Experiment 1 Chem 116 Laboratory- Physical Chemistry Laboratory Section 2 Department of Chemistry University of the Philippines
THE VAN DER WAALS EQUATION OF STATE, THE LAW OF CORRESPONDING STATES, AND THE COMPRESSION FACTOR
Laboratory Report DINA L. LACSON S in Chemistry !!! September "# 2$16
ARNOLD C. GAJE Laboratory !nstr%ctor
&bstract '''''''''''''''''''''''''''''''' ''''''''''''''''''''''''''''''''''''''''''''''''' '''''''''''''''''''''''' ''''''' E(%atio E(%ations ns of state state attemp attemptt to )escrib )escribee the relatio relationsh nship ip bet*ee bet*een n temper temperat% at%re re +,# +,# press%re +P# an) molar vol%me +v for a stan)ar) circ%mstances. ,he i)eal /as la* is the 1
simplest form of an e(%ation of state that can be consi)ere) as a lar/e (%antity of small molec%les that have no friction an) no attractive or rep%lsive forces. ,he i)eal /as la* is a reasonable approximation at lo* press%res an) hi/h temperat%res# b%t not at hi/her press%res an) lo*er temperat%res. 0an )er aals e(%ation is a mo)el of state e(%ation for real /as expresse) in terms of t*o parameters# one correspon)in/ to molec%lar attraction an) the other to molec%l molec%lar ar rep%ls rep%lsion ions. s. !t capt%r capt%re) e) the /eneral /eneral feat%res feat%res of the behavior behavior of real real /ases /ases incl%) incl%)in/ in/ their their critical critical behavi behavior or.. ,he ,he proper properties ties of real /ases /ases are then then coor)i coor)inat nate) e) by expressin/ their e(%ations of state in terms of re)%ce) variables. ,he %se of re)%ce) variables in comparison to the act%al variables *as performe) to verify the la* of correspon)in/ of states. icrosoft Excel sprea)sheet *as the soft*are-pro/ram employe) to comp%te for necessary val%es as *ell as in creatin/ /raphs# c%rves# tren) lines# scattere) points an) etc. ,*o s%bstances +Carbon monoxi)e an) n-pentane *ere st%)ie) in this experiment. Each of this this s%bs s%bstan tance ce *ere *ere exam examin ine) e) crit critica ically lly for for the the beha behavi vior orss an) an) )evi )eviat atio ions ns %sin %sin/ / the the compression factor 3.
!ntro)%ction '''''''''''''''''''''''''''''''' '''''''''''''''' ''''''''''''''''''''''''''''''''' '''''''''''''''''''''''' '''''''
2
simplest form of an e(%ation of state that can be consi)ere) as a lar/e (%antity of small molec%les that have no friction an) no attractive or rep%lsive forces. ,he i)eal /as la* is a reasonable approximation at lo* press%res an) hi/h temperat%res# b%t not at hi/her press%res an) lo*er temperat%res. 0an )er aals e(%ation is a mo)el of state e(%ation for real /as expresse) in terms of t*o parameters# one correspon)in/ to molec%lar attraction an) the other to molec%l molec%lar ar rep%ls rep%lsion ions. s. !t capt%r capt%re) e) the /eneral /eneral feat%res feat%res of the behavior behavior of real real /ases /ases incl%) incl%)in/ in/ their their critical critical behavi behavior or.. ,he ,he proper properties ties of real /ases /ases are then then coor)i coor)inat nate) e) by expressin/ their e(%ations of state in terms of re)%ce) variables. ,he %se of re)%ce) variables in comparison to the act%al variables *as performe) to verify the la* of correspon)in/ of states. icrosoft Excel sprea)sheet *as the soft*are-pro/ram employe) to comp%te for necessary val%es as *ell as in creatin/ /raphs# c%rves# tren) lines# scattere) points an) etc. ,*o s%bstances +Carbon monoxi)e an) n-pentane *ere st%)ie) in this experiment. Each of this this s%bs s%bstan tance ce *ere *ere exam examin ine) e) crit critica ically lly for for the the beha behavi vior orss an) an) )evi )eviat atio ions ns %sin %sin/ / the the compression factor 3.
