Exercise 8 Viscosity of Pure Liquids and Solutions CHEM 111.1 3L
Distor, edric! ". #lores $$$, Deo%racias C. &equiso, Princess '. Sison, &ic(ard Dean ). So*era, Eri!a +.
Date Perfor*ed Marc( - and 11, /1Date Su0*itted Marc( 18, /1-
Mr. Herin Errol ". Mendo2a La0oratory $nstructor
I. Introduction #luids flo flo in a *anner *anner in (ic( its co*4onents, co*4onents, (ic( (ic( can 0e ato*s ato*s or *olecules, sli4 4ast 4ast eac( ot(er and allo *oe*ent (eneer an adequate s(ear force is a44lied to its total or 4artial *ass. Viscosity, Viscosity, *a%nitud *a%nitude e of
η
η , is usually defined as t(e ease at (ic( t(e fluid is floin%. +s t(e
increases, t(e *ore difficult to flo it is for t(e fluid. $t is also so*eti*es
ter*ed as t(e 5t(ic!ness6 of t(e fluid 7&oussel, /19 t(e (i%(er t(e iscosity, t(e *ore 5t(ic!6 a fluid is. #or different fluids, (oeer, t(e ease at (ic( fluids flo is affected 0y certain factors suc( suc( as te*4e te*4erat rature ure,, t(e inter* inter*ole olecul cular ar force forces s 4resen 4resentt on t(e fluid fluid,, t(e struc structu ture re of t(e t(e co*4o co*4onen nents ts of t(e t(e liquid liquid,, for for exa*4l exa*4le, e, t(e len%t( len%t( of t(e *olecule *olecules s in solut solutio ion, n, and and t(e concentration, in case t(e fluid is a solution 7:arland, //3. Viscosity is usually ex4ressed in 4oise or %;c* s in C:S syste*, or in s;* or !%;* s in S$ units. Differen Differentt *et(od *et(ods s and a44arat a44aratuse uses s (ae (ae 0een 0een deelo4e deelo4ed d for t(e quantif quantificat ication ion and deter*ination of t(e alue of
η for liquids and solutions. + *et(od of *easurin% iscosity is
0y usin% r(eo*eters, (ic( consist of rotatin% concentric cylinders, an inner and outer cylinder, it( t(e torque of t(e inner cylinder o0sered (ile t(e outer cylinder 0ein% rotated 0y a *otor 7+t!ins, //<. $t *a!es use of t(e conce4t of s4rin% deflection, torque and an%ular elocities. =t(er *et(ods include t(e use of an ultrasonic 4ro0e, a *et(od inolin% t(e *easure*ent of 4ressure dro4 t(rou%( a friction tu0e, t(e use of a torque isco*eter, isco*eter, and a 4rocedure 4rocedure inolin% a ti*ed fall of 4iston on a cylinder. =ne of t(e *ost co**on 4rocedures, on t(e ot(er (and, inoles t(e *easure*ent of t(e flo ti*e of fluids in a isco*eter, (ic( is co*4osed of seeral ca4illary tu0es (ere fluid flo is o0sered. So*e isco*eters t(at are usually used are t(e =stald, Cannon>#ens!e, and ?00elo(de isco*eters. $n usin% isco*eters, t(e ti*e of donard flo of a olu*e of a fluid, (ic( is caused *ainly 0y %raitational 4ull is *easured and t(en calculated usin% t(e *odified and corrected Ha%en>Poiseuille equation
η= Aρt −
Bρ t
"(e alue of t(e standard :i00s@ free ener%y, ener%y,
∆ ´ G
78>1
o
can 0e co*4uted usin% a *odified
Eyrin%@s equation
´ = RT ln ∆G ln o
ηM h N A ρ
78>
"(e iscosity of 4oly*er solutions can 0e deter*ined usin% t(e Mar!>Houin! equation
[ η ] =k M a (ere
[ η]
is t(e intrinsic iscosity of t(e solution, and
78>3
k and
for a %ien 4oly*er>solent syste* at a certain te*4erature.
a are e*4irical constants
"(e o0Aecties of t(e t(e exercise are to 7a deter*ine deter*ine t(e iscosities of aqueous solutions of *et(a *et(ano noll 7CH3=H =H and and >4r >4ro4 o4an anol ol 77CH 77CH3CH=H CH=H at ario arious us conc concen entr trat atio ions ns99 70 70 t(e t(e t(er*ody t(er*odyna*i na*ic c 4ro4erti 4ro4erties es of t(e *ention *entioned ed test liquids liquids usin% usin% Eyrin%@s Eyrin%@s equation, equation, and9 and9 7c calc calcul ulat ate e t(e t(e *ole *olecu cula larr ei% ei%(t (t and and root root>* >*ea ean> n>sq squa uare re end> end>to to>e >end nd len% len%t( t( of sodi sodiu* u* car0oxy* car0oxy*et(y et(ylcell lcellulos ulose e 7aCMC 7aCMC at differe different nt salt concentra concentration tions s usin% usin% Mar!>Hou Mar!>Houin!@ in!@s s equation.
II. Materials and Methods A. Apparatus and Equipment Cannon>#ens!e and ?00elo(de isco*eters "(er*ostatted ater 0at( +s4irator "i "i*ers "(er*o*eter
"o4 loadin% 0alance Stirrin% rod Volu*etric flas!s )ea!ers Pycno*eter
B. Reagents Cali0ration liquids ater, et(anol, et(yl acetate "est "est liquids *et(anol, >4ro4anol Solent /.1/ M and /.//1 M aCl solutions Poly*eric salt sodiu* car0oxy*et(ylcellulose car0oxy*et(yl cellulose 7aCMC C. Procedure #or t(e %eneral 4rocedure of t(e use of Cannon>#ens!e 7and =stald isco*eters, t(e isco*eters ere initially as(ed it( deter%ent and ater, t(en rinsed it( dH = and acetone, and finally rinsed it( t(e sa*4le liquid for analysis. 1/ *L of t(e sa*4le liquid as 4laced into t(e isco*eter, (ic( as equili0rated afterards in t(e t(er*ostatted 0at( of recorded te*4erature. "(ree consecutie flo ti*es it( /.1>second difference ere o0tained, alloin% *ore trials (eneer it as necessary. Bit( t(e %eneral 4rocedure, t(e isco*eters ere cali0rated usin% t(e cali0ratin% liquids as sa*4le liquids at constant 0at( and roo* te*4erature9 *easured flo ti*es at /, , 3/, 3, -/, -, and / C usin% 4ure test liquid9 and *easured flo ti*es solutions of different concentrations. )inary solutions of t(e test liquid and H = ere 4re4ared at concentrations /.1/, /./, /.-/, /.
