5-CHAPTER_3b-PVT-EOS-STDN.pdf

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&2345671 78 ,5459 :&*,; 6< 41 92345671 5=45 >9?459< 5=9   @>9< @>9<<3 <3>>9A 59B 9B@9 @9>>45 453> 3>9A 9A 41 41C C 5= 5=9 9 B7 B7?4 ?4>> D7? 7?3B 3B9 9 :$E%; 78 4 <3F <3F<54 <541G9 1G9HH



%I@9< 78 &*, 6H

  +C9 +C94? 4? J4 J4<< :K4<;

66H

  E6>64 E6>64?? &* &*, , :K4<;

666H

E41 C9 C9>> L44 44?< ?< &* &*, , :K :K4< 4< M ?62 ?6236C 36C;;

6DH

'9C?6G=NOP71K '9C?6G=NO P71K &*, :K :K4< 4< M ?62 ?6236C 36C;;

Cubic EOS

&./01 203 !'+ !"# #  !"

$%&'(#$) *+, -#$)./ )./+01+ #23*)%0+ 04 $)*)# 405 $3-$)*+6#$ %+ )"# 7*$ '"*$# %$ )"# %,#*(.7*$ #23*)%0+ 04  $)*) $)*)#8 #8 !"%$ !"%$ #23*) #23*)%0+ %0+ '5#, '5#,%6 %6)$ )$ )"# )"# ! ." .#   -#" -#"*9 *9%0 %055 04 * 7*$ 7*$ 23%) 23%)# # *663 *6635* 5*))#(: #(: 1%) 1%)"%+ "%+ $0&# $0&# '50' '50'#5 #5(: (: $#(# $#(#6) 6)#, #, 5#7%0+8

Ideal gas equation of state  -7>  %=9

+Q&# +Q&#R R K4<9< 4<9
Z  

P V 

RT 



1

!D;0 !D ;094 9478 78 E7 E7> > => =>7? 7?/3 /333 60 601? 1?;1 ;109 09478 47833 &8 &8B B71 71B4 B48F 8F &. &./0 /011 203/3 203/3 -7> 6C94? K4< 61 41I G?7<9CN7G9< :CU V CL; W C. W !DC%

45

678 7839 3908 0899 :71 71;< ; =>7? 7?/3 /333 :U W . W !D C%;

445

67839 67 839089 089 => =>/3 /33; 3;>/ >/ @> @>7? 7?/3 /333 :  :U U W " W !@ C%; C%;

44455 444

&379A/ &379 A/>< ><01 01 => =>7? 7?/3 /333 :" W . W X; U W L 7> U W NL W '% ?1 EYZ ES W '% ?1 $SZ$Y

4B55 4B

$.40C0 $.40 C094 94? ? => =>7? 7?/3 /333 :C :CU U W X; :%YZ%S; :%YZ %S; W :ESZ :ESZEY; EY; 'Z!D %ENS W !7 !71< 1<5H 5H :%YZ%S; :%YZ %S; W :$YZ :$YZ$S; $S; 'Z!@ %$:SN;Z W !7 !71< 1<5H 5H :$YZ$S; :$YZ$ S; W :ESZ :ESZEY; EY; !@Z!D $E W !71<5H L W !D % W ' % Z  N S W ':%Y – %S;Z  N S W :$YEY – $SES  $SES;; Z  N S %S 41C EY 3<3 3<34?? 4??I I 31[17P1 31[17P1TT L W '%SZ  N S\:$Y S\:$Y Z$S; :NS;Z N S] W $SES $SESZ Z  N S\:$Y S\:$Y Z$S; :NS;Z N S]

B5

=71 71G9 G9> >7@4 @4? ? => =>7? 7?/3 /333 :$E δ W !71 !71<5H <5H;; :%YZ%S; :%YZ %S; W :ESZ :ESZEY; EY; 'Z!D %E δ NS W !7 !71< 1<5H 5H :%YZ%S; :%YZ %S; W :$YZ :$YZ$S; $S; 'Z!@ %$:SN δ;Z δ W !7 !71< 1<5H 5H LW\' LW\ '%SZ δ N S]\:$Y S]\:$Y Z$S; :δ NS;Z δ N S] UW ::δ – ; '%S; '%S; Z ::δ – S; :NS;; ::$Y ::$Y Z$S; :δ NS;Z δ – S;