!ntro)%ction '''''''''''''''''''''''''''''''' '''''''''''''''' ''''''''''''''''''''''''''''''''' '''''''''''''''''''''''' '''''''
2
,he perfect /as e(%ation of state is the approximate approximate e(%ation of state of any /as that becomes increasin/ly exact as the press%re of the /as approaches 4ero an) the simplest e(%atio e(%ation n that that )escrib )escribes es a relatio relationsh nship ip bet*ee bet*een n the press%re press%re p# p# mola molarr vol% vol%me me v an) the
pv = RT RT temperat%re T of a /as is the so-calle) i)eal /as la*5
. &s *e compare the pre)ictions
of this e(%ation of state *ith the experimental )ata# it *as fo%n) that it is only vali) in a very limi limite te) ) ran/ ran/ee of pres press% s%re ress an) an) vol% vol%me mess so for for a more more acc% acc%ra rate te )esc )escri ript ptio ion n of the the thermo)ynamic properties of /ases# an improve) e(%ation of state is re(%ire). !n this laboratory experiment# *e st%)y abo%t the van )er *aals e(%ation of state# the la* of correspon)in/ states# an) the compression factor. ,he van )er aals e(%ation is a mo)e mo)ell e(%a e(%ati tion on of stat statee for for a real real /as /as expr expres esse se) ) in term termss of t*o t*o para parame mete ters rs## one one correspon)i correspon)in/ n/ to molec%lar molec%lar attractions attractions an) the other to molec%lar rep%lsions. rep%lsions. !t capt%res capt%res the /eneral feat%res of the behavio%r of real /ases# incl%)in/ their critical behavio%r an) also the properties of real /ases are coor)inate) by expressin/ their e(%ations of state in terms of re)%ce) variables. ,his lab activity aims to establish the )ifference bet*een the i)eal /as e(%ation of state an) the van )er *aals e(%ation of state# an) the compression factor principle that )etermines the extent of )eviations from a perfect behavior# an) the la* of correspon)in/ states. ,hese concepts abo%t real /ases are important especially on learnin/ f%rther on the isotherms of a real /as that intro)%ces the concept of vapor press%re an) critical behavior. Real /ases )o not obey the perfect /as la* exactly except except in the limit of p approachin/ approachin/ to $# th%s the scope of this experiment is only limite) on the e(%ations of state for real /ases. &fte &fterr perf perform ormin in/ / this this expe experim riment ent## one one sho% sho%l) l) be able able to ans* ans*er er *het *hethe herr the the la* la* of correspon)in/ states is coherent *ith the van )er aals e(%ation of states an) *hat is the relationship bet*een critical press%re# temperat%re an) vol%me.
3
aterials an) etho) '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
,hro%/h the help of a )i/ital personal comp%ter# an excel sprea)sheet pro/ram *as %se) in analy4in/ )ata# constr%ctin/ tables# /eneratin/ /raphs an) solvin/ mathematical calc%lations specifically those relate) to van )er *aals e(%ation of state# la* of correspon)in/ states an) the compression factor. irst# *e chose t*o )ifferent s%bstances mentione) in table 1 of the sheet tables in a )esi/nate) sprea)sheet )oc%ment# then the names of the assi/ne) s%bstances +C"712 an) C8 for o%r /ro%p *ere copie) into the sheet pvT-calc. Usin/ excel soft*are# *e calc%late) the van )er aals parameters a an) b for these /ases by copyin/ their critical temperat%re T c# critical press%re pc an) critical molar vol%me vc into the sheet pvTcalc. ,hro%/ho%t this sprea)sheet# there is a bl%e colo%r co)e for inp%t variables *hich can
be chan/e) an) re) for the res%lt of a calc%lation *hich cannot be chan/e). 9ext# *e calc%late) ho* the press%re )epen)s on the molar vol%me for the assi/ne) t*o s%bstances %sin/ sheet pvT-calc. e )i) this for a fe* temperat%res for each /as consi)erin/ temperat%res that are above# sli/htly above# an) sli/htly belo* the critical temperat%re# *e then compare) the res%lts *e have obtaine) for the i)eal /as e(%ation of state an) the 0an )er aals e(%ation of state. e *ere able to calc%late the /as-li(%i) coexistence c%rve %sin/ the so-calle) ax*ell constr%ction + At the equilibrium pressure the areas that are enclosed by the line van der Waals loop and the line representing the equilibrium pressure are equal in size) ill%strate) belo*5
Figure 1: Gasliquid coe!istence curve using "a!#ell construction
4
&fter comparin/ the res%lts obtaine) for the i)eal /as e(%ation of state an) the 0an )er aals e(%ation of state# *e performe) the ax*ell constr%ction for the t*o s%bstances %sin/ the sheet Maxwell . or a /iven temperat%re the li(%i) phase is e(%ilibri%m *ith the vapo%r phase *hen &rea 1:&rea 2. e then calc%late) for fo%r temperat%res belo* ,c# the press%re p an) the molar vol%mes v +i.e.# v1 an) v; at *hich the li(%i) phase is in e(%ilibri%m *ith the /as phase for each of the t*o s%bstances assi/ne) to %s +C "712 an) C8. e a)<%ste) the Ptrial to have an area )ifference that is less than 2= an) sheet.
,his phase )ia/ram has several names5 the /as-li(%i) coexistence c%rve or the bino)al c%rve or the e(%ilibri%m c%rve. y choosin/ the insert pane in the c%stomi4e access toolbar of excel# *e /enerate) the scatter chart of the )ata# tracin/ a smooth line thro%/h the points to complete the bino)al c%rve an) labellin/ the c%rve. e have also /enerate) the p-0 phase )ia/rams for each of the five temperat%res +,c an) belo* ,c in workshee!Q" %sin/ the p-0 )ata from the ax*ell constr%ction an) the #$T%&'(& *or>sheet. ,he )ia/rams for each temperat%re *ere then s%perimpose) in one
/raph an) the bino)al c%rve *as locate) in the series of isotherms. & schematic s>etch of the pT phase )ia/ram of a realistic s%bstance that either can exist in the soli) phase# li(%i) phase an) the /as phase *as sho*n in the fi/%re belo*. p
soli) li(%i)
/as T
5
Figure $: The pressure temperature phase diagram o% a real substance&
Usin/ the five temperat%res +,c an) belo* ,c an) the correspon)in/ P trial an) = &rea )ifference for each temperat%re obtaine) in the previo%s calc%lations# *e *ere able to )erive the li(%i)-/as coexistence c%rve +in a pT )ia/ram. e then plotte) this c%rve in sheet workshee!Q) for both s%bstances +C "712 an) C8.
