AP3290_chapter_4_II_10

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Thermodynamics Y Y Shan

AP3290 58

To find the entropy change of this small 'universe', we add up the entropy changes

for its constituents: the surrounding room, and the ice+water.

0300273

>=+=K

dQ

K

dQdSdSdS

gssurroundinsystemuniverse

The total entropy change is positive; this is always true in spontaneous events in a

thermodynamic system and it shows the predictive importance of entropy: the final net

entropy after such an event is always greater than was the initial entropy.

Looking into another example: in a case of transfer heat of Q amount through a

system(gas or other working medium) from hotter reservoir at T1 to a cooler reservoir at T2,

the entropy change of the universe can be calculated:

Entropy change of the system: 0=system

S

Entropy change of the hot reservoir: entropyloosing,1

T

QS

reservoirhot

=

Entropy change of the hot reservoir: entropygainning,1

T

QS

reservoircold

+=

Entropy change of the Universe: ,0>universe

S that is

12

21

because,0 TTT

Q

T

QSSSS

chsu

++

=++=

When we apply the second law, its mathematical formulation of the entropy version can be

used:

0)( universeS

universe(isolated thermodynamic system) = a system + its surroundings. A

process can occur only if it increases the total entropy of the universe, or total entropy never

decreases.

Exercise: starting from this formulation of the second law, it can be deduced that the

maximum heat engine efficiency is Carnot efficiency:

H

C

Carnot

H

anyCarnotanyT

T

Q

W== 1,,

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2. Macroscopic viewpoint entropy: Disorder

For an isolated system, the natural course of events (i.e. spontaneous process) takes the

system to a more disordered state. This means that snapshots of a system taken at two

different times would show the state came later in time is more disordered, or the final state

is more disordered than the initial state.

Since entropy gives information about the evolution of an isolated system with time,

i.e., the total entropy increases. The final state has higher entropy that the initial state. Thus,

a more disordered state has higher entropy, in other words entropy measures disorder

And entropy is said to give us the direction of "time's arrow" .

Entropy as Time's Arrow:

One of the ideas

involved in the concept of entropy is that nature tends from order to disorder in isolated

systems. This tells us that the right hand box of molecules happened before the left.

"Disorder" should be defined if you are going to use it to understand entropy. A

more precise way to characterize entropy is to say that it is a measure of the "multiplicity"associated with the state of the objects. If a given state (like 7 dots below) can be

accomplished in many more ways, then it is more probabable than another state (like 2 dots)

which can be accomplished in only a few ways.

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For instance, when throwing a pair of dice, to get a state of 7-dots is more probable than to

get a state of 2-dots because you can produce seven in six different ways and there is only

one way to produce a two. The multiplicity for seven dots is six and the multiplicity for two

dots is just one. So seven-dots has a higher multiplicity than a two-dots, and we could say

that a seven represents higher "disorder" or higher entropy.

The water molecules in the glass of water can be arranged in many more ways; they have

greater "multiplicity" and therefore greater entropy.

Summary of entropy discussion

Like the concept of energy is central to the first law , which deals with the conservation of

energy. The concept of thermodynamic entropy is central to the second law , which deals

with physical processes and whether they occur spontaneously. Spontaneous changes occur

with an increase in entropy: 0)( universeS

Entropy:a state function, whose change is defined for a reversible process at

T where Q is the heat absorbed.Entropy: a measure of the amount of energy which is unavailable to do work.

Entropy: a measure of the disorder of a system.

Entropy: a measure of the multiplicity of a system.

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3. The fundamental thermodynamic relation

The first law for infinitesimal changes says the change of internal energy:

dWdQdU += .

With the introduction of the state function of entropy:

TdSdQ = , together with pdVdW = ,

an expression for internal energy can be formed :

PdVTdSdU = eqn4-3-1

This is called the fundamental thermodynamic relation. It involves only functions of state.

We can get the infinitesimal change of entropy from eqn4-3-1:

dVT

PdU

TdS +=

1 eqn4-3-2

4. Entropy of ideal gas

For processes with an ideal gas, the change in entropy from state A to state B can be

calculated from the relationship

Making use of eqn4-3-2, this can be written:

eqn4-3-3

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This is a useful calculation form if the T and V are known. Using the ideal gas law, you can

get the expression involving T, P. If you are working on a PV diagram it is preferable to

have it expressed in P,V terms. Using the ideal gas law nRT=PV

Then from eqn4-3-3

.