P"(V")^Cp/Cv = P!(V!)^Cp/Cv (P!/P") = (V"/V!)^Cp/Cv [(mRT!/V!)*(V"/mRT")] = (V"/V!)^Cp/Cv (T!V"/T"V!) = (V"/V!)^Cp/Cv  (T!/T") = (V"/V!)^Cp/Cv * (V"/V!)^-1 = (V"/V!)^[(Cv+R)/Cv - 1] = (V"/V!)^[1 - 1 + R/Cv] ! (T!/T") = (V"/V!)^ R/Cv not isobaric, can’t transform into enthalpy

mass, m can be cancelled, as it is fixed in closed-system Adiabatic, 0

! = Cp/Cv " Cp = !Cv (T!/T") = (V"/V!)^ R/Cv = Cv/Cv + R/Cv = [(mRT"/P")*(P!/mRT!)]^ R/Cv = 1 + R/Cv = (T"P!/T!P")^R/Cv = (T"/T!)^R/Cv * (P!/P")^R/Cv R = (!-1)Cv (T!/T")^(1+ R/Cv) = (P!/P")^R/Cv (T!/T")^(Cv/Cv+ R/Cv) = (P!/P")^R/Cv (T!/T")^(Cp/Cv) = (P!/P")^R/Cv (T!/T") = (P!/P")^ (!-1)Cv/!Cv (T!/T") = (P!/P")^[(R/Cv)*(Cv/Cp)] = (P!/P")^ (!-1)/! ! (T!/T") = (P!/P")^ R/Cp T!(P!)^-(!-1)/! = T"(P")^-(!-1)/! !T(P)^(1-!)/! = constant

!H0<@1/I J5K ,L5M +<49A N77OS B7? 78 46> 6< G7B@>9<<9C 8>7B 41 616564? <5459 78 S F4> 41C Y^7! 57 4 8614? <5459 78 ^ F4> 41C Y^7! FI   9A>// .4EE/>/89  /B/>34C1/ @>7?/33/3 61 4 G?7<9C < Y^7!
"94561K 45 G71<5415 D7?3B9 87??7P9C FI G77?61K 45 G71<5415  @>9<<3>9 +<75=9>B4? G7B@>9<<671 #C64F456G G7B@>9<<671 87??7P9C FI G77?61K 45 G71<5415  D7?3B9

#<<3B9 46> 57 F9 41 6C94? K4< P65= !D W :^ZY;' 41C !@ W :_ZY;'H -7> 94G= @>7G9<
%=9 P7>[ >9236>9C %=9 =945 5>41<89>>9C %=9 G=41K9< 61 6159>14? 919>KI %=9 G=41K9< 61 915=4?@I ,=7P 5=9 @>7G9<<9< 71 4 $D C64K>4B

+71;9478 4; "94561K 45 G71<5415 D7?3B9 87??7P9C FI G77?61K 45 G71<5415 @>9<<3>9

From Ideal gas law, find v1 and v3, v1 = v2 = 0.02479 m3/mol & v3 =0.004958m3/mol

F; +<75=9>B4? G7B@>9<<671

G; #C64F456G G7B@>9<<671 87??7P9C FI G77?61K 45 G71<5415 D7?3B9

DN ,=7P 5=9 @>7G9<<9< 71 4 $D C64K>4B

!H0<@1/I J5J ,L5M +<49A N77O#1 6C94? K4< 31C9>K79< 5=9 87??7P61K   3/D;/8?/ 7E  /B/>34C1/ @>7?/33/3 61 4 G?7<9C
->7B 616564? <5459 78 S F4> 41C _X7! 65 6< G7B@>9<<9C 4C64F456G4??I 57 S^X7! +5 6< 5=91 G77?9C 57 _X7! 45 G71<5415 @>9<<3>9 -614??IA 65 6< 9`@41C9C 6<75=9>B4??I 57 65< 7>6K614? <5459 6H 66H

!4?G3?459 LA UA .A " 87> 94G= 78 5=9 5=>99 @>7G9<<9< 41C 87> 5=9 9156>9 GIG?9H ,=7P 5=9 @>7G9<<9< 71 4 $D C64K>4BH

!"#$% &# ' ()*+,- .%/ &0 ' (1*+,-H

+71;9478 6N !4?G3?459 LA UA .A " 87> 94G= 78 5=9 5=>99 @>7G9<<9< 41C 87> 5=9 9156>9 GIG?9H 24B/8 ' W aH)Sb cZB7?HO

G@ W :)ZY;' W YXH_a^ cZB7?HO GD W :^ZY;' W SYHb_S cZB7?HO

%4[9 S B7?9 4< F4<6< 4; -7> 6C94? K4< 31C9>K761K 4C64F456G G7B@>9<<671A U W X

$>9<<3>9 $Y 6< 8731C 8>7B

F; !77?9C 57 _X7! 45 G71<5415 @>9<<3>9

G; -7> 6C94? K4<9< 31C9>K761K 41 6<75=9>B4? @>7G9<
*7> 9A/ /894>/ ?G?1/ U W XNSdd)VSbe^ W NSdac L W eea V dd^ – Sbe^ W Sdac  . W eeaNeeaV X W X  " W Sdd)NSdd) VX W X