9ext# *e constr%cte) a pr-0r )ia/ram %sin/ the re)%ce) press%re pr as a f%nction of the re)%ce) molar vol%me 'r at the same re)%ce) temperat%re T r for the t*o s%bstances assi/ne) to %s in a sin/le /raph in the sheet workshee!Q*,. e then too> the critical constants as calc%late) by the pro/ram in sheet pvT-calc. e then plotte) in the same *or>sheet the re)%ce) press%re pr as a f%nction of the re)%ce) molar vol%me ( r for the press%res an) vol%mes of the coexistence c%rve for the t*o s%bstances assi/ne) to %s. e have consi)ere) the la* of correspon)in/ states only theoretically %sin/ the 0an )er aals e(%ation of state in the previo%s steps. or the next step# *e investi/ate) ho* this la* *or>s for real /ases %sin/ the experimental )ata for ar/on an) ethane. ,he experimental critical constants *ere obtaine) from the /iven )ata in ,able 1 of the sheet tables. irst# *e compare) the experimental )ata *ith the pre)ictions of the van )er aals e(%ations by plottin/ p as a f%nction of 1v in sheet workshee!Q+ an) calc%late) the p for each of the experimental vol%mes %sin/ the van )er aals e(%ation of state. e then plotte) p vs. 1?0 )ia/rams for the &r/on an) ethane %sin/ the experimental )ata in a chart. e also plotte) the p vs. 1?0 )ia/rams for the /ases %sin/ the van )er aals pre)ictions in another chart an) ma)e a comparison bet*een the t*o. e teste) as *ell in sheet workshee!Q+ *hether the experimental )ata /iven in table 2 of the sheet tables obey the la* of correspon)in/ states by plottin/ pr as a f%nction of 1vr . y re*ritin/ the van )er aals e(%ation of state in terms of re)%ce) press%re# re)%ce) vol%me an) re)%ce) temperat%re# *e prove) mathematically the la* of correspon)in/ states. 6
,he compression factor 3 : P0 m?R, is a convenient meas%re of the )eviation from i)eal /as behavior. or i)eal /ases# 3:1 %n)er all con)itions@ )eviation of 3 from %nity is a meas%re of )eviations from the i)eal behavior. ,he extent an) si/nificance of this )eviation in )escribin/ the behavior of real /ases can be %n)erstoo) by st%)yin/ ho* 3 varies *ith press%re an) temperat%re. ,he variation of 3 *ith press%re an) temperat%re *as explaine) more effectively %sin/ the so-calle) oyle temperat%re +, as reference. or a van )er aals /as# , : a?bR. ,he val%es of a an) b *as calc%late) base) from the critical constants of the /as. ,he follo*in/ metho)s *ere performe) in another sprea)sheet )oc%ment save) as 3'Calc- C8.
•
irstly# the file 3-calc.xls *as opene).
•
Secon)ly# the van )er aals constants an) the , for the assi/ne) s%bstance *hich is the C8 /as *ere calc%late) in the %&'(& sheet.
•
,hir)ly# t*o other temperat%res5 , A , an) , B , *ere selecte).
•
o%rthly# the p from /iven molar vol%mes for specifie) temperat%re +, : , # , A , # an) , B ,. *ere /enerate) in the %&'(& sheet.
• •
9ext# the p an) 0m from 3-calc sheet *as copie) in sheet workshee . ,hen# p *as expresse) in atm to calc%late 3 val%es an) for each temperat%re# a 3 vs. p?atm )ia/ram *as prepare).
•
Lastly# the tren)s on ho* 3 varies *ith p at each temperat%re re/ion +, B , # ,:,# an) , A ,# an) ho* it reveals information abo%t intermolec%lar interactions in real /ases *as )escribe) in a concise statement.
