Kimia Fisik Castellan

5/28/2018 Kimia Fisik Castellan

1/21

13So lutio n sI . The Id ea l Sol ut i o n a n d

Col l iga t ive P ropert i e s

1 3 . 1 K IN D S O F SOLUTI O N S

A solution i s a homogeneous mixture of chemical species dispersed on a molecularscale. By this definition, a solution is a single phase. A solution may be gaseous, liquid,or solid. inar! solutions are composed of two constituents, ternar! solutions three,"uaternar! four. The constituent present in th greatest amount is ordinarily called thesolvent# while those constituents-one or more-present in relatively small amountsare called the solutes. The distinction between solvent and solute is an arbitrary one. Ifit is convenient, the constituent present in relatively small amount may be chosen as thesolvent. e shall employ the words solvent and solute in the ordinary way, reali!ingthat nothing funda mentaldistinguishesthem."xamplesof#indsofsolutionare listed in

Table 1 3 . 1 .$as mixtures have been discussed in some detaili n %hapter 1 1. The discussion in thischapter and in %hapter 1$ is devoted to liquid solutions. &olid solutions are dealt with asthey occur in connection with other topics.

Ta% l e1 3. 1

$aseous solutions'iquid so lutions

&olid so lutions$ases dissolved in solids'iquids dissolved in solids&olids dissolved in solids

(ixtures o fgases or vapors&olids, liquids, or gases, dissolved

in liquids

)!in palladium, *!in titanium(ercury in gold

%opper in gold, !inc in copper+brasses, alloys of many #inds


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&'(De) i n it ion o) the Ideal

*1

13. & D+F IN ITIO N O F T ,+ I D+- L SOLUTI ON

The ideal gas law is an important example of a liiting law. As the pressure approaches !ero, the behavior of any real gas approaches that ofthe ideal gas as a limit. Thusall real gases behave ideally at !ero pressure, and forpractical purposes they are ideal atlow finite pressures. rom this generali!ation ofexperimental behavior, the ideal gas isdefined as one that behaves ideally at any pressure.

e arrive at a similar limiting law from observing the behavior of solutions. orsimplicity, we consider a solution composed of a volatile solvent and one or more involatile solutes, and examine the equilibrium between the solution and the vapor . If a pureliquid is placed in a container that is initially evacuated, the liquid evaporates until the

space above the liquid is filled with vapor. The temperature of the system is #eptconstant. At equilibrium, the pressure established in the vapor is pO# the vapor pressureof the pureliquid +ig. / . a. If an involatile material is dissolved in the liquid, the equilibrium vaporpressure p over the solution is observed to be less than over the pure liquid +ig. /. b .

&ince the solute is involatile, the vapor consists of pure solvent. As more involatilematerial is added, the pressure in the vapor phase decreases. A schematic plot of thevap o r pressure of the so lvent against the mole fraction of the involatile solute in thesolution, / is shown by the solid line in ig. /.0. At 0& 1 #P 1 pO2 as 0& increases, pdecreases. The important feature of ig. /.0 is that the vapor pressure of the dilutesolution 0& near !ero approaches the dashed line connecting pO and !ero. 2epending onthe particular combina tion ofsolvent and solute, the experimental vapor-pressure curveat higher concentrations of solute may fall below the dashed line, as in ig. / .0, orabove it, or even lie exactly on i t . )owever, for all solutions the experimental curve istangent to the dashed line at 0& 1 # and approaches the dashed line very closely as thesolution becomes more and more dilute. The equation ofthe ideal line +the dashed lineis

p 1 pO4 pO0& 1 p3+ * x&.

If x is the mole fraction of solvent in the solution, then x 4 0& 1 , and the equation

5apor

T#pO

pO p

1

3a6

%6

Figur e 1 3. 1 5aporpressure lo7ering%!a n i nvolat i l e solute.


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o

&'(De) i n it ion o) the Ideal

p p

p5

66

66

66

66

6-66

6

666

o o I -x

Figure 1 3.& 5apor pr essu re

as a )un8t ion o)/07

Figure 1 3.3 9a ou l)s la7

) orth e solvent.

becomes 1 3 . 16

which is 9aoult:s la7 . It states that the vapor pressure of the solvent over a solution isequal to the vapor pressure of the pure solvent multiplied by the mole fraction of thesolvent in the so lution.