III. Results and Discussion
Viscosity,
η is a *easure of a resistance to flo of fluid9 *ore s4ecifically it quantifies t(e
resistance t(at a 4ortion of t(e fluid extends to t(e ot(er 4ortions of t(e sa*e fluid. $f to solid 4lates are se4arated 0y a t(in fil* of fluid, as s(on 0y t(e fi%ure 0elo, 0elo, and t(e u44er 4late is *oed *oed stead steadily ily in t(e x>direct x>directio ion n it( it( a %ien %ien eloci elocity ty,,
V o
, a force ill 0e needed to
oerco*e t(e friction 4resent due to t(e fluid 0eteen t(e 4lates. "(is required force aries for different fluids, elocities, 4late si2es, and t(e distances 0eteen t(e 4lates. )y *easurin% t(e τ , (oeer, t(e effect of different force 4er unit area of t(e 4late defined as t(e s(ear stress, 4late si2es considered is eli*inated.
#i%ure 8.1. "(e slidin% 4late ex4eri*ent.
+t lo alues of
V o
, t(e elocity 4rofile in t(e fluid 0eteen t(e 4lates is linear. $f a 4lot of
τ ersus t(e differential c(an%e in t(e elocity it( res4ect to t(e ertical distance of t(e 4arallel 4lates,
dV dy , co**only called as t(e s(ear rate or elocity %radient, is constructed
for different fluids at constant te*4erature and 4ressure, t(e folloin% fi%ure ould 0e o0sered
#i%ure 8.. Plot of
τ ersus
dV dy for different fluids at constant te*4erature and 4ressure.
"(e 0e(aior t(at is *ostly ex(i0ited 0y *any fluids in nature is t(at re4resented 0y t(e strai%(t line 4assin% t(rou%( t(e ori%in. "(ese fluids are called netonian fluids, (ic( o0eys t(e eton@s la of iscosity, ex4ressed *at(e*atically as 7De eers, 11
dV τ = μ dy
78>-
(ic( i*4lies t(at t(e s(ear stress ex4erienced 0y t(e liquid is directly 4ro4ortional to t(e elocity %radient, (ere t(e 4ro4ortionality constant is t(e a0solute iscosity of t(e fluid,
μ .
eton@s la of iscosity conce4tuali2es a fluid as 0ein% arran%ed into layers of 4lanes of *olecules (erein a 4late is !e4t static (ile t(e ot(er 4arallel 4late is *oin% steadily at a certain elocity, as ex(i0ited 0y t(e slidin% 4late ex4eri*ent. + *oin% 4late 4ulls a layer of liquid as it *oes at a certain len%t(, creatin% a transfer of *o*entu* to t(e ot(er layers. "(e sa*e is also ex(i0ited 0y t(e ot(er layers located a0oe t(e ot(er layers. )ecause of t(e frictional forces due to t(e in(erent c(aracteristics of t(e fluid, t(e *a%nitude of t(e elocity of eac( layer ill 0e less t(an t(e s4eed of t(e layer a0oe it. "(e layers of t(e fluid offer frictional η . forces of resistance t(at %ies rise to t(e iscosity of t(e fluid, %enerally sy*0oli2ed as Hoeer, t(is is only alid for la*inar flos c(aracteri2ed 0y s*oot( strea*lines and (i%(ly ordered *otion, (ic( occurs as a series of t(in 4lates of liquid s*oot(ly slidin% 4ast eac( ot(er at different elocities. La*inar flo is in contrast it( a tur0ulent flo, (ic( ex(i0its eddy for*ation and is c(aracteri2ed 0y elocity fluctuations and (i%(ly disordered *otion. "ur0ulence is deter*ined 0y different *odels and equations, since it does not o0ey t(e eton@s la of iscosity 7#oust, Ben2el, Clu*4, Maus G +ndersen, 1
#i%ure 8.3. 7a La*inar and tur0ulent flo. #luids can 0e %enerally classified as eit(er netonian or non>netonian fluids. etonian fluids usually ex(i0it la*inar flo at lo elocities. "(e alues of
η is constant at certain
te*4eratures and 4ressures and is inde4endent of t(e elocity %radient. So*e exa*4les of netonian fluids include all %ases, all liquids it( si*4le c(e*ical for*ulas, and *ost solutions η alues, (ic( are of si*4le *olecules. =n t(e ot(er (and, non>netonian fluids (ae de4endent on t(e alues of
τ , (ic( is defined as t(e force 4arallel to t(e area 0ein%
considered, as o44osed to nor*al forces and stresses t(at acts 4er4endicularly to an area. Co**on ty4es of non>netonian fluids are )in%(a* fluids, (ic( can resist s*all s(ear forces infinitely 0ut flo easily under lar%e s(ear stresses. Exa*4les are slurries, Aellies and toot(4aste. Pseudo4lastics li!e 0lood and ot(er 4oly*er solutions ex(i0it iscosities t(at decrease it( increasin% elocity %radient. "(ose considered unco**on are dilatants fluids li!e starc( solutions, (ic( (ae alues of iscosity t(at increases as t(e elocity %radient increase. "(e *et(od used in t(e ex4eri*ent is t(e ca4illary rise *et(od, (ere t(e iscosity of fluids are deter*ined 0y *easurin% t(e ti*e of flo of a %ien olu*e,
V , of a liquid t(rou%( a
ertical ca4illary tu0e under t(e influence of %raity. Ca4illary isco*eters consist of a ?>s(a4ed %lass tu0e (eld ertically9 in one ar* of t(e ? is a ertical section of 4recise narro 0ore, t(e ca4illary. +0oe t(is is a 0ul09 it( it is anot(er 0ul0 loer don on t(e ot(er ar*. "(e liquid is dran into t(e u44er 0ul0 0y suction, and t(en alloed to flo don t(rou%( t(e ca4illary into t(e loer 0ul0. "o *ar!s, one a0oe and one 0elo t(e u44er 0ul0, indicate a !non olu*e. =stald and Cannon>#ens!e isco*eters are so*e a44aratuses t(at *a!e use of t(is idea. +n =stald isco*eter (as a strai%(t tu0e cured at t(e 0otto* 4art and to 0ul0s, (ic( are located at t(e u44er and loer ends of t(e tu0e as s(on in t(e folloin% fi%ure
#i%ure 8.-. +n =stald isco*eter. )ecause t(e tu0e is strai%(t, t(e flo ti*e of a %ien liquid is lon%er co*4ared to t(at of t(e Cannon>#ens!e isco*eter, (ic( is also co*4osed of to 0ul0s, 0ut ex(i0its a slanted 4art of t(e tu0e, as s(on 0y t(e fi%ure 0elo.