Res%lts ''''''''''''''''''''''''''''''''''''''''''''''''''''''''
7
,he follo*in/ )ata presents the res%lts obtaine) from the experiment on the van )er *aals e(%ation of state# the la* of correspon)in/ states# an) the compression factor. ,he variables an) the correspon)in/ val%es *ere entere) on a sprea)sheet )oc%ment in forms of tables an) the /raphs *ere a%tomatically /enerate) by the excel pro/ram. ,he van )er *aals e(%ation of state pre)icts that the relationship bet*een the press%re p# molar vol%me v an) temperat%re T is /iven by5
p + a ( v − b ) = RT v 2
7ere R is the /as constant an) a an) b are n%merical constants that can be obtaine) by analysin/ the experimental )ata on the press%re# vol%me an) temperat%re. 8ne of the properties of the van )er aals e(%ation is that it is capable in pre)ictin/ a /as-li(%i) transition. ,he /as-li(%i) coexistence c%rve can be calc%late) %sin/ the so-calle) ax*ell constr%ction *herein the critical point is characteri4e) by the critical temperat%re T c# the critical press%re pc an) the critical molar vol%me vc& ,he van )er aals constants can be calc%late) from these critical properties follo*in/ these parameters5
a=
2D R
2
T c2
b
6C pc
=
RT c E p c
# ,he critical properties +e./.# press%re# temperat%re# an) vol%me of the t*o s%bstances +carbon monoxi)e an) n-pentane assi/ne) to %s are /iven in the table belo*5
Substance p(cr)/[Pa] T(cr)/[K] v(cr)/[m3.mol-1] *arbonmono!ide ;F"6F" 1;2.F F.;1E-$" n+entane ;;$$$$ 6F.6 ;.$E-$ Table 1: *ritical properties o% carbon mono!ide and npentane
8
,he follo*in/ /raphs sho* the comparison of i)eal /as an) van )er *aals e(%ations of state an) the relationship of molar vol%me *ith press%re. a. ,he res%lts obtaine) for the i)eal /as e(%ation of state an) van )er *aals e(%ation of state of carbon monoxi)e are plotte) comparatively for fe* temperat%res that are above# sli/htly above# an) sli/htly belo* the critical temperat%re.
160 140 120 100 80 60 40 20 0
Pressure (100000Pa) ideal gas
9 5 7 1 7 6 0 8 van der Waals 2 3 0 3 8 2 4 8 7 2 8 2 6 4 8 0 7 1 3 1 0 6 1 0 . 4 .
Molar volume (0.0001m3/mol)
Figure ,: Temperature at T-1./ 0Tc2 %or carbon mono!ide
9
Pressure (100000Pa) ideal gas
160 140 120 100 80 60 40 20 0 -20
9 8 7 2 van der Waals 7 8 8 5 3 5 3 6 8 2 8 3 2 5 2 0 6 9 8 7 5 3 1 5 6 2 . 2 0 4 .
Molar volume (0.0001m3/mol)
Figure 3: Temperature at T- 134 0Tc2 %or carbon mono!ide
10
160 140 120 100 80 60 40 20 0
Pressure (100000Pa) ideal gas
9 6 7 0 7 4 8 5 van der Waals 8 3 3 3 8 8 5 1 2 9 2 6 6 0 8 3 6 3 1 8 6 8 9 0 . 3 .
Molar volume (0.0001m3/mol)
160 140 120 100 80 60 40 20 0
Pressure (100000Pa) ideal gas
9 9 7 6 7 6 8 3 van der Waals 7 3 1 3 8 2 1 8 0 2 3 2 9 6 3 8 1 9 1 3 4 6 9 0 . 3 .
Molar volume (0.0001m3/mol) Figure 4: Temperature at T-132.9 K -Tc2 %or carbon mono!ide
11
Figure .: Temperature at T- 1$5 06Tc2 %or carbon mono!ide
200 150 100 50 0
Pressure (100000Pa) ideal gas
3 5 4 5 9 7 7 van der Waals 6 1 6 3 6 8 1 2 4 0 1 7 1 5 0 9 2 6 5 0 1 3 6 . 2 . 3 1
Molar volume (0.0001m3/mol)
Figure 7: Temperature at T- 1$/ 06Tc2 %or carbon mono!ide
12
160 140 120 100 80 60 40 20 0
Pressure (100000Pa) ideal gas
9 3 7 5 7 3 2 8 van der Waals 3 3 3 2 8 2 8 7 2 6 2 2 6 6 8 9 5 3 1 6 6 4 1 0 . 4 .
Molar volume (0.0001m3/mol)
b.
,he
res%lts
obtaine) for the i)eal /as e(%ation of state an) van )er *aals e(%ation of state of n pentane are plotte) comparatively for )ifferent temperat%res.
200 150 100 50 0
Pressure (100000Pa) ideal gas
3 9 3 5 9 7 9 7 van der Waals 9 6 2 3 8 7 2 4 4 1 8 0 5 8 9 6 5 1 7 4 3 4 . 2 . 1 4
Molar volume (0.0001m3/mol) Figure 8: Temperature
at T- 354 0Tc2 %or npentane
13
Figure 5: Temperature at T- 38/ 0Tc2 %or npentane
200 150 100 50 0
Pressure (100000Pa) ideal gas
3 9 5 2 9 7 7 van der Waals 9 1 6 3 3 8 4 4 6 9 0 1 5 5 7 9 6 5 3 1 5 3 0 . 2 . 5 1
Molar volume (0.0001m3/mol)
Figure 1/: Temperature at T- 3.5&. 0- Tc2 %or npentane
14
180 160 140 120 100 80 60 40 20 0
Pressure (100000Pa) ideal gas
9 3 5 2 7 9 7 9 1 6 van der Waals 3 3 8 4 6 4 9 1 0 5 5 7 6 9 5 3 5 1 0 3 2 . 5 . 1
Molar volume (0.0001m3/mol)
Figure 11: Temperature at T- 34/ 06 Tc2 %or npentane
200 150 100 50 0
Pressure (100000Pa) ideal gas
3 7 5 3 9 7 2 7 van der Waals 5 7 6 3 8 1 1 4 3 1 5 0 3 5 4 9 9 5 1 1 6 3 . 2 . 1 4
Molar volume (0.0001m3/mol)
15
Figure 1$: Temperature at T- 3,/ 066 Tc2 %or npentane
&s observe)# the /as behave) i)eally at hi/her temperat%res. ,he /raph of the /ases in the 0an )er aals )ra*s near the /raph of i)eal /as. &t lo*er temperat%res# loops are observe). ,his is an in)ication sho*in/ the pro)%ct of the mathematical artefact. !n correctin/ this loops# the experimenters %se the ax*ell constr%ction *here &1 : &2.