66N

,=7P 5=9 @>7G9<<9< 71 4 $D C64K>4BH

:4>401 !'+ 

-7> >94? K4<9
  .1C9>   2.345.3$/

67%/"3"7%A E6>64? &*, G41 F9 3<9C 57 C959>B619 5=9 B7?4> D7?3B9 78 #.075 08.2$ *0Rg



%=9 E6>64? &*, 6< F4<9C 71 5=9 +C94? J4< &*,T ~

Z  

P V 

RT 



1

!"# %#&'( #) *+, -.&./0 123/*.#4 7#'6&,((.8.0.*9 )/7*#&

-.&./0 ,56/4(.#4( :B7>9 4GG3>459 68 $⤒ F35 f  ;

•

56 7   41C <7 71 4>9 5=9 E6>64? !79886G6915<

•

Virial coefficients are function of T only

!H0<@1/I J5P ,L5M +<49A'9@7>59C D4?39< 87> 5=9 D6>64? G79886G6915< 78 6<7@>7@417? D4@7> 45 YXX h! 4>9i 5 W N)aa GB)B7?NS

7 W NYdXXX GBdB7?NY

!4?G3?459 E 41C j 87> 6<7@>7@417? D4@7> 45 YXX h ! 41C SX F4> FIi 4;

%=9 6C94?NK4< 92345671

F;

E6>64? 92345671 61 59>B< 78 $ 5>31G459C 57 Y 59>B
G;

E6>64? 92345671 61 59>B< 78 E 5>31G459C 57 ) 59>B
Given T = 200 + 273.15 = 473.15K P = 10 bar = 10000 kPa B = -)aa GB)B7?NS C W NYdXXX GBdB7?NY a) The Ideal gas equation V = Videal = RT/P

where Z = 1 for ideal gas

Videal =[ 1 (83.14GB) F4>H B7?NS HONS) (473.15 K) ]/ (10 bar) Videal = 3934 cm3/mol Z = PV/RT = 1 for ideal gas 

,C- :4>401 /D;09478 48 9/><3 7E = 9>;8?09/. 97 K 9/><35 ~

PV 

R T  ~

V 

~

V 

~

V 

~

V 



1

 RT  

P  RT 







 Z  

 BP  RT 

[1 

 BP  RT 

]

  B

P (83.1483.14cm3 bar / mol.K)( 473.15K ) 10 bar 



( 388cm3 / mol )

3546cm3 / mol PV 

RT



(10bar )(3546cm3 / mol ) (83.14cm3 bar / mol.K)(473.15K )



0.901

,?- :4>401 /D;09478 48 9/><3 7E : 9>;8?09/. 97 J 9/><35

~

 Z 

~

V 



PV 

R T 

 B

C 

 1 ~  ~2 V 

V 

 B C    1 ~  ~2    P  V  V    RT 

Facilitate iteration.; where i  iteration number & ~ ~ ~  when i  0 , V i  V 0  V ideal

V i 1

(83.14)( 473.15)  388 26000   1  ~  ~ 2  10 V i V i  

(83.14)(473.15)  388 26000  V i 1  1  ~  ~ 2  10 V i  V i  1st  iteration ; i  0;

V 01 V 1

2nd  iteration ; i  1;

4th iteration ; i  3;

5th iteration ; i  4;

~

V 5

~

 V 4  0 or

 Z  

PV 



~

V 5



(83.14)(473.15)  388 26000  ~ 1  ~  ~ 2  where V 2  3495cm3/mol 10 V 2  V 2 



(83.14)(473.15)  388 26000  ~ 1  ~  ~ 2  where V 3  3489cm3/mo l 10 V 3  V 3 

 3488 cm3/mol

V 41 V 5

(83.14)(473.15)  388 26000  ~ 1  ~  ~ 2  where V 1  3539cm3/mol 10 V 1  V 1 

 3489 cm3/mol

V 31 V 4



 3495cm3/mol

V 21

V 3

(83.14)(473.15)  388 26000  ~ ~ 1  ~  ~ 2  where V 0  V ideal  3934 cm3/mol 10 V 0  V 0 

 3539 cm3/mol

V 11

V 2

3rd  iteration ; i  2;