#i%ure 8.. + Cannon>#ens!e isco*eter. Measure*ents o0tained usin% an =stald isco*eter is de4endent on t(e olu*e of t(e liquid used 0ecause t(e 4ressure (ead aries it( different solutions. )y usin% t(e Cannon>#ens!e isco*eter, a *odification of t(e =stald isco*eter, *ini*u* c(an%es in t(e 4ressure (ead since t(e loer 0ul0 is located directly 0elo t(e u44er 0ul0 is 4roduced resultin% to t(e ertical ali%n*ent in t(e isco*eter. "(e differences in 4ressure (eads and ot(er in(erent errors, (ic( are *ostly caused 0y t(e %eo*etry of t(e tu0es, are corrected 0y t(e Cannon>#ens!e. +not(er a44aratus used in t(e ex4eri*ent is t(e sus4ended leel or ?00elo(de isco*eter, (ic( is co*4osed of t(ree tu0es, and 0ul0s located and desi%ned in suc( a ay t(at errors due to certain conditions, suc( as t(e 4ressures actin% a0oe, and ot(er in(erent errors are corrected. "(e liquid initially dran into t(e s*all u44er 0ul0 is not connected to t(e reseroir as it flos don t(e ca4illary durin% *easure*ent. "(e ca4illary is rat(er sus4ended a0oe t(e reseroir. $t (as a t(ird ar* extendin% fro* t(e end of t(e ca4illary and o4en to t(e at*os4(ere, (ic( ensures t(at t(e only 4ressure difference 0eteen t(e to4 of t(e 0ul0 and t(e 0otto* of t(e ca4illary is t(at due to t(e (ydrostatic 4ressure or t(e ei%(t of t(e liquid. "(e 4ressure
(ead of t(e sus4ended liquid on t(e ca4illary tu0e is inde4endent of t(e liquid ori%inally 4laced on t(e isco*eter. "(is ty4e of isco*eter is usually used for iscous and 4oly*er solutions 7Daniels, Matt(es, Billia*s, )ender G +l0erty, 1<.
#i%ure 8.<. +n ?00elo(de isco*eter.
"(e flo of t(e liquid in t(e ca4illary is %oerned 0y t(e Ha%en>Poiseuille equation deelo4ed 0y 'ean Louis Poiseuille and :ott(ilf Heinric( Ha%en, ex4ressed as 7)ird, Steart G Li%(tfoot, // 4
πr t∆ P V = 8 ηl
(ere
r is t(e tu0e radius,
l is t(e tu0e len%t(,
t is t(e ti*e of flo, and
78>
∆ P is t(e
4ressure difference at t(e tu0e ends. Since t(e to isco*eters (ae tu0es o4en at 0ot( ends, ∆ P can 0e calculated as t(e alue of
∆ P= ρgh
)y su0stitutin% t(e equation for
78><
∆ P , t(e Ha%en>Poiseuille equation could 0e ritten as 4
π r tρgh V = 8 ηl
78>F
"(e Ha%en>Poiseuille equation does not consider t(e 4ressure dro4 due to t(e *otion of t(e fluid itself9 t(e !inetic ener%y correction are necessary for accuracy. "(e equation s(ould 0e corrected for tu0e>end effects 0ecause t(e liquid flo could not 0e considered entirely la*inar, 0ecause a layer of a liquid is slidin% relatie to one anot(er 0ot( in t(e entrance and exit re%ions, resultin% to eddy for*ations, (ic( i*4lies tur0ulence. $n t(e entrance re%ion of t(e ∆ P 9 in t(e exit re%ion, t(e ener%y tu0e, t(e acceleration of t(e fluid decreases t(e alue of due to t(e deceleration of t(e fluid is dissi4ated as (eat. "(e alue of t(e correction factor, ex4ressed as
ρV 8 πlt . "o si*4lify calculations, constants
A and
B are introduced in t(e
equation, reducin% it into t(e folloin% relation
η= A ρt −
Bρ t
B as
ρV 8 πlt . "(e alues of t(e constants
4
(ere A is defined as and
π r gh and 8Vl
78>8
A
B can 0e calculated usin% t(e infor*ation of t(e di*ensions of t(e isco*eters used,
0ut so*e errors cannot 0e aoided due to so*e irre%ularities suc( as t(e non>unifor*ity of t(e ca4illary tu0e. #or furt(er si*4lification of t(e calculations, t(e isco*eters to 0e used s(ould 0e A and B can 0e deter*ined 0y t(e deter*ination of ρ and t cali0rated. Constants for liquids it( !non
η . )y rearran%in% t(e equation a0oe, η B = A − 2 ρt t
)y 4lottin% equal to
η ρt ersus
78>
1 2
t
, a strai%(t line it( a y>interce4t equal to
A and a slo4e
– B can 0e o0tained, as su%%ested 0y t(e *odified equation. ?sin% t(e liquids
ater, et(yl acetate and et(anol, different alues ere o0tained (ic( ere used for t(e cali0ration of t(e isco*eters. "(e folloin% data ere o0tained for t(e cali0ration of t(e Cannon>#ens!e isco*eters, (ic( ere used in t(e ex4eri*ent. "a0le 8.1. Cali0ration of t(e Cannon>#ens!e isco*eter used for 1>4ro4anol. η A&erage time o' )iscosity !" ρ % Calibrating Liquid Density !" #g$m 'lo( t !" s cP Bater .F383 1/.F /.88 Et(yl +cetate 88.13 F. /.-3 Et(anol F8.33F F. /.31 Slo4e fro* linear re%ression 7 B , *
.8<-x1/ ><
y>interce4t fro* linear re%ression 7 A , *;s Correlation coefficient 7
1./38-x1/ >F
r
/.81--</3
"a0le 8.. Cali0ration of t(e Cannon>#ens!e isco*eter used for >4ro4anol. A&erage time o' )iscosity η !" Calibrating Liquid Density ρ !" g$cm% t !" s 'lo( cP Bater /.F383 /.F1 /.88 Et(yl +cetate /.8813 3-.8< /.-3 Et(anol /.F833F .31 /.31 Slo4e fro* linear re%ression 7 B , c*
A , c*;s
y>interce4t fro* linear re%ression 7 Correlation coefficient 7
-.88
r
/./
0 0 0
f(x) = - 0x + 0 R² = 0.96
0 0
η/ρt (m2/ s2)
0 0 0 0 0 0.01
0.01
0.01
0.01
0.02
0.02
0.02
1/t2 (1/s2)
#i%ure 8.F. Cali0ration data for t(e Cannon>#ens!e isco*eter for *et(anol and /.1 M aCl> 4oly*er solution.