8aoult 7slaw is another example of a limiting law. 8eal so lutions follow 8aoult 7slaw

more closely as the solution becomes more dilute .The ideal solution is defined as one thatfollows 8aoult 7s law over the entire range of concentrations. The vapor pressure of thesolvent over an ideal solution ofan involatile solute is shown in ig. 13.3. All real so lutionsbehave ideally as the concentration of the solutes approaches !ero.

rom "q. 13.16 the vapor pressure lo7ering#po* p# can be calculated 9

po4p 1po4 /po1 1 4 /6pO#

po4p 1 /;po. 1 3 .&6

The vapor pressure lowering is proportional to the mole fraction of the solute. If severalsolutes, 3# . . . , are present, then it is still true that p 1 /po2 but, in this case, 1 * / 10 ; < 0 3 < . . .

and

13.36In a solution containing several involatile solutes, the vapor pressure lowering depends onthe sum of the mole fractions of the various solutes. *ote particularly that i t does notdepend on the #inds of solutes present, except that they be involatile. The vaporpresssure depends only on the relative numbers of solute molecules.

In a gas mixture, the ratio of the partial pressure of the water vapor to the vaporpressure of pure water at the same temperature is called the relative huidit!. henmultiplied by 1#it is the per8ent relative huidit!. Thus

8.). 1 Ppo

and :8.). 1P

16.P

5ver an aqueous solution that obeys 8aoult 7slaw, the relative humidity is equal to the molefraction ofwate r in the so lution.


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flliq1 flap< 9T In p5< 9T In x.

1 3.3 -N -L=T I C-L FO 9 > O F T, + C , + > I C- L

POT+NTi-L I N I D+- L LI?UI D SOLUT

I O N S

As a generali!ation of the behavior of real solutions the ideal solution follows 8aoult 7slaw over the entire range of concentration. Ta#ing this definition of an ideal liquidsolution and combining it with the general equilibrium condition leads to the analyticalexpression of the chemical potential of the solvent in an ideal solution. If the solution isin equilibrium with vapor, the requirement ofthe second law is that the chemical potentialofthe so lvent have the same value in the solution as in the vapor, or

flliq1 flvap, + / .;

where flliq is the chemical potential of the solvent in the liquid phase, flvap the chemicalpotential of the solvent in the vapor. &ince the vapor is pure solvent under a pressure p,the expression for flvap is given by "q. + 3.; I C- L PO T+ N T I- L O F T , + S O LU T + I N - IN - 9 = I D +- L

SOLUTI O N 2 - P PL IC-TIO N O F T ,+ @ I S*D U , +> + ? U-TI

O N

The $ibbs-2uhem equation can be used to calculate the chemical potential ofthe so lutefrom that of the solvent in a binary ideal system. The $ibbs-2uhem equation, "q. + .@A, for a binary system T#p constant is

+ /.

The symbolswithoutsubscripts in "q. + /. refer to the solvent= those with the subscript

&


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Sol utions I&A&

0

refer to the solute. rom "q. + /.A, dB0 1 - +nn0dB= or, since nBn0 1 xx!, we have

2ifferentiating "q. + / . ? #eeping T and p constant, we obtain for the solvent dB 1%8Tx dx, so that dB0 becomes

4 dxdB 1 -8T-.C!

)owever, x 4 C0 1 , so that dx 4 dx! 1 # or dx 1 -dx0. Then dB0 becomes

dx!dB018T-.C0

Integrating, we haveB0 1 8T In C0 4 C# + /.lation, C can be a function of T andp and still be a constant for this integration. Ifthe value of C0 in the liquid is increased until it is unity, the liquid becomes pure li"uidsolute, and B0 must beB. , thechemicalpotential ofpure li"uidsolute. &o ifx01 , B0 1 B.. >sing these values in "q. +/.