(83.14)(473.15)  388 26000  ~ 1  ~  ~ 2  where V 4  3488cm3/mol 10 V 4  V 4 

 3488 cm3/mol ~

 V 4

(10)(3488)

stop

iteration;

 0.8867

Final

~

V 

~

 V 5  3488 cm3/mol

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DE

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27   

64 

; =

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8 

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

  %=9 59>B V  6< E41 C9> Waal’s 87>G9< P=6G= 4GG7315 87> 5=9 455>4G56D9 87>G9< F95P991 B7?9G3?9< 2

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  =ψ

∝( )   

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 

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

'9C3G9C 59B@9>453>9 9 > M >9C3G9C @>9<<3>9 : > T r



T  

T c

,

Pr

P 

Pc

Q986165671 78 4G915>6G 84G57> ω 6 1.0  log

   

P  sat

r

T r  0.7

:0@7> Z :0@7>R14O/ >7793 7E 9A/ F/8/>4? ?;C4? !'+ 

#@@?I 87> D4@7> <3F<541G9

   +1D7?D9

659>45671< P65= 616564? D4?39 78  [ \ S

 =      ≡Ω = 

− + (+)

 

Ѱ∝  Ω

'9?459C @4>4B959>< 4D46?4F?9 61 %4F?9 )HS :@K ea;

]4D;4. Z ]4D;4.R14O/ >7793 7E 9A/ F/8/>4? ?;C4? !'+   #@@?I

87> ?6236C <3F<541G9

   +1D7?D9

659>45671< P65= 616564? D4?39 78  [ \ β +−

 =         ( ≡Ω = 



)

 

Ѱ∝  Ω

'9?459C @4>4B959>< 4D46?4F?9 61 %4F?9 )HS :@KH ea;

!H0<@1/ J5W^L5M +<49A @F WP_ J6D91 5=45 5=9 D4@7> @>9<<3>9 78 (;8#%$(- 45 )^X O 6< eHb^_) F4>A 861C 5=9 B7?4> D7?3B9< 78 i :4; ,453>459CND4@7> 1NF35419 :F; ,453>459CN?6236C 1NF35419 45 5=9<9 G71C65671< 4< K6D91 FI 5=9 '9C?6G=ZOP71K 92345671H

2/8/>014`/. 67>>/109478 *7> 203/3 

%9G=16239< 57 861C 5=9 ji

SH

$65m9> !7>>9?45671 87> 5=9 G7B@>9<<6F6?65I G=4>5ZC454

YH

$65m9> !7>>9?45671 87> 5=9 <9G71C D6>64? G79886G6915 

)H

$65m9> !7>>9?45671 87> 5=9 5=6>C D6>64? G79886G6915 

S- =49`/> 67>>/109478 E7> 9A/ 67<@>/334C4149G ?A0>9a.090 

J919>4?6m9C G7B@>9<<6F6?65I 84G57> i

 Z    Z 

0

0

1

  Z  , Z 

   Z 

1

– G41 F9 8731C 61 #@@91C6` &A %4F?9 &SN &b

Z!7B@>9<<6F6?65I !=4>5  

 

– G41 F9 8731C 61 #@@91C6` /A 54F?9 /S

%A/ ?7<@>/334C4149G ?A0>9 

Thermodynamics for Chemical Engineers EP205, L4

$@@/8.4H N I %0C1/ NS

 $@@/8.4H ! I %0C1/ !S – !X

K- =49`/> 67>>/109478 E7> 9A/ +/?78. :4>401 67/EE4?4/89    '9G4??

5=9 ,9G71C D6>64? 92345671i

~

 Z  

PV 

R T 



1

 BP

 1   B ˆ

R T 

Pr  T r 

P=9>9

ˆ  B    B  B  0

0

B 



0.083 

1

0.422 1 .6

T r 

,

1

B 



0.139 

0.172 4.2

T r 

J- =49`/> 67>>/109478 E7> 9A/ %A4>. :4>401 67/EE4?4/89    '9G4??

5=9 5=6>C D6>64? 92345671i ~

Z  

P V 

RT 

B 



1 ~

ˆ

P=9>9 C 

0

C 

1

C 

V 



C  

P r    ˆ   ˆ    1  B   C    T  r  Z    T  r  Z    P r 

~2

V 

0

C 

2

1

  C 

 0.01407 

0.02432

 0.02676 

T r



0.05539 2.7

T r

0.00313



10.5

T r

,

0.00242 10.5

T r

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4.

Analyze the molar volume of   n-pentane  in cm)/g at 237 oC and 25 bar, using the generalized Pitzer correlation for third virial  coefficient method.