η/ρt (cm2/s2)
0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0 0 0
f(x)==0.82 R² - 4.89x + 0.02
0
0
0
0
0
0
0
0
0
0
0
1/t2 (1/s2)
#i%ure 8.8. Cali0ration data for t(e Cannon>#ens!e isco*eter for >4ro4anol and /.//1 M aCl>4oly*er solution. +fter t(e cali0ration of t(e isco*eters, t(e alues of
η of t(e 4ure test liquids *et(anol
and >4ro4anol at different te*4eratures ere deter*ined 0y *easurin% t(e corres4ondin% flo ti*es for t(e 4articular te*4erature. Viscosities ere co*4uted usin% t(e *odified Ha%en> Poiseuille equation. "(eoretically, as t(e te*4erature of t(e liquid increases, its iscosity decreases. B(en t(e te*4erature is increased, t(e aera%e !inetic ener%y and t(e rando* *otion of t(e *olecules are also increased, t(ere0y reducin% t(e ti*e of interaction and t(e distance 0eteen t(e*, reducin% t(e s(ear stresses ex4erienced 0y a stratified layer of *olecules and t(e 0ul! iscosity of t(e fluid 7:arland, i0ler G S(oe*a!er, //3. "(e folloin% data ere %at(ered for t(e effect of te*4erature on t(e alue of
η .
"a0le 8.3. Viscosities of 4ure *et(anol at arious te*4eratures. ρ !" A&erage time o' 'lo( Density *emperature" + t !" s % #g$m 3.1 F-.3-
"a0le 8.-. Viscosities of 4ure >4ro4anol at arious te*4eratures. ρ !" A&erage time o' 'lo( Density *emperature" + t !" s % g$cm 3.1 /.F8<11F 1F<.
)iscosity
η !"
#g$m s /.///<1
)iscosity
η !"
g$cm s
8.1 3/3.1 3/8.1 313.1 318.1 33.1
/.F81-/.FF</< /.FF/ /.F
1.3 13-.3 11F.1 1/3.F 1. 81.
3.18F8 /.8F<<< --1.88F<8 388.F8<8 3-.<FF 3//.883--
"(e ex4eri*ental data %enerally s(os t(at as t(e te*4erature is increased, t(e iscosity of t(e liquid decreases and a%rees it( t(e t(eoretical result as can 0e seen fro* t(e fi%ures and ta0les a0oe. Eyrin%@s "(eory, 0ased on t(e "ransition State "(eory, (ic( ex4lains c(e*ical reactions and ot(er *olecular 4rocesses, 4roides an ex4lanation on iscous flo, 4ro4erties of liquids and solutions, and a 0asis on correlation of iscosity data. #or iscous flo, Eyrin%@s equation is ex4ressed as
η=
∆ ´ G RT
o
h N A
e
´ V
78>1/
)y rearran%in% t(e equation, ´ ∆G
(ere
∆ ´G
o
o
= RT ln
ηM h N A ρ
78>11
h is t(e Planc!@s constant
is t(e standard *olar free ener%y for iscous flo,
(ic( is equal to <.<3- ' s,
´ V is t(e *olar olu*e of t(e su0stance and ´ ∆ H
o
+o%adro@s nu*0er. +ssu*in% t(e *olar ent(al4y c(an%e for flo, entro4y of actiation,
∆ ´S
o
´ ∆G
are 0ot( inde4endent of te*4erature,
o
N A
is t(e
and t(e *olar
can 0e ex4ressed
as
´ −T ∆ ´S ∆ ´ G =∆ H o
)y 4lottin%
´ ∆G
o
o
∆ ´S
o
T , t(e alues of
ersus
o
and
78>1 ´ ∆ H
o
can 0e calculated usin%
o −∆ S´ and y>interce4t equal to
linear re%ression, it( slo4e equal to
´ ∆ H
o
. "(e folloin%
alues ere calculated on t(e ex4eri*ent "a0le 8.. Co*4uted *emperature" +
´ ∆G
o
,
´ ∆ H
o
and
Density ρ !" #g$m
%
∆ ´S
o
of 4ure *et(anol. )iscosity η !" #g$m s
´ ∆G
o
" ,$mol
3.1 F-.3-
´ -tandard molar entropy" ∆ S
o
o
*emperature" +
´ ∆G
o
o
and
Density ρ !" %
g$cm 3.1 /.F8<11F 8.1 /.F81-3/3.1 /.FF</< 3/8.1 /.FF/ 313.1 /.F
´ -tandard molar entropy" ∆ S
o
!" ,$mol +
´ -tandard molar enthalpy" ∆ H
o
>/./
!" ,$mol
´ ∆ H
,
8.--/8.F1-1/ ./F881F .3-F< .<F1/ .-/-8 3/.F1FF
/./
!" ,$mol +
´ -tandard molar enthalpy" ∆ H
"a0le 8.<. Co*4uted
/.///<1
!" ,$mol
1/.<
∆ ´S
o
of 4ure >4ro4anol. )iscosity η !"
´ ∆G
g$cm s
o
" ,$mol
F/.1//.3 /-F
30.500000 f(x) = 0.06x + 10.65 R² = 1
30.000000
29.500000
ΔG°, J/mol
29.000000
28.500000
28.000000
27.500000 290
295
300
305
310
315
320
325
Temperature, K
#i%ure 8.. Plot of
∆ ´ G
o
ersus te*4erature for 4ure *et(anol.
86000 85000 84000
f(x) = 200.39x + 20472.3 R² = 1
83000 82000 ΔG, J/mol
81000 80000 79000 78000 77000 76000 290
295
300
305
310
Temperature, K
315
320
325
#i%ure 8.1/. Plot of
´ ∆G
o
ersus te*4erature for 4ure >4ro4anol.
