2 . 5 R e v e r s i b l e A d i a b a t i c P r o c e s s e s

A d i a b a t i c p r o c e s s q = 0 wwqU

F o r e x p a n s i o n 0w 012 UUU 0 T

c o m p r e s s i o n 0w 012 UUU 0 T i . e . T h e w o r k i s d o n e b y c o n s u m i n g i n t e r n a l e n e r g y o f t h e s y s t e m . C o n s i d e r o n l y o n e m o l e o f i d e a l g a s . wdTcdU v R e v e r s i b l e e x p a n s i o n PP op

PdVdTc v i d e a l g a s o n e m o l e V

RTP

dVV

RTdTc v

V

dVR

T

dTc v

F r o m s t a t e 1 s t a t e 2 2

1

1

2

V

VRl

T

Tlc nnv o r vc

R

V

V

T

T)()(

2

1

1

2

F o r i d e a l g a s Rcc vP , v

Pc

c 1

vcR

1

2

1

1

2 )(

V

V

T

T a n d )(

2

1

2

1

1

2

1

2

V

V

V

V

T

T

P

P o r constantPV

2 . 6 R e v e r s i b l e I s o t h e r m a l P r o c e s s o r V o l u m e C h a n g e s o f a n I d e a l G a s

wqdU I s o t h e r m a l 0dT

i d e a l g a s dTT

UdU v)(

0)(

TV

U

0dU dVV

RTPdVwq ( o n e m o l e o f i d e a l g a s )

F r o m s t a t e 1 s t a t e 2 :

)()(2

1

1

2

P

PRTl

V

VRTlqw nn

P V ＝ c o n s t a n t ( f o r i d e a l g a s a n d i s o t h e r m a l p r o c e s s ) 21 PP ad iabeticiso th erma l ww

Numerical Example 10 liters, monoatomic ideal gas, 25℃, 10 atm 1 atm

Rcv 2

3

Calculate w, q, U and H for (a) isothermal reversible expansion (b) adiabatic reversible expansion

Solution:

First calculate the number of moles of the gas (ideal gas)

molesKmoleKatmliter

literatmRTVP

na

aa 09.4)(298)(08206.0

)(10)(10

(a) isothermal reversible (a b)

0U 1

22

1

2

1 VV

nRTldVVnRT

PdVwq n

V

V

V

V

11

22

1

2

PRTPRT

VV TTT

21 2

1

1

2

PP

VV

kJlnPP

nRTlnwq 3.231

10298314.809.4

2

1 xxx

0U

02

1 TT pdTCnH (or 0)( 1122

VPVPUH )

(b ) a d ia b a tic re v e rs ib le (a c )

constantPV 3

5

23

25

R

R

c

c

v

p fo r id e a l g a s

liters 101 V atm 11 P atm 102 P

35

35

2 )10(101 V litersV 8.39)1010( 53

35

2

KnR

VPT 119

08206.009.4

8.391222

kJnRTTncdTncwU v

T

T v 12.9)298119(2

3)( 12

2

1

kJTTnRH 2.15)298119(314.85.209.4)(2

521

0q O th e r e x a m p le s in c lu d e : ( i) iso th e rm a l p ro c e s s fo llo w e d b y a c o n s ta n t v o lu m e p ro c e s s (a e c ) ( i i) c o n s ta n t v o lu m e p ro c e s s fo llo w e d b y a n iso th e rm a l p ro c e s s (a d c ) ( i i i) iso th e rm a l p ro c e s s fo llo w e d b y a c o n s ta n t p re s su re p ro c e s s (a b c ) ( iv ) c o n s ta n t v o lu m e p ro c e s s fo llo w e d b y a c o n s ta n t p re s su re p ro c e s s (a f c ) (v ) c o n s ta n t p re s su re p ro c e s s fo llo w e d b y a c o n s ta n t v o lu m e p ro c e s s (a g c )

kJU 12.9 kJH 2.15 i n d e p e n d e n t o f t h e p a t h s ( i ) – ( v )

I n e a c h o f t h e p a t h s ( i ) t o ( v ) , t h e h e a t a n d w o r k e f f e c t s d i f f e r , a l t h o u g h i n e a c h c a s e t h e d i f f e r e n c e q - w e q u a l s – 9 . 1 2 k J . ( i ) w ＝ a r e a a e i h q ＝ - 9 . 1 2 ＋ a r e a a e i h ( i i ) w ＝ a r e a d c i h q ＝ - 9 . 1 2 ＋ a r e a d c i h ( i i i ) w ＝ a r e a a b j h － a r e a a b j i q ＝ - 9 . 1 2 ＋ a r e a a b j h － a r e a c b j i ( i v ) w ＝ a r e a f c i h q ＝ - 9 . 1 2 ＋ a r e a f c i h ( v ) w ＝ a r e a a e i h q ＝ - 9 . 1 2 ＋ a r e a a g i h H o m e w o r k C h a p . 2 2 . 1 , 2 . 3 , 2 . 5 , 2 . 7 D u e 3 / 1 0

2.7 Introduction to the Second Law of thermodynamics Reversible and Irreversible Processes Spontaneous or Natural Processes

－If the initial state of the system is not the equilibrium state, the system will spontaneously move toward its equilibrium state.

－A process which involves the spontaneous movement of a system from a non equilibrium state to an equilibrium state is called a natural or spontaneous process.

－The reverse process of a spontaneous process will never occur spontaneously.

Entropy－quantification of irreversibility Consider a weight-pulley-heat reservoir

Lewis and Randall considered three processes:

(1) Heat reservoir at temperature T2. The weight is allowed to fall, performing work w, and the heat produced q enters the heat reservoir.

