1a Fundamental Equations1

8/18/2019 1a Fundamental Equations1

1/39

FUNDAMENTAL EQUATIONS

Chapter 1


2/39

HOMOGENEOUS CLOSED SYSTEM

(Q and/or W

(Q and/or W


3/39


4/39

• A !"o#ed #$#te% one that doe# note'!hane %atter )&th &t# #*rro*nd&n#+

a"tho*h &t %a$ e'!hane ener$,

dn& - .(& - 1+ + 0,, (1,1

• No &nterna" ener$ tran#ported a!ro##o*ndar$,

• A"" ener$ e'!hane et)een a !"o#ed#$#te% and &t# #*rro*nd&n# appear# a#

heat and )or2,

• The tota" ener$ !hane o3 the#*rro*nd&n# e4*a"# the net ener$

tran#3erred to or 3ro% &t a# heat and


5/39

t and #e!ond "a)# o3 Ther%od$na%&!#5

Pd TdSdU

(1,

or re6er#&"e pro!e##5

7dTdSdU

(1,9

hT dS - dQre6 5 heat a#ored $ the #$#te%

7 d8 - dWre6 5 )or2 done $ the #$#te%

he &ntera!t&on o!!*r# &rre6er#&"$5

Pd TdSdU

(1,;


6/39

The &nterna" ener$ !hane !an e!a"!*"ated $ &nterat&n e4, (1,5

∫

1

1

1

V

V

S

S

Pd TdSUUU (1,


7/39

De>n&t&on5

(1,?&@erent&at&n e4, (1,? $&e"d#5

8d7d8dUdH

8d7d87d8TdSdH

8dTdSdH

Co%&n&n the ao6e e4*at&on )&th e4,(1,9 "ead# to

a !"o#ed #$#te% at !on#tant S and 7 5

.dH7+S ≤

(1,

(1,B

ENTHAL7Y


8/39

the He"%ho"t 3ree ener$ (A ather%od$na%&! potent&a" that %ea#*re#the *#e3*" )or2 ota&na"e 3ro% a

!"o#ed #$#te% at a !on#tant te%perat*reand 6o"*%e,

∆A - the %a'&%*% a%o*nt o3 )or2

e'tra!ta"e 3ro% a ther%od$na%&!pro!e## &n )h&!h te%perat*re and6o"*%e are he"d !on#tant,

Under the#e !ond&t&on#+ &t %&n&%&ed at

LMHOLT FEE ENEGY


9/39

De>n&t&on5

TUA

(1,1.&@erent&at&n e4, (1,1. $&e"d#5

SdTdSdUdA

SdTdS7d8TdSdA

7dSdTdA

Co%&n&n the ao6e e4*at&on )&th e4,(1,9 "ead# to

a !"o#ed #$#te% at !on#tant T and 8 5

.dA 8+T ≤

(1,11

(1,1


10/39

IS FEE ENEGY

G&# 3ree ener$ (G ather%od$na%&! potent&a" that %ea#*re#the J*#e3*"J or pro!e##:&n&t&at&n )or2ota&na"e 3ro% a ther%od$na%&! #$#te%at a !on#tant te%perat*re and pre##*re(other%a"+ oar&!,

The G&# 3ree ener$ the %a'&%*%a%o*nt o3 non:e'pan#&on )or2 that !ane e'tra!ted 3ro% a !"o#ed #$#te%K th

%a'&%*% !an e atta&ned on"$ &n a


11/39

De>n&t&on5

THG (1,19&@erent&at&n e4, (1,19 $&e"d#5

SSTHG

dSdST8d7dSTdG

Sd8d7dG

Co%&n&n the ao6e e4*at&on )&th e4,(1, "ead# to

a !"o#ed #$#te% at !on#tant 7 and T 5

.dG 7+T ≤

(1,1;

(1,1


12/39

AWELL EQUATION

- F('+ $+ the tota" d&@erent&a" o3 F

d$

Fd'

'

FdF

'$

∂

∂

∂

∂

=

NdMd'dF

$'FM

∂

∂=

'$

FN

∂

∂

=

)&th

(1,1=


13/39

rther d&@erent&at&on $&e"d#

$'

F

$

M

' ∂∂=

∂

∂

$'

F

'

N

$ ∂

∂

=

∂

∂ $' '

N

$

M

∂

∂

=

∂

∂

Hen!e 3ro% e4*at&on5 NdMd'dF

)e ota&n5

$' '

N

$

M

∂

∂

=

∂

∂

(1,1?


14/39

e#*%e5

7dTdSdU

(1,9

8dTdSdH (1,

7dSdTdA (1,1

1

Sd8d7dG (1,1

;

NdMd'dF

$' 'N

$M

∂

∂=

∂

∂


15/39

!ord&n to e4, (1,1?5

8S S78T

∂

∂

∂

∂

(1,1

7S S

8

7

T

∂

∂

=

∂

∂

(1,1B

8T T

7

8

S

∂

∂

=

∂

∂

(1,.