,he table belo* presents the five temperat%res +,c an) belo* ,c of carbon monoxi)e *ith their respective vol%mes +01#02#0;# Ptrial# an) = &rea Difference. T90)
( m,mol
+trial + a
; Area
")-*-*./*
C'00o 1e 2eer340e2
01: .FE-$" 02: 1.22E-$ 0;: 1.E-$ 01: .1FE-$" 02: 1.2FE-$ 0;: 2.E-$
;$$16$
1.39
22F$"$
1.21
01: 6.2E-$" 02: 1.;;E-$ 0;: ;.1E-$ 01 : 6.1E-$" 02 : 1.;E-$ 0; : .$6E-$
1F$26$
1.$
162$"F
$.
132.
1$8
1$/
114
11/&5
Table $: Five temperatures 9Tc and belo# Tc) o% carbon mono!ide 16
Usin/ the )ata /iven in the tables above# the /raphs of the p-0 )ia/rams of Carbon monoxi)e for five temperat%res *ere compresse) into one as constr%cte) belo*.
Figure 1,: *ompressed p( diagrams o% *arbon mono!ide %or %ive temperatures Usin/ excel# a scatter chart of the )ata *as /enerate). ,he bino)al c%rve of carbon monoxi)e *ith respect to the press%re an) vol%me is sho*n belo*.
17
Binodal Curve
4 3.5 3 2.5 p /10^5 Pa 2
V/m^3/mol
1.5 1 0.5 0 0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04 3.50E-04 4.00E-04 4.50E-04
Figure 13: =inodal curve o% *arbon mono!ide #ith respect to v and p ,he table belo* presents the five temperat%res +,c an) belo* ,c of n-pentane *ith their respective vol%mes +01#02#0;# Ptrial# an) = &rea Difference. T/K
!/ m#3/mol
Ptr$al / P a
% &rea '$erence
18
*.* 3./
344
33/
3,4
C'00o 1e 2eer340e2
""56666 ;1$$6$
01 : ;.;6E-$ 02 : .1E-$ 0; : 6.$$E-$ 01 : ;.1FE-$ 02 : ."E-$ 0; : 6."6E-$ 01 : 2.6E-$ 02 : ."E-$ 0; : .1E-$ 01 : 2.E-$ 02 : .61E-$ 0; : .E-$
1.
2F6""$$
1.
2"1"$$
1.6F
2611$$
1.;;
Table ,: Five temperatures 9Tc and belo# Tc) o% npentane
Usin/ the )ata /iven in the tables above# the /raphs of the p-0 )ia/rams of Carbon monoxi)e for five temperat%res *ere compresse) into one as constr%cte) belo*.
p-V Diagrams 100 90 80
p/100000 Pa
70
Tc
60
T=435 K
50
T=440 K
40
T=455 K
30
T=460 K
20 10 0
0
0
0
0
0
0
0
0
0
0
Vm/m^3 mo-1
Figure 14: *ompressed p( diagrams o% npentane %or %ive temperatures
19
Binodal Curve 40 35 30 25
!sa"/10^5
20 15 10 5 0 2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
V/m^3/mo
U sin/ excel# a scatter chart of the )ata *as /enerate). ,he bino)al c%rve of carbon monoxi)e *ith respect to the press%re an) vol%me is sho*n belo*. Figure 1.: =inodal curve o% *arbon mono!ide #ith respect to v and p ,he follo*in/ )ata in the table belo* *as %se) to plot the li(%i)-/as coexistence c%rve of carbon monoxi)e in a pT )ia/ram. T0 11/&5 114 1$/ 1$8 132.
+9+a) 162$"F 1F$26$ 22F$"$ ;$$16$ ")-*-*./*
; Area
Table 3: Temperatures o% carbon mono!ide #ith their pressures and ; area di%%erences
20
#$ p-T Diagram 4000000 3500000 3000000 2500000
!sa"/105 !a
2000000 1500000 1000000 500000 0 105 110 115 120 125 130 135
T/ %K& ,he li(%i)-/as coexistence c%rve of carbon monoxi)e *as plotte) %sin/ the previo%s )ata for press%re an) = area )ifference as sho*n in a p-, )ia/ram belo*.