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I

pI

I

I

I ff im

w

F i gu r e1 3.$ Co l l igati vepr op er ti es .

involatile, it does not appear in the gas phase, so the curve for the gas is the same asforthe pure gas. Ifwe assume that the solid contains only the solvent, then the curveforthesolid is unchanged .)owever, because the liquid contains a solute, the F.l ofthe solvent islowered at each temperature by an amount - 8T In /. The dashed curve in ig. /.;+ais the curve for the solvent in an ideal solution. The diagram shows directly that theintersection points with the curves for the solid and the gas have shifted. The newintersection points are the free!ing point, T7.r, and the boiling point, T, ofthe solution. Itis apparent that the boiling point of the solution is higher than that of the pure solvent+boiling-point elevation, while the free!ing point of the solution is lower +free!ing-point

depression. rom the figure it is obvious that the change in the free!ing point is greaterthan the changein the boilingpoint for a solution of the same concentration.The free!ing-point depression and boiling-point elevation can be illustrated on the

ordinary phase diagram of the solvent, shown for water by the solid curves in ig. / .;+b. If an involatile material is added to the liquid solvent, then the vapor pressure islowered at every temperature as, for example, from point a to point % . The vapor-pressure curve for the solution is shown by the dotted line .The dashed line shows thenew free!ing point as a function ofpressure. At atm pressure, the free!ingpoints andboiling points are given by the intersections of the solid and dashed lines with thehori!ontal line at atm pressure. This diagram also shows that a given concentration ofsolute produces a greater effect on the free!ing point than on the boiling point.

The free!ing point and boiling point of a solution depend on the equilibrium of theso lvent in the so lutionwith pure solid solvent or pure solvent vapor. The remainingpossible equilibrium is that between solvent in solution and pure liquid solvent. This

equilibrium can be established by increasing the pressure on the solution sufficiently toraisethe F.l of the solvent in solution to the value of the F.l of the pure solvent. Theadditional pressure on the solution that is required to establish the equality ofthe F.l ofthesolvent both in the solution and in the pure solvent is caled the osoti8 pressure of theso lution.

1 3 . T , + F 9 ++ I N @ * P O IN T D+P9+

S S I O N

%onsider a solution that is in equilibrium with pure solid solvent. The equilibrium condition requires that

F.l+T,p# / 1 F.lso lid+T, p6# + / . 3


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Solutions E&A$

Bl x BlsoliiTBlsolid

+/.

, - ,

x, +oTBoxp

4

F

1 F dT.

x1x )fus)fus*

where eT,p# is the chemical potential of the solvent in the solution, p6 is thechemical potential of the pure so lid .&ince the solid ispure# does not depend on anycomposition variable .In "q. + / . 3, T is the equilibrium temperature, the free!ingpoint

ofthe solution = from the form of "q. + / .3, T is some function ofpressure andx, the molefraction ofsolvent in the solution. If the pressure is constant, then T is a function only ofx.

If the solution is ideal, then"q. + / . 3becomes

BleT,p# xin the solution is given by "q. +/.?, so that

8earrangement yields

8T

+ / .

&ince is the chemical potential of the pure liquid, p6 p6 where$fus isthe molar $ibbs energy offusion ofthepure solvent at the temperature T."quation

ln x 1 -

**

..8

$T

f

us

+ / . 0

To discover how T depends on we evaluate 2ifferentiating "q. +/ . 0with respect to x,p being constant, we obtain

x 8 aT p oxp>sing the $ibbs-)elmholt! equation, "q. +3.?;, Ho+$BTBoTIp1 - l.)BT0,we obtain

)fusIn "q.+ / . /,

)fus - 8T0 oxp7 +/ . /

is the heat offusion ofthepure solvent at the temperature T.The procedure is now reversed and we write "q. + / . / in differential form and integrate

9

/dx T1 x To 8T

+ / . ;

The lower limit corresponds to pure solvent having a free!ing point To. The upper7limit corresponds to a solution that has a free!ing point T. The first integral can beevaluated immediately = the second integrat ion is possible if is#nown as afunction oftemperature. or simplicity we assume that is a constant in the temperature rangefrom Toto T= then "q. + / . ;becomes

In x1 -..)8

fusT To

+ / .


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Solutions E&A$

8 In x

?

This equation can be so lved for the free!ing point T, or rather more conveniently for fT,

T To- ..)fus7 +/ . Awhich relates the free!ingpoint ofan ideal solution to the free!ing point ofthe pure solvent,To, the heat offusion ofthe solvent, and the mole fraction ofthe solvent in the solution, x.


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1 f

efdef

f

The relation between free!ing point and composition of a solution can be simplifiedconsiderably ifthe solution is dilute. e begin by expressing the free!ing-point depression* dT in terms of the total molality of the solutes present, #where 1 ; <

3< . . . .