+not(er condition t(at 4ossi0ly affects t(e alue of iscosity is t(e concentration. $ts effect as tested usin% 0inary solutions of t(e test liquids and ater it( different concentrations. "(e olu*e of t(e 4ure test liquid needed to 4re4are a olu*e of t(e solution to 0e tested as co*4uted usin% t(e folloin% for*ula
V A V T
(ere
=
ρB χ A M A ρ A M B− ρ A χ A M B + ρ B χ A M A
78>13
χ is t(e *ole fraction of t(e test liquid, + is t(e 4ure test liquid and ) is t(e solent
(ic( is ater in t(is case. "(e densities of t(e solutions ere calculated usin% t(e *easure*ents o0tained fro* a 4ycno*eter usin% t(e folloin% equation
ρsol =
! "y# + sol −! "y# ! "y# + H $−! "y#
% ρ H $ 2
78>1-
2
Viscosity alues, t(eoretically, are lar%ely de4endent on t(e nature of t(e 0inary solution considered, t(e inter*olecular forces of attraction, $M#+, 4resent 0eteen t(e solute and solent 4articles, and t(e *ole fraction of t(e solute in t(e solution, (ic( i*4lies its concentration. Stron% $M#+ i*4edes flo and increases t(e iscosity of t(e liquid. $f t(e solute> solute and solent>solent interactions are *ore occurrin% t(an t(e solute>solent interactions, as t(e *ole fraction of t(e solute increases, t(e iscosity decreases. $f solute>solent interactions are *ore occurrin% t(an t(e solute>solute or solent>solute interactions, as t(e *ole fraction decreases, iscosity increases. $t is also found out t(at as t(e iscosity increases, t(e 0oilin% 4oint of t(e solution rises, to%et(er it( t(e *olar ent(al4y of a4ori2ation 7Leine, //. "(e folloin% alues ere o0tained for t(e effect of concentration on t(e iscosity of a liquid. "a0le 8.F. Viscosities of *et(anol>ater solutions at arious concentrations. Mole 'raction o' η !" A&erage time o' )iscosity ρ % Density !" #g$m χ A solute" 'lo( t !" s #g$m s /. 81.1-/ .- /.//3-<1 /.8 8<-.831/ 31.3 /.//F183 /.< //./1 3/. /.//F88- /.3.-<-8 3.8 /.//31F4ro4anol>ater solutions at arious concentrations.
Mole 'raction o'
χ A
solute"
Density
/. /.8 /.< /./. /.1
ρ
!" g$cm%
)iscosity η !"
A&erage time o' t !" s 'lo(
/.<-1F8/1 /.F183/.F11F /.8F-<83 /.-F/88< /.3-/88F
g$cm s 11.-8<8
1
0 f(x) = 0x + 0 R² = 0.69
0
0
η (k/m s)
0
0
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mole fraction of solute
#i%ure 8.11. Plot of iscosity ersus concentration of solute of 1>4ro4anol.
0.9
1
800 700 f(x) = - 151.17x + 701.86 R² = 0.4
600 500 η ( /cm s)
400 300 200 100 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mole fraction of solute
#i%ure 8.1. Plot of iscosity ersus concentration of solute of >4ro4anol.
Sodiu* car0oxy*et(ylcellulose, a00reiated aCMC, a 4oly*er it( *any uses in food sciences, 4(ar*aceuticals, oil>drillin% industry, 4rotein 4urification and *any ot(er industrial a44lications, (as t(e folloin% structure
#i%ure 8.13. + sodiu* car0oxy*et(ylcellulose *olecule.
=n t(e ex4eri*ent, t(e 4oly*er as dissoled usin% to different concentrations of t(e solent to inesti%ate on t(e effect of t(e 4oly*er concentration on t(e iscosity of liquids, 4articularly a 4oly*er solution9 /.1 and /.//1 M sodiu* c(loride, aCl solution ere used.
So*e deried iscosity *easure*ents and functions 4resented 0elo can 0e coneniently used in t(e deter*ination of t(e aera%e *olecular ei%(t and *olecular ei%(t distri0ution of a 4oly*er 7CHEM 111 Lecturers and La0oratory $nstructors, /1. Viscosity alues %ie infor*ation a0out t(e s(a4e and si2e of t(e 4oly*er *olecules.
(ere
η t = ηo t o
&elatie iscosity
ηrel=
S4ecific iscosity
ηs" =
&educed iscosity
η¿ =
#
78>1F
$n(erent iscosity
1 η η&h = ln # ηo
78>18
$ntrinsic iscosity
lim η 1 η #( 0 s" ln = [ η ] = lim ηo # #( 0 #
78>1
#
η −η o ηo
=
78>1
t −t o 78>1<
t o
ηs"
is t(e concentration of t(e solution, (ile
η ,
ηo
t , and
,
t o
are t(e
iscosities and flo ti*es for 4oly*er solution and 4ure solent, res4ectiely. + 4oly*er is coiled it(out a solent. $n a 4oly*er solution, as t(e concentration increases, t(e iscosity also increases 0ecause *ore 4oly*er c(ains *ean stron%er attraction 0eteen *olecules. :enerally, as t(e concentration of t(e solent increases, t(e iscosity of t(e 4oly*er solution decreases 0ecause t(e solent co*4etes for interaction. +t lo concentrations of t(e solent, solation or dissolution, (ic( is t(e 4rocess of association and attraction 0eteen t(e *olecules of t(e solute and solent, increases. "(is ino!es uncoilin% of t(e 4oly*er, t(ere0y increasin% t(e iscosity of t(e 4oly*er solution. "(e folloin% ta0ulated alues ere calculated usin% /.1 M and /.//1 M aCl solent. "a0le 8.. S4ecific iscosities of aCMC solutions at arious concentrations usin% /.1// M aCl solution as solent. Concentration g$dL!
A&erage 'lo( time" s
1.// /.F /./ /. /.1/ /./
3.< .- 1.8 1. 1./ 1./
-peci'ic &iscosity 3.8<<<
ηs"
!
"a0le 8.1/. S4ecific iscosities of aCMC solutions at arious concentrations usin% /.//1 M aCl solution as solent. Concentration g$dL!
A&erage 'lo( time" s
1.// /.F /./ /. /.1/ /./
1.1 1.3 1.F .83. /.
