clausius inequality

8/9/2019 Clausius Inequality

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The Clausius Inequality

Expressions of inequality/equality relating toheat ow at a xed temperature.

The expression is required for the derivationof an equation for entropy which is our nextmain topic.

erived from a !thought experiment" using#arnot engines acting in a series.


2/13

$olume

%ressure

a

b

d

T1

Q1

Carnot Cycle

Q2

V

nRTP

1=

V

constP

.=

T2c

Q=0Q=0

V

nRTP

2=


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Heat Flows in a Carnot Cycle

&ot 'eservoir( T)

#old 'eservoir( T*

CW

Q1

Q2

2

1

2

1=

Q

Q

T

T

2

1

21= T

T

QQ


4/13

+uch that 02

2

1

1=

T

Q

T

Q

,ne could also consider the small amount of

reversi-leheat owQrevthat ows at atemperature Tat each point in the cycle. The netheat ow is equal to the sum of the dierentialows

02

2

1

1 =

+=

T

Q

T

Q

T

Q

cycle

rev

0or a #arnot cycle( some of the heat into the cycleis converted to wor1 so that Q) 2 Q*. 3e also

1now that2

2

1

1

T

Q

T

Q=

0rom the denition of an integral( we nd for theentire cycle that

0= T

Q rev

4ut is this generally true for anycycle5


5/13

erivation of the #lausius 6nequality

Principal reservoir T~

C1

T1

Auxiliary

reservoirsW1

Q1

Q1

T~

11

~

T

TQ

1

1

~

T

TQQ

~

Q~

Working

subsance in

a cycle

1 2


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C1

T1

W1

Q1

Q1

T~

11

~

T

TQ

11

~

T

TQ

Working

subsance in

a cycle

1 2

W2

Q2

Q2

C2

T2

T~

!

22

~

T

TQ

2

2

~

T

TQ

Add NCarno

engines in oalCo"posi

e

#yse"


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nalysis of Composite Device

The total heat supplied -y the reservoir to

the composite device in a cycle with Nengines is

6n an entire cycle( Ufor the wor1ingsu-stance and the composite device is 7.Then the 0irst 8aw says

U9 7 9 W: Q

;n important conclusion is thatQ

9


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%rincipalreservoir

T~

#ompositesystem

i

Q

T

TQ

1

~

= i

$

iWW =i

$

Q= - W

Q = -W2 7

$iolates =elvinstatement ofthe *nd 8aw>

3hat can-eallowed5


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#ompositesystem

Q

W

?9 < 3 @7

;llowed


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Co"posie

syse"

Q 9 < W9 7

Q

W

;llowed

%#u" o& 've and

(ve Qand W)


11/13

Results Allowed by SecondLaw

0~

=

1=N

i i

i

T

QTQ

Q@ 0or ? 9 7( thus

0~T4ut * so we can divide -yT~

0

1N

i i

i

T

Q

=

6n the limit of smallQi* we can integrate over the

entire cycle

0

oT

dQ

where Tois the temp. of the reservoir Aexternal heat

sourceB and the circle represents integration over the

entire cycle.

ClausiusInequality


12/13

Reversible Cycles6f the cycle at the centre of the compositedevice is reversible( then it can -e run in

reverse.The opposite result is then o-tained. 6n thecomposite device( wor1 is done ,C the system( andheat is given ,00 to the reservoir.

T~

Q

W

;llowed

7 Q9 < W


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Reversible Cycles

Thus( there are restrictions on Qwhen the cycle isoperated in reverse.

4ut the cycle can still also -e operated in theforward direction such that

0~

== 1=

N

i i

i

T

QTQW

,nly one solution satises -oth requirementsfor forward and reverse cycles

0=~

= 1=

N

i i

i

T

QTQ

6ntegrating

6n a reversi-le process( the temperature of thesystem and reservoir are equal T.

0= TdQR

ClausiusEquality

clausius inequality

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