Figure 17: >iquidgas coe!istence curve o% carbon mono!ide plotted in a pT diagram
,he follo*in/ )ata in the table belo* *as %se) to plot the li(%i)-/as coexistence c%rve of n-pentane in a pT )ia/ram. T0 *.* 3./ 344 33/ 3,4
p+a "".5 ;1.$$6 2F.6"" 2".1" 2.611
; Area
Table 4: Temperatures o% npentane #ith their pressures and ; area di%%erences
,he li(%i)-/as coexistence c%rve of n-pentane *as plotte) %sin/ the previo%s )ata for press%re an) = area )ifference as sho*n in a p-, )ia/ram belo*.
'-!('"a'( p-T Diagram 40 35 30 25
!sa"/105 !a
20 15 10 5 0 430435440445450455460465470475
T/ %K&
Figure 18: >iquidgas coe!istence curve o% npentane plotted in a pT diagram
Re)%ce) %nits of a li(%i) or /as are %nits that are scale) by their critical constants# i.e. pr - pp c2 2 'r - ''c an) T r TT c& ,he chart sho*in/ press%re as a f%nction of molar vol%me *as plotte) in a /raph of the p-0m )ia/ram of carbon )ioxi)e an) n-pentane.
22
p - Vm Diagram 600 500 400
!r(ss*r(
#$
300
'-!('"a'( 200 100 0 0.00 E+00
5.00 E-04
1.00 E-03
1.50 E-03
2.00E -03
)oar Vo*m(
Figure 15: p(m diagram o% carbon mono!ide and npentane
,he /raph belo* sho*s the re)%ce) press%re pr as a f%nction of the re)%ce) molar vol%me ( r for the press%res an) vol%mes of the coexistence c%rve.
pr - Vr Diagram 16 14 12 10
Reduced Pressure
#$
8
'-!('"a'(
6 4 2 0
0
1
2
3
4
5
6
Reduced Volume
Figure $/: pr(r diagram o% carbon mono!ide and npentane ,he /raph belo* sho*s the re)%ce) press%re pr as a f%nction of the re)%ce) molar vol%me ( r for the press%res an) vol%mes of the coexistence c%rve.
23
Coexistence Curve Vr - Pr Diagram 1.5
(,*c(, !r(ss*r( pasa"/10^5!a
0.15
1
0.1
0.5 0 0.00E+00
0.05
2.00E+00
4.00E+00
'-!('"a'( #$
0 6.00E+00
(,*c(, o*m( Vm/m3mo-1
Figure $1: *oe!istence curve o% (r+r diagram %or carbon mono!ide and npentane &ccor)in/ to the La* of Correspon)in/ States# s%bstances at correspon)in/ states behave ali>e at same re)%ce) states. ,he /raph above represents that *hen 0 an) P of a s%bstance s%ch as n-pentane an) Carbon monoxi)e are re)%ce)# the behavior of the s%bstances still correspon)s to that of the ori/inal 0 an) P. ,he experimental critical constants for ethane an) ar/on are /iven in the table belo*. Substance ?thane
p(cr)/[Pa] T(cr)/[K] ;6 ;$".
v(cr)/[m3.mol-1] $.$$$1
Argon 6;"1 1"$.2 .";E-$" Table .: ?!perimental critical constants %or ethane and argon
T(K) "$$
2
Tr (K) 1.6;1F 1.6;
7 Re)%ce) temperat%res at *hich these )ata of Ethane an) &r/on are collecte) at 1.6;1FG an) 1.6;G respectively. Derive) by )ivi)in/ ,+G over ,+cr in G.
24
p s 1/V Diagram Ep(rim('"a 8.00E+07 6.00E+07 rgo'
!r(ss*r(/10^5 !a
4.00E+07
E"a'(
2.00E+07 0.00E+00 0.00E+00
1.00E+04
2.00E+04
3.00E+04
1/V
,he follo*in/ /raphs sho* comparison bet*een the experimental )ata an) the pre)ictions of the van )er aals e(%ations by plottin/ p as a f%nction of 1v.
p s 1/V Diagram a' ,(r aas 600 500
!r(ss*r(/10^5 !a
400
rgo'
300
E"a'(
200 100 0 0.00E+00
1.00E+04
2.00E+04
3.00E+04
1/V
Figure $$: ?!perimental p vs 1( diagram o% argon and ethane Figure $,: (an der #aals p vs 1( diagram o% argon and ethane &s it is observe)# the behavior of both the experimental an) theoretical /raphs a/rees *ith each other. ,his means that as the val%e of 1?0 increases the val%e of press%re also increases obeyin/ oyleHs La* *hich states that P I 1?0.