-peci'ic &iscosity
ηs"
!
/.1<1F/.1/8<< /./1F313 ./8<< 3.313/-3 /
:ood solents, (en *ixed it( 4oly*ers, results in t(e uncoilin% and extension of 4oly*ers in solution 0ecause t(e stron% 4oly*er>solent attraction 4redo*inates. "(is increases t(e iscosity of t(e solution. "(e (eat of reaction or *ixin% of a 4oly*er it( a %ood solent is ne%li%i0le or 2ero since ener%y is not needed and t(e 4rocess is s4ontaneous. $n contrast, 0ad solent results to *ore coilin% of t(e 4oly*er considered (ic( reduces t(e iscosity of t(e 4oly*er solution. + 4ositie alue of (eat of reaction or *ixin% is o0tained (en a 0ad solent is *ixed it( a 4oly*er, 0ecause *ore ener%y is required to for* t(e solution. +queous sodiu* c(loride is a %ood solent for aCMC at lo concentrations9 at (i%( concentrations, t(e saltin% out effect alters t(e 4oly*er>solent interactions since ater solates aCl instead of t(e 4oly*er, (ic( results to *ore 4oly*er coilin% 7+ndrady, //8. "(e equations for t(e intrinsic iscosity *ay 0e considered linear at lo concentrations and t(e alue of a%ainst
[ η]
can 0e o0tained as t(e y>interce4t of a line 0y 4lottin% t(e reduced iscosity
# or t(e in(erent iscosity a%ainst
considered alid since t(e li*it as
# and extra4olatin% to
# =0 . "(is *ay 0e
# ( 0 is ta!en.
+ relations(i4 0eteen iscosity alues and *olecular ei%(t of a 4oly*er is found out to 0e
ηs" #
(ere
k
=kM
is a constant for class si*ilar 4oly*ers. +t infinite dilution, t(at is,
78>/
# =0 , t(e
alue of t(e intrinsic iscosity *ay 0e o0tained as
[ η ] =kM
78>1
+ relations(i4 t(at also relates iscosity and t(e *olecular ei%(t of a 4oly*er is t(e Mar!> Houin! or Mar!>Houin!>Staudin%er>Sa!urada 7MHSS equation, (ic( is deried ta!in% into account t(e root>*ean>square end>to>end distance of a linear *olecule. #or 0ranc(ed 4oly*ers it( co*4licated %eo*etries, t(e Mar!>Houin! equation s(ould 0e corrected for 0ond an%le corrections, restriction to free rotation a0out 0onds and for t(e finite olu*e of t(e c(ain se%*ents for accurate results 7+ndrady, //8. $n equation for*,
´a [ η ] =k M
)y rearran%in% t(e equation, t(e alue of
´ M can 0e calculated as
( )
´= M
(ere
78>
[η]
1 a
78>3
k
´ M is t(e iscosity aera%e *olecular ei%(t, and
k and
a
are e*4irical
constants t(at can 0e deter*ined ex4eri*entally for a %ien 4oly*er>solent syste* at a certain te*4erature. S4ecifically, t(e constant solution. "(e alue of
a
is a function of t(e %eo*etry of t(e 4oly*er
a is /. for ell>connected or ti%(tly curled 4oly*er, (ile for ri%idly
extended 4oly*ers, its alue is 1.F. "(e folloin% ta0le 4resents t(e literature alues for calculations, calculated alues for t(e iscosity alues to%et(er it( ot(er 4ara*eters and ´ ) M
"a0le 8.11. Literature alues and constants used for calculations. -ol&ent$Parameter /.1// M /./1/ M /.//1 M
*+ 10
5
a
dL$g
1.3 /.<-< /.1//
#lory@s constant 7 ,
/.1 1./ 1.-/ .8-x1/1 dL;*ol c* 3
"a0le 8.1. Viscosity alues of aCMC solutions at arious concentrations usin% /.1// M aCl solution as solent. )iscosity )alues Concentration g$dL! η η Inherent &iscosity &h ! Reduced &iscosity ¿ ! 1.// /.F /./ /. /.1/ /./ -lope o' the line m! yintercept b! Correlation coe''icient r! Intrinsic &iscosity
[ η ]¿
1.8-/ 1.F83< 1.8/F3 ./-33/ 3.3<-F <.F-- >3.F
3.8<<<<<<.-3-/8<1 .1888< /.-<3<-
-.1-3-8<
.1888<
´ A&erage molar mass M
1/181./-1
1/3F1.
!" g$mol
"a0le 8.13. Viscosity alues of aCMC solutions at arious concentrations usin% /.//1 M aCl solution as solent. )iscosity )alues Concentration g$dL! η η Inherent &iscosity &h ! Reduced &iscosity ¿ ! 1.// /.F /./ /. /.1/ /./ -lope o' the line m! yintercept b! Correlation coe''icient r! Intrinsic &iscosity
[ η] ! ´ M
A&erage molar mass
3.F383FF -.1<83<1 .F188F .-3F13 1.81/883 / >3.-333.<3/-8 >/.--/3-/3
3.13/-3-F8 .1/8<< 3.3.333<- -.-31>/.F/383/-8
3.<3/-8
-.-31-
13<.3F<
!" g$mol
9 8 7 6 5
f(x) = - 2.44x + 5.16 R² = 0.21 f(x) = - 3.77x + 4.51 R² = 0.5
4 3 2 1 0
0
0.2
0.4
0.6
0.8
1
1.2
#i%ure 8.1-. Plot for t(e deter*ination of t(e intrinsic iscosity of 1>4ro4anol usin% in(erent and reduced iscosities.
6
5 f(x) = - 3.34x + 5 R² = 0.5
4
f(x) = - 3.59x + 4 R² = 0.89
3
Reduced viscosity i!e"# (Reduced viscosity) $!%e#e!t viscosity i!e"# ($!%e#e!t viscosity)
2
1
0 0
0.2
0.4
0.6
0.8
1
1.2
#i%ure 8.1. Plot for t(e deter*ination of t(e intrinsic iscosity of >4ro4anol usin% in(erent and reduced iscosities. "(e alues of t(e e*4irical constants
k and
a can 0e deter*ined ex4eri*entally 0y
*odifyin% t(e Mar!>Houin! equation ´ log [ η ] = log k + a log M
)y 4lottin% t(e alues of
log [ η ] ersus
alue of a slo4e equal to
a and it( y>interce4t
78>
´ log M and t(rou%( linear re%ression, a line it( a log k is o0tained.