7T T

8

7

S

∂

∂

∂

∂

(1,1


16/39

"p$ and Entrop$ a# F*n!t&on# o3 T

(1,

(1,9

ntha"p$

3*n!t&on o3 7 and T+ )e %a$ e'pre##5

7+THH=

d&@erent&a" o3 the ao6e e4*at&on

d7

HdT

T

HdH

T7

∂

∂

∂

∂

=

∂T7 ota&ned 3ro% the de>n&t&on o3 C75

7

7

T

HC

∂

∂

=


17/39

(1,


18/39

8dTdSdH

8d7dST

d7dH

a"a* T 2on#tan+

d7

dST

7

H

TT

=

∂

∂


19/39


20/39

(1,B

(1,

T7 der&6ed 3ro% 3*nda%enta" e4*at&on5

8dTdSdH (1,

ent&at&on )&th re#pe!t o3 T at !on#tant 7 $&e"

77 T

ST

T

H

∂

∂

=

∂

∂

%&n&n e4, (1,9 )&th (1,5

!&n e4#, (1,1 and (1,B &nto e4, (1,? re

d

T

8

T

dTCdS

7

7

∂

∂

(1,9.

T

C

T

S 7

7

=

∂

∂


21/39

IDEAL GAS

7

8&

=

7

T

8

7

&

=

∂

∂

(1,91

dT

8T8dTCdH

7

&&&

7&

∂

∂

[ ]d88dTCd77

T8dTC &&&7

&&7

dCdH

&

7

&

=

ntha"p$ o3 &dea" a#

Fro% e4, (1,=

(1,9


22/39

ntrop$ o3 &dea" a#

(1,99

Fro% e4, (1,9.

dT

8

T

dTCdS

7

&&7

&

∂

∂

7d

TdTCdS &7

&


23/39

a"a* #te% %ena"a%& pro#e# dar& 2eadaan(T1+ 71 2e (T+ 7+ %a2a per*ahan entrop$n$a

ada"ah5

∫

1

1

&

&1

7

7

T

T

&7

S

S

&

7

d

T

dTCdS

&2a C7 2on#tan %a2a

=

1

1

&

7

&

1

&

&

7

7

"nT

T

"nCSSS


24/39

G&# Ener$ a# a Generat&n F*n!t&o

The 3*nda%enta" propert$ re"at&on# 3or

ho%oeneo*# *&d# o3 !on#tant !o%po#&t&on&6en $ E4#, (1,9+ (1,+ (1,11+ and (1,1;#ho) that ea!h o3 the ther%od$na%&! propert&e#U+ H+ A+ and G 3*n!t&ona""$ re"ated to a #pe!&a"pa&r o3 6ar&a"e#, In part&!*"arSd8d7dG (1,1;

pre##e# the 3*n!t&ona" re"at&on5

T+7GG=

Th*# the #pe!&a"+ or !anon&!a" 6ar&a"e# 3or theG&# ener$ are te%perat*re and pre##*re,

S&n!e the#e 6ar&a"e# !an e d&re!t"$ %ea#*redand !ontro""ed+ the G&# ener$ a

ther%od$na%&! propert$ o3 reat potent&a"


25/39

An a"ternat&6e 3or% o3 E4, (1,1;+ a3*nda%enta" propert$ re"at&on+ 3o""o)#

3ro% the %athe%at&!a" &dent&t$5d

T

GdG

T

1

T

Gd

S*#t&t*t&on 3or dG $ E4, (1,1; and 3orG $ E4, (1,19 &6e#5

dT

TSHdTSd78

T

1

T

Gd

dT

Hd7

T

8

T

Gd

(1,9;

Sd8d7dG


26/39

=

T

1Gd

1

T

Gd

T

1dGdG

T

1

1

dTTGdG

T1

1

dT

GdG

T

1


27/39

Fro% e4, (1,9;

T7

TG

T

8

∂

∂=

(1,9


28/39

d

d

T

H

T

8

d7

dT

T

H

d7

d7

T

8

d7

TGd

dT

Hd7

T

8

T

Gd

Unt*2 T 2on#tan+ dT - .