25
¬her /raph *as constr%cte) to test *hether the experimental )ata /iven in table 6 obey the la* of correspon)in/ states by plottin/ pr as a f%nction of 1vr .
pr - 1/Vr Diagram 1.60E+01 1.40E+01 1.20E+01 1.00E+01 rgo'
8.00E+00
!r/!a
E"a'(
6.00E+00 4.00E+00 2.00E+00 0.00E+00 0.00E+00
5.00E-01
1.00E+00
1.50E+00
2.00E+00
1/Vr/m3mo-1
Figure $3: pr as a %unction o% 1(r %or argon and ethane
,o prove the la* of correspon)in/ states mathematically# *e re*rite the van )er aals e(%ation of state in terms of re)%ce) press%re# re)%ce) vol%me an) re)%ce) temperat%re. G4$e08 T r =
T
P r=
T c
P Pc
Pc=
Vc =3 b
V r =
a 27 b
V V c Tc =
2
8a 27 Rb
So(94o08 P =
RT Vm −b
PrPc=
Pr
[
−
2
Vm
R ( TrTc ) VrVc −b
( )[ a
27 b
a
2
=
] [( −
R Tr (
a 2
VrVc )
8a 27 Rb
Vr ( 3 b )− b
)
]
][ −
a Vr
2
2
(3 b )
] 26
Pr
( )[ a
27 b
=
2
(
8a 27 Rb
)
Vr ( 3 b )−b
][ −
a 2
2
Vr 9 b
]
8 aTr 2
27 b
a
a
27 b
[ [
Pr =
Pr =
Pr =
2
Pr =
27
Vr 3 b −b
( − )(
8 aTr
(
−b )
27 Vr 3 b
8 aTr
(
27 b Vr 3
8 Tr 3 Vr
−1
−
1
27 b
2
a 27 b
a
2
−
a
27 b
2
V r 9b
)]−[ )]−[
2
a
a 2
2
Vr 9 b a 2
Vr 9 b
2
2
( (
27 b
2
a 27 b
a
2
)] )]
3
Vr
2
,he variation of 3 *ith press%re an) temperat%re can be explaine) more effectively %sin/ the so-calle) oyle temperat%re +, as reference. or a van )er aals /as# , : a?bR. &s state) above# the val%es of a an) b can be calc%late) from the critical constants of the /as. Usin/ the )ata for Carbon monoxi)e# *e *ere able to calc%late the van )er aals constants an) the ,# for the s%bstance assi/ne) to %s in the 3-calc sheet. or each temperat%re# a p vs 3 )ia/ram *as prepare).
Caron monoxide !p vs " Diagram# 2.5 2
T=T
1.5
Compression $actor/"
1
TT
0.5
TT
0
0
500
1000
1500
P/atm
27
Figure $4: p vs @ diagram o% carbon mono!ide %or each temperature &s presente) by the /raph above# the compression factor +3-calc varies )irectly as the press%re increases. !n the ,:,b /raph# the 3-calc an) p sho*s a c%rve in an increasin/ tren). Same thin/ happens *ith ,A,b an) ,B,b /raphs. !n a))ition# the above /raph reveale) that *hen temperat%re is increase) the 3-calc an) press%re also increases an) *hen temperat%re is )ecrease)# the 3-calc an) press%re also )ecreases.
Disc%ssion '''''''''''''''''''''''''''''''''''''''''''''''''''''''' ,his laboratory experiment *as performe) to )emonstrate a verifie) )escription an) mathematical comparison of ho* real /ases behave )ifferently a/ainst i)eal /ases %sin/ the i)eal /as e(%ation an) van )er aals e(%ation of state. ase) on the /raph sho*n in fi/%re ; for carbon monoxi)e an) fi/%re -12 for n-pentane# it can be observe) that the /raphs of van )er aals e(%ation of state *as oscillate) compare) to that of the i)eal /as e(%ation *hich forme) a smooth c%rve. ,hese oscillations are )%e to the loops *hich can be correcte) %sin/ ax*ell constr%ction. ,he loop implies that if press%re increases# the vol%me also increases# th%s not obeyin/ empirical la*5 the oyleHs La*. Upon the res%lts of the /raphs of molar vol%me as a f%nction of press%re# fo%n) in i/%re 1; an) 1"# an inverse proportionality *as )etermine) for both /ases at )ifferent temperat%res. ,he molar vol%mes 01 an) 0; *ere obtaine) from the ax*ell constr%ction. &s the loops *ere correcte)# a hori4ontal line from the isotherms can be seen. Usin/ ax*ell constr%ction# an ass%mption that &1 : &2 or the = &rea )ifference not excee)in/ 2= for the /raph *as consi)ere) to be /oo) eno%/h. &s observe)# both /ases +carbon monoxi)e an) n pentane behave) i)eally at hi/her temperat%res. ,he /raph of the /ases in the 0an )er aals )ra*s near the /raph of i)eal /as +see fi/%re an) fi/%re 12 above. &t lo*er temperat%res# 28
loops *ere observe). ,his is an in)ication sho*in/ the pro)%ct of the mathematical innovations. !n correctin/ this loops# the ax*ell constr%ction *as %tili4e) *ith an ass%mption that &1 : &2. ,he re/ion of the c%rve by *hich the press%re an) vol%me are constant is calle) the critical point *hich *as also observe) in i/%re 1; an) 1". ,he bino)al c%rve of /ases is the re/ion *herein t*o phases coexist. !n this lab activity# it is also calle) the li(%i)-/as phase. ,his happens *hen press%re an) temperat%re in a re/ion remains %nchan/e)# allo*in/ the compression or expansion of the /ases in a certain point. ,he bimo)al c%rves of carbon monoxi)e an) n-pentane is sho*n in fi/%re 1 an) 16 respectively.