+fter t(e calculation of t(e intrinsic iscosity, t(e alue of
´ M , si2e, radius and len%t( of t(e
4oly*er ere calculated. +ssu*in% t(e 4oly*er s(a4e is s4(erical, t(e radius, 4oly*er is ex4ressed as
r , of t(e
r=
(ere
⟨r ⟩ 1 2
V s"
(
3 M V s"
)
78><
is t(e s4ecific olu*e of t(e 4oly*er. "(e root>*ean>square end>to>end len%t(,
, (ic( ca*e fro* t(e ir!ood>&ise*an t(eory, is calculated as
⟨ r ⟩ =( 1 2
(ere
4 π
1 3
∅
)
´ 13 [ η ] M ∅
78>F
is equal to .8-x1/ 1 dL;*ol c* 3, and is called t(e #lory@s constant. "(e folloin%
ere t(e calculated alues for t(e *entioned 4ara*eters.
"a0le 8.1-. Calculated 4ara*eters for aCMC usin% /.1 M aCl as solent. /sing inherent &iscosity /sing reduced &iscosity Parameter η&h η¿ ! !
´ +era%e *olar *ass 7 M ,
1/181./-1
1/3F1.
.3/8-3F3-
-.1/131FF
-./<< x1/ >3
-.//1 x1/ >3
1.-FF-F88x1/ >8
1.8--8-33x1/ >8
%;*ol Molecular radius 7
r , c*
Molecular radius 7 r , c*;*ol &oot>*ean>square end to>end len%t( 7
⟨r ⟩ 1 2
"a0le 8.1. Calculated 4ara*eters for aCMC usin% /.//1 M aCl as solent. /sing inherent &iscosity /sing reduced &iscosity Parameter η&h η¿ ! !
´ +era%e *olar *ass 7 M ,
13<.3F<
1./3<
./11/1
3.<-F8-31 x1/ >3
3.<81<- x1/ >3
-.18//3F3<
-.F-188<
%;*ol Molecular radius 7 r , c* Molecular radius 7
r
,
c*;*ol &oot>*ean>square end to>end
len%t( 7
⟨r ⟩ 1 2
#or accuracy of t(e *easure*ents done on t(e ex4eri*ent, certain conditions *ust 0e satisfied. "(e flo of t(e fluid in t(e isco*eter *ust follo t(at of t(e etonian fluid. "(e liquid *ust 0e 4ure, free of certain i*4urities, (ic( *ay affect t(e flo ti*e t(at it ould ex(i0it. "(e te*4erature of t(e syste* *ust 0e *aintained constant, since t(e iscosity alues of t(e liquid, or a fluid, in %eneral, is affected 0y te*4erature, as su%%ested 0y t(e exercise. So*e sources of errors in t(e ex4eri*ent can alter t(e results and reduce t(e accuracy of t(e data o0tained. "(e in(erent assu*4tion of t(e ex4eri*ent t(at t(e flo of t(e liquid in t(e ca4illary tu0e is co*4letely la*inar leads to so*e erroneous results. "e*4erature fluctuations, (ic( ere *ini*i2ed 0y t(e t(er*ostatted ater 0at( can 4roduce in eit(er increase or decrease in t(e flo ti*e of t(e fluid in t(e isco*eter, de4endin% on t(e fluctuations ex4erienced. B(en i*4urities are 4resent in t(e tu0es, draina%e clo%%in% occurs and t(e flo ti*e of t(e fluid is altered. +not(er error t(at affects flo ti*e is t(e tiltin% of t(e isco*eter as t(e liquid flos9 t(e isco*eter s(ould stand ertically for accurate results. "(e use of ti*ers can also 0e erroneous 0ecause eery indiidual (as different res4onse ti*es for different instances. $f t(ere are occurrences of ea4oration of solent or increase in t(e concentration of t(e solution, t(ere are c(an%es in t(e sa*4le co*4osition and t(e corres4ondin% flo ti*es and densities *easured are also altered. =t(er errors suc( as s4illa%e of sa*4le and not folloin% of 4rocedures are ot(er sources of errors.
I). -ummary and Conclusions Viscosity is a 4ro4erty of a fluid t(at c(aracteri2es a fluid@s resistance to flo and is affected 0y *any conditions. eton@s la of iscosity conce4tuali2es a fluid as 0ein% arran%ed into layers of 4lanes of *olecules (erein a 4late is !e4t static (ile t(e ot(er 4arallel 4late is *oin% steadily at a certain elocity. "(e *et(od for iscosity deter*ination used in t(e ex4eri*ent is t(e ca4illary rise *et(od, (ere t(e iscosity of fluids are deter*ined 0y *easurin% t(e ti*e of flo of a %ien olu*e,
V , of a liquid t(rou%( a ertical ca4illary tu0e
under t(e influence of %raity. "(e flo of t(e liquid in t(e ca4illary is %oerned 0y t(e Ha%en> Poiseuille equation, (ic( as used entirely in t(e calculations of t(e iscosities and ot(er 4ara*eters used in t(e ex4eri*ent. +s t(e te*4erature of t(e liquid increases, its iscosity decreases due to t(e addition of ener%y caused 0y t(e te*4erature rise, oerco*in% t(e inter*olecular forces of attraction, (ic( (inders flo and increases iscosity. "(e effect of concentration de4ends u4on t(e solation effect of t(e solent in t(e solute. Deried iscosity *easure*ents and functions can 0e coneniently used in t(e deter*ination of t(e aera%e *olecular ei%(t and *olecular ei%(t distri0ution of a 4oly*er. Viscosity alues %ie infor*ation a0out t(e s(a4e and si2e of t(e 4oly*er *olecules. So*e sources of errors li!e
in(erent assu*4tions, ex4eri*ental conditions, and syste*atic and rando* in t(e ex4eri*ent can alter t(e results and reduce t(e accuracy of t(e data o0tained. #or 0etter understandin% of t(e conce4ts, trends and effects of 4ro4erties and conditions, it is reco**ended t(at t(e ex4eri*ent *ay use *ore co*4ounds and solution>solent 4airs.
). -ample Calculations Calibration o' )iscometer
η B = A − 2 ρt t
Calibrating Liquid Bater Et(yl +cetate Et(anol )y linear re%ression,
ρ !" #g$m% Density .F383 88.13 F8.33F
A&erage time o' 'lo( t !" s 1/.F F. F.