Since specific heatsare related by CP-CV= nR,

.

Since entropy is a state variable, just depending upon the beginning and end states, these

expressions can be used for any two points that can be put on one of the standard graphs.

5 TS diagram

(a) conjugate variables, generalizedforcesand generalized displacements

For a mechanical system, a small increment of energy, dW, is the product of a force,

Ftimes a small displacement dx .

FdxdW=

A very similar situation exists in thermodynamics. An increment in the energy of a

thermodynamic system can be expressed as the sum of the products of certain generalized

"forces"which, when imbalanced cause certain generalized "displacements"

TdSdQ

PdVdW

=

=

So that

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F-x(Force - displacement)

P-V(Pressure- Volume)

T-S(Temperature- Entropy)

are called pairs of Conjugate variables. Another important pair of conjugate variable is

Chem. potential / Particle number )( N .

Pressure P, temperature T, and chemical potential are called the generalized forces,

driving the changes in volume V, entropy S, and particle number N, respectively. V, S, and

N are called generalized displacements.

In thermodynamics, pairs of conjugate variablescan be used to express the internal

energy of a system (the fundamental thermodynamic relation) and other thermodynamic

potentials. For e.g. PdVTdSdU = .

(b) TS diagram

Like in PV-diagram, energy (work) is calculated from the pair of P-V conjugate variables

PdVdW = ,

+==1

2

2

1

2112 PdVPdVWWW

net,

In a TS-diagram, with entropy S as the horizontal axis and temperature T as the vertical axis,

energy (heat) can be calculated from the pair of T-S conjugate variables.

TdSdQ =

==1

2

2

1

2112 TdSTdSQQQ

net

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Any point on a TS graph, will represent a particular thermodynamic state. A thermodynamic

process will consist of a curve connecting an initial state (1) and a final state (2 ). The area

under the curve will be:

==2

1

2

1

12 TdSdQQ

which is the amount of heat energy transferred in the process. If the process moves to larger

entropy, the area under the curve will be the amount of heat absorbedby the system in that

process. If the process moves towards lesser entropy, it will be the amount ofheat removed.

The area inside the cycle will then be the net heat energy added to the system, but since the

internal energy, a state function, its integral over any closed loop is zero, i.e.

netnet

netnet

WQ

WQdUU

=

=+== 0

So the net heat (the area inside the loop on a T-S diagram) is equal to the net work performed

bythe system if the loop is traversed in a clockwise direction, and is equal to the netl work

doneonthe system as the loop is traversed in a counterclockwise direction.

Thermodynamic processes in PV diagram can be described in TS diagram

accordingly. For an isothermal process (T is a constant), it can be represented by a horizontal

line in TS diagram.

For an adiabatic process ( 0therefore,0 ===T

dQdSdQ ), entropy is constant. An

adiabatic process is equivalent to an isentropic process, which can be represented by a

vertical line in TS diagram

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Carnot cycle in PV diagram Carnot cycle in TS diagram

(c) Carnot cycle described in temperature-entropy (TS) diagram

The behavior of a Carnot engine or refrigerator can be best understood by using a TS

diagram

General engine cycle and Carnot cycle in TS diagram:

For a generalized thermodynamic cycle taking place between a hot reservoir at THand a cold

reservoir at TC, by the second law of thermodynamics, the cycle cannot extend outside the

temperature band from TCto TH.

The area in red QCis the amount of heat energy released to the cold reservoir.

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The area in white is the net heat, equal to the amount of work energy exchanged by the

system with its surroundings. If the system is behaving as an engine, the process moves

clockwise around the loop, and moves counter-clockwise if it is behaving as a refrigerator.

The efficiency of the cycle is the ratio of the white area (work) divided by the sum of

the white and red areas (total heat):

==

==

1

2

,S

S

C

S

S

H

H

CH

H

net

TdSQTdSQ

Q

QQ

Q

W

B

A

For a Carnot cycle, taking place between a hot reservoir at THand a cold reservoir at TC,

Evaluation of the above integral is particularly simple.

The total amount of heat energy absorbed from the hot reservoir is:

)(ABH

S

S

H SSTTdSQ

B

A

==

The total amount of heat energy released to the cold reservoir is:

)(BAC

S

S

C SSTTdSQ

B

A

==

The efficiency is defined to be:

H

C

ABH

ABCH

H

CH

H

net

CarnotT

T

SST

SSTT

Q

QQ

Q

W=

=

== 1

)(

))((

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AP3290_chapter_4_II_10