8

d7

TGd=

(T 2on#tan

d7TGd

T8

=

(T 2on#tan

T7

TG

T

8

∂

∂

=


29/39

Th*#+ )hen )e 2no) ho) G/T (or G re"ated to &t# !anon&!a" 6ar&a"e#+ T and 7+

&,e,+ )hen )e are &6en G/T - (T+ 7+ )e!an e6a"*ate a"" other ther%od$na%&!propert&e# $ #&%p"e %athe%at&!a"operat&on#,

The G&# ener$ )hen &6en a# a3*n!t&on o3 T and 7 there3ore #er6e#

a# a enerat&n 3*n!t&on 3or the otherther%od$na%&! propert&e#+ and

&%p"&!&t"$ repre#ent# !o%p"etepropert$ &n3or%at&on,


30/39

ESIDUAL 7O7ETY

Un3ort*nate"$+ no e'per&%enta" %ethod 3or the

d&re!t %ea#*re%ent o3 n*%er&!a" 6a"*e# o3 G orG/T 2no)n+ and the e4*at&on# )h&!h 3o""o)d&re!t"$ 3ro% the G&# ener$ are o3 "&tt"epra!t&!a" *#e,

Ho)e6er+ the !on!ept o3 the G&# ener$ a# aenerat&n 3*n!t&on 3or other ther%od$na%&!propert&e# !arr&e# o6er to a !"o#e"$ re"atedpropert$ 3or )h&!h n*%er&!a" 6a"*e# are read&"$

ota&ned,Th*#+ $ de>n&t&on the re#&d*a" G&# ener$

& GGG

)here G and G& are the a!t*a" and the &dea":a#6a"*e# o3 the G&# ener$ at the #a%e

te%perat*re and pre##*re,

(1,9?


31/39

Other re#&d*a" propert&e# are de>ned &n anana"oo*# )a$,The re#&d*a" 6o"*%e+ 3or e'a%p"e+

7

7T

7T8888 &

17

T8

de>n&t&on 3or the ener&! re#&d*a" propert$

& MMM

Where M the %o"ar 6a"*e o3 an$ e'ten#&6ether%od$na%&! propert$+ e,,+ 8+ U+ H+ S+ or G,Note that M and M&+ the a!t*a" and &dea":a#

propert&e#+ are at the #a%e te%perat*re andre##*re,

(1,9

(1,9B


32/39

E4*at&on (1,9;+ )r&tten 3or the #pe!&a" !a#e o3 an&dea" a#+ e!o%e#5

d

T

Hd7

T

8

T

Gd

&&&

!t&n th e4*at&on 3ro% E4, (1,9; &t#e"3 &6e#5

d

T

Hd7

T

8

T

Gd

Th 3*nda%enta" propert$ re"at&on 3or re#&d*a"propert&e# app"&e# to *&d# o3 !on#tant!o%po#&t&on, U#e3*" re#tr&!ted 3or%# are5

(1,;.

T

7TG

T8

∂

∂= (1,;1

7

T

TGT

T

H

∂

∂

(1,;


33/39

dT

Hd7

T

8

T

Gd

dT

Hd7

T

8

T

Gd

&&&

(

dT

HHd7

T

88

T

GGd

&&&

=

dTHd7

T8

TGd


34/39

In add&t&on+ the de>n&n e4*at&on 3or the G&#ener$+G - H TS+ %a$ a"#o e )r&tten 3or the #pe!&a"

!a#e o3 an &dea" a#+G& - H& TS&K $ d&@eren!e+ TSHG

e re#&d*a" entrop$ there3ore5

G

T

H

S (1,;9


35/39

Th*# the re#&d*a" G&# ener$ #er6e# a# aenerat&n 3*n!t&on 3or the other re#&d*a"propert&e#+ and here a d&re!t "&n2 )&th e'per&%ent

doe# e't, It pro6&ded $ E4, (1,;1+ )r&tten5

d7

TGd

T

8 =

(!on#tant T

dT8

TGd

=

(!on#tant T

at&on 3ro% ero pre##*re to ar&trar$ pre##*re 7 $&e"

∫

7

.

dT

8

T

G(!on#tant T

)here at the "o)er "&%&t G/T e4*a" to ero

e!a*#e the ero:pre##*re #tate an &dea":a#


36/39

T

7

TG

T

8

∂

∂

=

d7

TGd

T

8 = (T 2on#tan

(1,;1


37/39

∫

=

7

.

G

G

d

T

8

T

Gd

&

∫

7

.

&

dT

8

T

G

T

G

∫

7

.

d

T

8

T

G

1

7

T8

17

1

T

8


38/39

∫

7

.

dT

8

T

G

17

T8

∫

=

7

.

d17T

T1

TG

∫

7

.

7d1

TG


39/39

In 6&e) o3 E4, (1,95

∫

7

.

7

d1

T

G(!on#tant T (1,;;

D&@erent&at&on o3 E4, (1,;; )&th re#pe!t tote%perat*re &n a!!ord )&th E4, (1,; &6e#

∫

∂

∂

7

. 7

7

d

T

T

T

H(!on#tant T (1,;

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1a Fundamental Equations1