have

1 1C

H

C

H

Te

Q T

Q

b a c dV V V V

ln1

lnc d

b a

V V

V V

(Carnot Cycle only)

(T must be in K)

Efficiency of the Carnot Cycle

1 Ccarnot

H

Te

T (Carnot Cycle)

General Comments: Higher efficiency if either TC is lower and/or TH is higher. For any realistic thermal process, the cold reservoir is far

above absolute zero, i.e., TC > 0. Thus, a realistic e is strictly less than 1! (No 100%

efficient heat engine) Realistic heat engines must take in energy from the high T

reservoir for the work that it produces AND some heat energy must be released back to the lower T reservoir.

(Kelvin-Planck’s Statement)skip

Internal Combustion EngineThe Otto CycleA fuel vapor can be compressed, then detonated to rebound the cylinder, doing useful work.

animationhttp://auto.howstuffworks.com/engine1.htm

The Otto Cycle

a b b, c c d

2 ( ) 0H V c bQ U nC T T

4 ( ) 0C V a dQ U nC T T

r is the compression ratio (8 to 13)

intakeexhaust

intake exhaust

For the two constant V processes: 2 and 4, we can calculate,

The Otto Cycle

Applying the definition of efficiency,

1 1 1V a dC a d

H V c b c b

nC T TQ T Te

Q nC T T T T

Now, we can utilize the two adiabatic processes: 1 and 3,

1 1a a b bT V T V and 1 1

c c d dT V T V

1 1a bT rV T V

11c dT V T rV

1a bT r T 1

c dT T r

The Otto Cycle

Substituting Tb and Tc into the efficiency equation, we have,

1 1 11 1 1a d a

d

a

c b da

d d

a

T T T T T Te

T T T r T r T T r

1

11e

r

Using a typical value for the compression ratio r = 8 and = 1.40 gives,

0.56 (or 56%)e Note: This is a theoretical value. Realistic gasoline engine typically has e ~ 35%.

The Diesel Cycle

Key difference: No fuel in cylinder at the beginning of the

compression stroke (process 1) Fuel is injected only moments before

ignition in the power stroke No fuel until the end of the adiabatic

compression can avoid pre-ignition Compression ratio r value can be higher (15

to 20) Higher temperature can be reached during

the adiabatic compression Higher e and no need for spark plugs

Dsteam engine

EntropyRecall from a Carnot Cycle, we have derived the following relationship:

C C

H H

Q T

Q T 0H C

H C

Q Q

T T

Formally, we can rewrite this as,

0cycle

Q

T (We have absorbed the explicit

sign back into the variable Q.)

where Q represents the heat absorbed/released along the isotherm at temp T.

Entropy

Any reversible cycles can be approximated as a series of Carnot cycles !

Now consider any reversible cycles…

Entropy

This suggests that the following generalization to be true for any reversible cycles,

0r

cycle

dQ

T

where, dQr is the infinitesimal heat absorbed/released by the system at an

infinitesimal reversible step at temp T.

denotes the integration evaluated over one complete cycle.cycle

Entropy

We have seen this property previously,

0cycle

dU Changes in the internal energy U over a closed cycle is zero!

This is a consequence of the fact that U is a state variable and dU for any processes depends on the initial and final states only.

Thus the result indicates that there is another state variable Ssuch that,

0r

cycle

dQ

T

and 0cycle

dS dQdS

T

This new state variable S is called the entropy of the system.

From our previous derivation, the quantity was from the isothermal branch of the infinitesimal Carnot cycle. Let look at an isothermal expansion of an ideal gas microscopically:

Entropy: DisorderRecall that the 2nd Law of Thermodynamics is a statement on nature’s preferential direction for systems to move toward the state of disorder.Let see how Entropy is a quantitative measure of disorder.

Intuitively, as the gas expands into a bigger volume, the degree of randomness for the system increases since molecules now have more choices (spaces) for them to move around. One can associate the increase in randomness to the ratio:

V dVor

V V

/d Q T

(T stays the same avg. KE stays the same)

Entropy: DisorderSince this is an isothermal process, we have the following relation from the 1st Law:

dQ dW 1 dQ dV

T VnR

So, the newly introduced macroscopic variable S (entropy),

dQdS

T

is a quantitative measure of the degree of disorder of the system.

f

i

dQS

T

/S J K

dS is an infinitesimal entropy change for a reversible process at temperature T. For any finite reversible process, the total entropy change S is,

0dU

nRTPdV dV

V

Entropy Changes for Different Processes

1. General Reversible Processes:

V

i

f

P

f f

r

i i

dQS dS

T

NOTE: in most applications, it is the change in entropy Swhich one typically needs to calculate and not S itself.

Note: S is a state variable, S is the same for all processes (including irreversible ones) with the same initial and final states!

Entropy Changes for Different Processes

2. Reversible Cycles:

0rcycle

cycle cycle

dQS ds

T

( , ) ( , )i i f fT V T V

1st Law gives,

r

r V

r V

dU dQ dW

dQ dU dW nC dT PdV

nRTdQ nC dT dV

V

(Note: Thru the Ideal Gas Law, Pis fixed for a given pair of T & V.)

3. Any Reversible Processes for an Idea Gas:

Entropy Changes for Different Processes

Dividing T on both sides and integrating,

f fVr

i i

nC dTdQ dVS nR

T T V

We have,

ln lnf fV

i i

T VS nC nR

T V

Entropy: Disorder (General Reversible Process)

In our derivation of for an ideal gas through a general reversible process, we just derived the following relation,

Vr nC dTdQ dVdS nR

T T V

S

dTS

T

Thermal agitations

dVS

V

Availability of space

4. Calorimetric Changes:

Entropy Changes for Different Processes

f f

i i

dQ mcdT

dQ mcdTS

T T

If c is constant within temperature range, ln f

i

TS mc

T

f

i

c T dTS m

T If is a function of T, c T

Entropy Changes for Different Processes

5. During Phase Changes (or other isothermal Processes):

1dQS dQ

T T

(T stays constant during a phase change.)

Q mLS

T T

Entropy Changes for Different Processes6. Irreversible Processes:

Although for a given irreversible process, we cannot write dS = dQr/T, S between a well defined initial state a and final state b can still be calculated using a surrogate reversible process connecting a and b. (S is a state variable!)

Example 20.8: (adiabatic free expansion of an ideal gas)

Initial State a: (V,T) Final State b: (2V,T)

Since Q=W=0, U=0.For an ideal gas, this means that T=0 also.

Although Q=0, but S is not zero!

,V T

2 ,V T

,V T

2 ,V T

S in a Adiabatic Free ExpansionImportant point: Since S is a state variable, S is the same for any processes connecting the same initial a and final b states.

In this case, since T does not change, we can use an surrogate isothermal process to take the ideal gas from state a (V,T) to state b (2V,T) to calculate S.

Applying our general formula to the surrogate isothermal expansion,

ln lnf fV

i i

T VS nC nR

T V

we have,

lnV

TS nC

T

2ln ln 2 5.76 /

VnR nR J K

V (n=1)

Surrogate Isothermal Expansion

,V T

2 ,V T

2nd Law (Quantitative Form)

“The total entropy (disorder) of an isolated system in any processes can never decrease.”

0

( 0 ; 0 )

tot sys env

tot tot

S S S

S reversible S irreversible

“Nature always tends toward the macrostate with the highest S(disorder) [most probable] in any processes.”

system

environmentQ

W