)iscosity g$cm s /.88 /.-3 /.31
η !"
Slo4e I >.8<-x1/ >< * Jalue for >)K y>interce4t I 1./38-x1/ >F *;s Jalue for +K r I /.81--</3
E''ect o' *emperature on )iscosity
η= A ρt −
Bρ t
#or 4ure *et(anol at 3.1 ,
η= ( 1.02384 + 10 ! )( 794.3460 kg / ! −7
η= 0.000616715
2
3
) ( 9.85 s) −
( 2.28624 + 10−6 !2 )( 794.3460 kg / !3 ) 9.85 s
kg !s
Calculation o' *hermodynamic Data
∆ ´G = RT ln o
(
ηM h N A ρ
o ∆ ´G = 8.314
)
kg !s
)(
−3
60.096 + 10
kg !ol
)
- ( 293.15 * ) ln !ol * ( 6.626 + 10−34 - s ) ( 6.023 + 10 23) ( 794.3460 kg / !3 )
∆ ´G =28.440994 o
(
0.000616715
- !ol
´ −T ∆ ´S ∆ ´G =∆ H o
o
o
*emperature" + 3.1 8.1 3/3.1 3/8.1 313.1
´ ∆G
o
" ,$mol
8.--/8.F1-1/ ./F881F .3-F< .<F1/
318.1 33.1
.-/-8 3/.F1FF
)y linear re%ression, Slo4e I /./
∆ ´S
K
´ ∆ H
K
o
o
y>interce4t I 1/.< ';*ol Jalue for r I /.81--</3
Preparation o' -olutions 'or E''ect o' Concentration
V A V T
=
ρB χ A M A ρ A M B− ρ A χ A M B + ρ B χ A M A χ A =0.95
#or 1>4ro4anol solution it(
V A =
( (
0.8053
g 3
#!
)(
0.8053
18.016
g #!
3
,
) (
)
#!
(
Calculation o' Densities o' -olutions
! "y# +sol −! "y# ! "y# + H $−! "y#
% ρ H $ 2
2
#or 1>4ro4anol solution it(
ρsol =
χ A =0.95
,
24.291 g−16.2521 g g % 995.738539 3 26.072 g−16.2521 g #!
ρsol =815.1450159
g #!
3
) (
g [ 1 −0.95 ] + 0.995738539 g 3 ( 0.95 ) 60.096 g !ol !ol #!
V A =31.80437147 !.
ρsol =
)
( 0.95 ) 0.995738539 g 3 ( 50 !. )
)
Calculation o' -peci'ic )iscosity
ηs" =
t −t o t o
#or /.1// M and aCMC 1.// %;dL solution,
ηs" =
3.65 s − 0.75 s =3.866667 0.75 s
Calculation o' Reduced )iscosity
η¿ =
ηs" #
#or /.1// M and aCMC 1.// %;dL solution,
η¿ =
3.866667 d. =3.866667 g g 1 d.
η¿ =3.866667
d. g
Calculation o' Inherent )iscosity
1 η η&h = ln # ηo #or /.1// M and aCMC 1.// %;dL solution,
η&h =
1 3.65 s ln g 0.75 s 1 d.
η&h =1.582409
d. g
Calculation o' Intrinsic )iscosity ?sin% t(e alues for reduced iscosities of /.1// M and aCMC solution, Concentration g$dL! 1.// /.F /./ /. /.1/ /./
Reduced &iscosity
η¿
!
3.8<<<<<<
)y linear re%ression, Slo4e I >.-3-/8<1 y>interce4t I .1888< Jalue for intrinsic iscosityK r I /.-<3<-
Calculation o' A&erage Molecular Mass
( )
´= M
[η]
1 a
k
?sin% t(e alue of t(e intrinsic iscosity fro* reduced iscosities of /.1// M and aCMC solution,
´ = 5.15888695 M
(
)
´ =103791.6765 M
g !ol
12.3 + 10
−5
1 0.91
Calculation o' Molecular Radius
r=
r=
(
(
3 M V s" 4 π
)
1 3
(
g 3 103791.6765 !ol
)
)
)
)
)(
#! 0.565 g
)(
#! 0.565 g
3
4 π
1 3
r =24.10131775 #!
r=
(
(
g 3 103791.6765 !ol
3
4 π
23
r = 4.00221 + 10
1 3
1
( 6.02 + 10 23 ) −
#! !ol
Calculation o' RootMean-quare EndtoEnd Length o' 0aCMC
⟨ r ⟩= 1 2
(
(
g 5.15888695 103791.6765 !ol 2.84 + 10
!ol#!
⟨ r ⟩=1.284484332 + 10 1 2
d.
21
−6
#!
3
)
)
1 3
)I. Literature Cited +ndrady, +.L. 7//8. Science and Technology of Polymer Nanofibers . ?S+ 'o(n Biley and Sons. +t!ins, P.B. 7//<. Physical Chemistry , 8t( edition. =xford, En%land =xford ?niersity Press. )ird, &.)., Steart, B.E. G Li%(tfoot, E.. 7//. Transport Phenomena. ?S+ 'o(n Biley and Sons, $nc. CHEM 111 Lecturers and La0oratory $nstructors. 7/1. Exercise 8 Viscosity of 4ure liquids and solutions. Physical Chemistry I Laboratory Manual . La%una ?PL). Daniels, #., Matt(es, '.H., Billia*s, '.B., )ender, P. G +l0erty, &.+. 71<. Experimental Physical Chemistry , t( edition. ?S+ Mc:ra>Hill )oo! Co*4any, $nc. De eers, .D. 711. Fluid Mechanics for Chemical Engineers, nd edition. Sin%a4ore Mc:ra Hill, $nc.
#oust, +.S., Ben2el, L.+., Clu*4, C.B., Maus, L.M. G +ndersen, L.). 718/. Principles of nit !perations, nd edition. Sin%a4ore 'o(n Biley and Sons, $nc. :arland, C.B., i0ler, '.B. G S(oe*a!er, D.P. 7//3. Experiments in Physical Chemistry , 8t( edition. e or! Mc:ra>Hill. Leine, $.. 7//. Physical Chemistry , Hill Hi%(er Education. &o%ers, D.B. /11. Concise Physical Chemistry . e 'ersey 'o(n Biley and Sons. &oussel, &.M. 7/1. " Life Scientist#s $uide to Physical Chemistry . London Ca*0rid%e ?niersity Press.