'et n and > be the number ofmoles and molar mass ofthe solvent, respectively= then themass of solvent is n>. Then ; 1 n;Gn>2 31 n3Gn>2 . . . = or n; 1 n>;2 n3

1

n>32 . . . The mole fraction of the solvent is given by

n nn < n; < n3< . . . n < n>3;< 3< . ..

< >

Ta#ing logarithms and differentiating, we obtain In x 1 - In + < >6# and

d In x 1> d

< >

+ / . 9 T d . + / . @Ll,)us + < >6

If the solution is very dilute in all solutes, then approaches !ero and T approaches To#and "q. + / . @becomes

4

a p. m 1 5

>9T1

Ll,)us1

Jf . +/.03The subscript, 3, designates the limiting ofthe derivative, and K is the free!ingpoint depression constant. The free!ing-point depression 1 To* T# 1 * dT#so for dilute solutions we have+-oaefp,m1 5 1 Kf 7 +/.0

which integrates immediately, if is small, to

ef1 Kf. +/.00The constant K depends only on the properties of the pure solvent. or water,

> 1 3.3D3?0 #gmol, To1 0


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Solutions I&A

I

4 9>

Ta % l e1 3. &

Free;ing*poi nt depress ion 8onstants

%ompound (+#9gmol tJKC JIB+J #gmol

ater 3.3 D3 3 .DA

Acetic acid 3.333 A .A /.?singthese datawe findfrom"q. +/ .0? that the ideal solubility x I 3.0; at 03 K%. The measured solubilities invarious solvents aregiveninTable/ ./ .

The ideal law of solubility is frequently in error i f the temperature of interest is farbelow the melting point of the solid, since the assumption that il,)us is independent oftemperature is not a very good one in this circumstance.The law is never accurate for

Ta % l e1 3.3

! perission )ro . ,. ,ilde%rand and 9. L. S8ott# The Solu%ilit! o)

Nonele8trol!tes # 3ded.Ne7=orP 9einhold# 1 (B#p . &A3.


12/21

Sol ution s I&AA

.

-

1 1

solutions of ionicmaterials inwater, since the saturated solutions of these materials arefar from being ideal and are far below their melting points. As the table of solubilitiesof naphthalene shows, hydrogen-bonded solvents are poor solvents for a substancethat cannot form hydrogen bonds.

13. A + L+5-TIO N O F T ,+ O ILI N@ POIN T

%onsider a solution that i s in equilibrium with the vapor o fthe pure solvent.The equilibrium condition is that

If the solution is ideal,

and

Fl3T# p# x 1FlvapT#p6.

FlOT#p6 < 8T In x 1FlvapT#

p6#

In x -LFlvap * FlOT#

p6

8T

1 3 .&'6

The molar $ibbs energy of vapori!ation is

Ll@vap 1FlvapT#p6 * FlOT#p6#so that

In xLLl@vap

. 31 3 .&A68T

*ote that "q. 13.&A6 has the same functional form as "q. 31 3 .1&6 except that the signis changed on the right-hand side. The algebra which follows is identical to that used for

the derivation of the formulas for the free!ing-point depression except that the sign isreversed in each term that contains either Ll@ or Ll,. This difference in sign simplymeansthatwhile the free!ing point is depressed, the boiling point is elevated.

e can write the final equations directly. The analogues of"qs. 31 3 . 1B6 and 13.16 are

In x 1 Ll,vap8

+GTL

To,

:

or

1 1 8Inx*T

1*To


13/21

Sol ution s I&AA

1 1 1or water, > .1A1B& #gmol, To 3 ' 3 .1B K# and il,vap $ B Fmol, thenJb 1.B1&(( K #gmol. The relation, "q. 31 3 . 316# between boiling-point elevation and

the


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%ompound

ater

(+#gmol

3.3D3

tb=a%

33

JbB+J#gmol

3.?

(ethyl alcohol 3.3/03 A;.< 3.DA"thyl alcohol 3.3;


15/21

Figure 1 3. Sipl e osoti8

pre ssure e/perient.