,he p-, )ia/ram also presente) a relationship# press%re bein/ the in)epen)ent variable an) temperat%re bein/ a )epen)ent variable. ,he /raphs that *ere /enerate) )isplay a transition phase *here /as an) li(%i) coexist an) )escribes one of the characteristics of a van )er aals e(%ation of state *here attractive an) rep%lsive interactions are in e(%ilibri%m# th%s permittin/ the coexistence of other phases. ,he branch that is plotte) by the /as phase sho*n in fi/%re 2 of the materials an) metho) part )isplays that there is an existin/ critical point *here soli)# li(%i) an) /as phase is possible.
&ccor)in/ to the La* of Correspon)in/ States# s%bstances at correspon)in/ states behave ali>e at same re)%ce) states. ,he /raph of p-0 )ia/rams in the res%lts sho* that *hen 0 an) P of a s%bstance s%ch as n-pentane an) Carbon monoxi)e are re)%ce)# the behavior of the s%bstances still correspon)s to that of the act%al 0 an) P. ,his is sho*n in i/%re 2$ an) fi/%re 21. ,his also applies *hether the reciprocal of the molar vol%me is re)%ce) since an i)entical c%rve has also forme).
,he compression factor 3# the ratio of its meas%re) molar vol%me 0m: 0?n is a meas%re of )eviation from the perfect behavior of a /as. & val%e of 3:1 )enotes that the /as 29
behaves i)eally or perfectly an) is ass%me) to be the basis for a /as to classify as perfect or not. &s presente) by the i/%re2" above# the compression factor +3-calc varies )irectly as the press%re increases. !n the ,:,b /raph# the 3-calc an) p sho*s a c%rve in an increasin/ tren). Similarly# it also happens *ith ,A,b an) ,B,b /raphs. !n a))ition# the above /raph reveale) that *hen temperat%re is increase)# the 3-calc an) press%re also increases an) *hen temperat%re is )ecrease)# the 3-calc an) press%re also )ecreases.
C89CLUS!89 '''''''''''''''''''''''''''''''''''''''''''''''''''''''' Jenerally# at hi/h temperat%res# the meas%re of the behavior of real /as )eviation from perfect /as can be )etermine) *hen s%fficient information is /iven to /enerate isotherms from the van )er aals e(%ation of state. !n correctin/ van )er aals /as c%rves# *here there is a presence of oscillations pro)%ce) by loops# the mathematical e(%ation is %se). ith the ass%mption that the areas %n)er the c%rve are e(%al that is# &1 : &2# a small )ifference not excee)in/ 2= *as also consi)ere). ,he %se of the bino)al c%rves simply ill%strates the re/ions *here /as an) li(%i) coexist. Dia/rams of p-, mainly exhibit the transition phase *here the /as an) li(%i) also coexist. ,he la* of correspon)in/ states /enerally sho*e) that *hen the experimental val%es for a s%bstance# specifically a /aseo%s compo%n) *ere plotte)# it /enerates a c%rve. &lso *hen the re)%ce) val%es of the experimental *ere plotte)# an i)entical c%rve *as /enerate) sho*in/ the same behavior as that of the experimental val%es. ith the %se of the compression factor 3# *e *ere able to establish an) pict%re o%t the )ifference in the behavior of real an) i)eal /as.
ost
importantly# this experiment sho*e) that real /ases act%ally )eviate from i)eal /ases. !n this 30
sprea)sheet experiment# it *as also )etermine) ho* real /ases behave from i)eal ones %sin/ /raphical vis%ali4ations an) calc%lations. ,he %se of compression factor 3# La* of correspon)in/ states an) van )er aals e(%ation of states acc%rately ill%strate) the molec%lar interactions pre)ictin/ the relationship of variables of the state.
References '''''''''''''''''''''''''''''''''''''''''''''''''''''''' K1
&t>ins# P..# )e Pa%la# M. +hysical *hemistry +2$1$. Chapter 15 ,he properties of
/ases. Physical Chemistry Fth E)ition +pp. 1F-;. 8xfor) University Press# 2$1$ K2
Rei)# R.C.# Pra%snit4# M.. # Polin/# .E.# The properties o% Gases >iquids2 3th
?dition2 cJra*-7ill# 1F K;
Reichl# L.E.# A modern course in Btatistical +hysics2 E)*ar) &rnol) L,D# 1F$
K
cN%arrie# D.&.# Btatistical "echanics# University Science oo>s# 2$$$
K"
Silbey# R.M.# &lberty# R.&.# a*en)i# .J. +hysical *hemistry# th e)ition# iley#
2$$" K6
Jarlan)# C..# 9ibler# M..# Shoema>er# D.P.# ?!periments in +hysical *hemistry# Fth
e)ition# cJra*-7ill# 2$$F
31