(2) Heat reservoir at temperature T2 placed in thermal contact with a heat reservoir at a lower temperature T1, and the same heat q is allowed to flow from T2 reservoir to T1 reservoir.

(3) Heat reservoir at temperature T1. The weight is allowed to fall, performing work w, and the heat produced q enters the reservoir.

Figure 3.1 A weight-pulley-heat reservoir arrange-ment in which the work done by the falling weight is degraded toheat, which appears in the heat reservoir.

Each of these processes is spontaneous and hence irreversible, and degradation occurs in each of them. However, as process (3) is the sum of the processes (1) and (2), the degradation occurring in process (3) must be greater than the degradation in each of the processes (1) and (2). Thus it can be said that process (3) is more irreversible than either process (1) or process (2). How to quantify the irreversibility? Both the amount of heat produced q, and the temperatures between which this heat flows are important in defining a quantitative scale of irreversibility.

The quantity Tq is thus taken as being a measure of the degree of irreversibility of the

process, and the value of Tq is called the increase in entropy, S.

i.e. the entropy produced by the system S is given by Tq .

Figure 3.2 A thermostalled piston and cylinder containing water and water vapor.

2.8 An Illustration of Reversible and Irreversible Processes

Would it be possible for the process to be conducted that the degree of irreversibility is minimized, or ultimately, zero? Take water evaporation / condensation as an example: The system is at equilibrium when )(

2TPP OHext and when the temperature of the water

and water vapor in the cylinder equals the temperature T of the heat reservoir. If PPP extext (pressure decreases suddenly) piston pushes rapidly out of the cylinder water vapor pressure OHP 2

decreases due to volume expansion water spontaneously evaporates to re-establish equilibrium between water and its vapor

water temperature decreases due to endothermic evaporation; heat flows spontaneously from reservoir to keep the temperature constant

If, when 1 mole of water has evaporated, extP is instantaneously increased to the original value, evaporation ceases, the flow of heat ceases, and complete equilibrium is re-established

The work done by the system during this process equals )( PPext V, where V is the molar volume of water vapor at )(

2TP OH .

Similar but opposite process occurs when the pressure is increased to PPext .

If the magnitude of extP is decreased by an infinitesimal amount P , the resulting minute imbalance

between the pressures acting on the piston causes the cylinder to move slowly out of the piston. The slow expansion of the water vapor decreases its pressure, and when the pressure has fallen by an

infinitesimal amount below the saturation value, evaporation of the water begins. The evaporation sets up an infinitesimal temperature gradient between the heat reservoir and the cylinder,

down which flows the required latent heat of evaporation of the water. The smaller the value of P , then the slower the process, the smaller the degree of unstauration of the water vapor, and the smaller the temperature gradient. The more slowly the process is carried out, then the greater the opportunity afforded to the evaporation and

heat flow to keep up with equilibrium. If, after evaporation of 1 mole of water, the external pressure is instantaneously increased to its original value

extP , then the work done by the system equals VPPext )( . If the external pressure is then increased by

P , then work VPPext )( , is done on the system to condense 1 mole of water vapor, and the permanent

change in the external agency equals the work done on the system minus work done by the system during the cyclic process. The smaller the value of P , the more nearly equal are the two work terms, and in the limit that they are equal, no permanent change occurs in the external agency, and hence the cyclic process has been conducted

reversibly.

2.9 Entropy and Reversible Heat Consider only the evaporation process The work done by the system during the evaporation of 1 mole is seen to have its

maximum value )0( P VPw extmax , and the maximum amount of heat, revq , is transferred from the heat reservoir to the cylinder, when the process is conducted

reversibly. maxwUqrev

Any irreversible evaporation process performs less work, VPPw ext )(

and less heat q is transferred from the reservoir to the cylinder, wUq

The difference between reversible process and irreversible process is )()( max wwqqrev

If evaporation is conducted reversibly, the change in the entropy of heat reservoir

is T

qS revreservoir , and the change in the entropy of water and water vapor in

the cylinder is T

qS rev

vaporwater

The total change of entropy is thus 0 vaporwaterreservoirtotal SSS If evaporation is conducted irreversibly, the change in the entropy of heat reservoir is

T

qS reservoir , while the total heat entering the cylinder equals the heat q transferred

from the reservoir plus the heat that is produced by degradation of work, i.e. )()( max qqww rev

the change in the entropy of the cylinder is T

q

T

qq

T

qS revrev

vaporwater

The total change of entropy is 0

T

qq

T

q

T

qS revrevtotal

totalS increases if the process is carried out irreversib ly. (Entropy is created as a result of the irreversible process.)

I t i s i m p o r t a n t t o n o t i c e t h a t , n o m a t t e r w h e t h e r t h e p r o c e s s i s c o n d u c t e d r e v e r s i b l y o r i r r e v e r s i b l y , t h e c h a n g e s o f t h e e n t r o p y o f t h e w a t e r a n d w a t e r v a p o r a r e t h e s a m e .

T h e d i f f e r e n c e i n e n t r o p y b e t w e e n t h e f i n a l a n d i n i t i a l s t a t e s i s i n d e p e n d e n t o f w h e t h e r

t h e p r o c e s s i s c o n d u c t e d r e v e r s i b l y o r i r r e v e r s i b l y .

i . e . E n t r o p y i s a s t a t e f u n c t i o n . I n g o i n g f r o m s t a t e A t o s t a t e B .

T

qS

T

qSSS rev

irrAB

T h e c h a n g e i n e n t r o p y c a n b e d e t e r m i n e d b y t h e m e a s u r e m e n t o f h e a t f l o w f o r r e v e r s i b l e p r o c e s s e s .