16/21

Sol utions I&(

of purewater. The level ofthe sugar solution in the tube is observed to riseuntilitreachesa definite height, which depends on the concentration ofthe solution. The hydrostaticpressure resulting from the difference in levels of the sugar solution in the tube and thesurface ofthe pure water is the osoti8 pressure of the solution. 5bservation shows thatno sugar has escaped through the membrane into the pure water in the bea#er. Theincrease in volume of the solution that caused it to rise in the tube is a result of thepassage ofwate r through the membrane into the bag. The collodion functions as aseiperea%le membrane, which allows water to pass freely through it but does notallow sugar to pass. hen the system reaches equilibrium, the sugar solution at anydepth below the level of the pure water is under an excess hydrostatic pressure due tothe extra height of the sugar solution in the tubing. The problem is to derive therelation between this pressure difference and the concentration of the so lution.

1 3 . ( . 1 The va n: t , o )) + "uat ion

The equilibrium requirement is that the chemical potential of the water must have thesame value on each side ofthe membrane at every depth in the bea#er. This equality ofthe chemical potential is achieved by a pressure difference on the two s ides of themembrane. %onsider the situation at the depth h in ig. 13.C. At this depth the solventis under a pressure p#while the solution is under a pressure p < n.IfpeT#p < n#x isthe chemical potential of the solvent in the solution under the pressure p < n# and poeT#

p6 that of the pure solvent under the pressure p# then the equilibrium condition is

peT#p < n#x 1poeT#p6# 1 3 . 3&6and

1 3 .33 6

The problem is to express the p of the solvent under a pressurep < n in terms of the psolvent under a pressure p. rom the fundamental equation at constant T# we havedpo1


17/21

!o .n!8T

since n! n in the dilute solution. Then "q. 1 3.36 becomes

n!8Tn 1

n O5.

1 3. 3 '6

By the addition rule the volume ofthe ideal solution is O 1 nO5< n! O .If the solutionis dilute, n! is very small, so that O n This result reduces "q. 13.3'6 to

n 1 **O

or n 1 c8T. 1 3 . 3 A 6

In "q. 13.3A6# c 1 n!O,the concentration of solute +molm/in the solution. "quation 13.3A6 is the van7t )off equation for osmotic pressure.

The stri#ing formal analogy between the van7t )off equation and the ideal gas lawshould not go unnoticed.In the van7t )off equation, n! is the number of moles of solute.The solute molecules dispersed in the solvent are analogous to the gas molecules dispersedin empty space. The solvent is analogous to the empty space between the gas molecules .In the experiment shown in ig. 13.'# the membrane is attached to a movable piston. Asthe solvent diffuses through the membrane, the piston is pushed to the right = thiscontinues until the piston is flush against the right-hand wall. The observed effect is thesame as ifthe solution exerted a pressure against the membrane to push it to the right.The situation is comparable to thefreeexpansion ofagasinto vacuum. If thevolume ofthe solution doubles in this experiment, the dilution will reduce the final osmotic pressureby half, Nust as the pressure of a gas is halved by doubling its volume.

In spite of the analogy, it is deceptive to consider the osmotic pressure as a sort ofpressure that is somehow exerted by the solute. 5smosis, the passage of solvent throughthe membrane, is due to the inequality of the chemical potential on the two sides of the

membrane. The #ind of membrane does not matter, but it must be permeable only to thesolvent. *or does the nature of the solute matter= it is necessary only that the so lventcontain dissolved foreign matter which is not passed by the membrane.

The mechanism by which the solvent permeates the membrane may be different foreach different #ind ofmembrane. A membrane could conceivably be li#e a sieve that allowssmall molecules such as water to pass through the pores while it bloc#s larger molecules .Another membrane might dissolve the solvent andso bepermeated byi t ,whilethe soluteis not soluble in the membrane. The mechanism by which a solvent passes througha membrane must be examined for every membrane-solvent pair using the methodsof chemical #inetics. Thermodynamics cannot provide an answer, because theequilibrium result is the same for all membranes.

Seiperea%le e%rane

. . . . . .

. . . . . . .

. . . . .

Solution Pgas 6

Pure solvent* va8uu6

Piston

Figure 1 3. ' Osoti8 ana )og

o) the o u l e e/per ient.


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Solutions I&(&

1 3 . ( . & > easu reen t o) Os ot i8 P r essure

The measurement of osmotic pressure is useful for determining the molar massesof materials that are onlyslightly soluble in the solvent, orwhichhave very high molarmasses +for example, proteins, polymers ofvarious types, colloids. These are convenientmeasure ments because of the large magnitude of the osmotic pressure.

At 0? K%, the product 9T 99999990;D3FmoG. Thus, for a molN' solution 38 1 333 molNm/, we have

n 1 89T 1 0.;D C 3A Pa 1 0;.?atm.

This pressure corresponds to a height ofa column ofwater ofthe order ofD33 ft9 &implyto #eep the experiment in the laboratory, the solutions must be less than 3.3 molar, andare preferably ofthe order of3.33 molar. This assumes that we are using an apparatusofthe type shown in ig. /.A. Oery precise measurements of osmotic pressures up to

several hundred atmospheres have been made by ). *. (orse and F. %. . ra!er,and by 'ord Ber#eley and ". $. F. )artley using special apparatus of different design.In a molar mass determination, if H& is the mass of solute dissolved in the volume, 5#

then n 1 7&9TI>& 5#or

"ven when H& is small and >&large, the value ofn is measurable and can be translatedinto avalue of >&M5smosis plays a significant ro le in the function oforganisms. A cell that is immersed inpure water undergoesplasmolysis. The cellwallpermits water to flow into it = thereuponthecell becomes distended, the wall stretches until it ultimately ruptures or becomeslea#yenough to allow the solutes in the cellular material to escape from the interior. 5n theother hand, if the cell is immersed in a concentrated solution ofsalt, the water from thecellflows into the more concentrated salt solution and the cell shrin#s. A salt solution which is

Nust concentrated enough so that the cell neither shrin#s nor is distended is called an

isotoni8so lution.

5smosis might be called the principle of the prune. The s#in of the prune acts as amembrane permeable to water. The sugars in the prune are the solutes. aterdiffusesthrough the s#in and the fruit swells until the s#in ruptures or becomes lea#y. 5nlyrarelyare plant and animal membranes strictly semipermeable. requently, their function in theorganism requires that they pass other materials, as well as water. (edicinally, theosmoticeffect is utili!ed in, for example, the prescription of a salt-free diet in some cases of ab

normally high fluid retention by the body.

? U+STI O N S

13 .1 Is the lowering of the chemical potential of a solvent in an ideal solution, "q. +/.?, anenthalpy effect or an entropy effectQ "xplain.

13.& Interpret +a free!ing-point depression and +b boiling point elevation in terms of )l as a measureof 6escaping tendency.6

13.3 )ow does the temperature dependence of the solubility of a solid in a liquid illustrate'e%hatelier7s principleQ

13.$ 9everse osmosis has been suggested as a means of purifying sea water +roughly an *a%l-)!5


19/21

Solutions I&(&solution .)ow could this be accomplished with an appropriate membrane, with special attentionplaced on the required pressure on the solution Q


20/21

Solutions I&(&

P 9 O L + > S

13.1 Twenty grams of a solute are added to 33 g of water at 0? 3c. The vapor pressure ofpure water is 0/., must be dissolved in @3 g of water to produce a solu

tion over which the relative humidity is D3 :Q Assume the solution is ideal.

13.3 &uppose that a series of solutions is prepared using D3 g of )03 as a solvent and 3 g ofan involatile solute. hat will be the relative vapor pressure lowering if the molar mass of thesolute is9 33 gmol, 033 gmol, 3,333 gmolQ

13.$ a or an ideal solution plot the value ofpGpoas a function of C0, the mole fraction of thesolute.

b etch the plot ofpGpoas a function ofthe molality of the solute, ifwater is the solvent.c &uppose the solvent +forexample, toluene has a higher molar mass. )ow does this affect theplot ofpGpoversus mQ )ow does it affectpGpoversus C0

Q

d "valuate the derivative ofpo* ppowith respect to m, as m --4 5.

13.B A stream of air is bubbled slowly through liquid ben!ene in a flas# at 03.3 K% against anambient pressure of 33.?A #Pa. After the passage of ;.D 3 ' of air, measured at 03.3 K% and33.? #Pa before it contains ben!ene vapor, it is found that .


21/21

Pro%les &(3

13.13 %alculate the boiling-point elevation constant for each of the following substances.

Su%stan8e t%t8 illivapFg

Acetone, +%)/h%5 ?A. ?03.@

Ben!ene, %A)A D3.0 /@;.A

%hloroform, %)%l/ A . ? 0;

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