The Thermodynamic Properties of Solutions

Chapter 7

The Thermodynamic Properties of Solutions

With the definitions of activities, standard states, ideal solutions, and partial molar properties given in Chapters 5 and 6, we are ready to describe the properties of solutions. There are many topics of interest, but we will limit our discussion in this chapter to four the change in the thermodynamic properties of nonelectrolyte solutions resulting from the mixing process, the calculation of the thermodynamic properties of solutions containing strong electrolyte solutes using the DebyeHtickel theory, the calculation of the change in the thermal properties of solutions due to mixing, and the effect of concentration on the osmotic pressure.

7.1

Change in the Thermodynamic Properties of Nonelectrolyte Solutions Due to the Mixing Process

We are interested in describing and calculating AmixZ, the change in the thermodynamic variable Z, when liquids (or solids) are mixed to form a solution. We will begin by deriving the relationship for calculating /kmixG. Changes in the other thermodynamic properties can then be obtained. For the process nlA + n 2 B - Solution the change in Gibbs free energy is given by AmixG -- Gsolution - G ~:- G so that /kmixG - nl/-tl -4- n2H2

hilt ~

n2/_t*

(7.1)

where #l and #2 are the chemical potentials of components A and B in the solution and #]~ and /z* are the chemical potentials of the pure components.

))

--.i

9

-

9

~

m~

~

II

-...

p

N

~

--m.

~

~ " go

o

.

__~

~"

~

~

~.

~"

m

,-,.

0

~I

~

cr

~

~;

Cm

~

II

~

~

.~

.'~

..~.

-.

II

I>

0

C;I

~

0

"~

n

~,

tD

~'--' ~"

0

~

~

--

~

~

~

o

.

"~ N ' G

=.~~

9

9

q-

~

I:>

"

'-,

o

~

m"

~

~

9

elm

~,.,~.

cT

~

o "~" --..3

~.,.

9

~.

9

,..0

•

b..~.

0

-,

~

~

I~

~

~

I:>

0

~.-.

~<

~

="

m"

9

"~"

+

~

II

I>

~O ~

~

~m

=~

~

~

0

~

~

~:

--

-

~:

m" lm

0 m"

o-"

~

O~

~

~

--,

~

~

9

I

+

I>

<

~-"-

--

o

.....

>

Im

,q m-'

lm

m"

The Thermodynamic Properties of Solutions

327

The entropy change to form an ideal mixture from the pure components is obtained by differentiating equation (7.7) with respect to T. Since xi is independent of T, the result is

I ~ /XmixG id lp

__ _AmixS id m_ R

m

OT

xilnxi

9 11

or

/kmixSimd - - R

~

xi

In x i .

(7.8) a

i

The ideal enthalpy of mixing is easily obtained from equations (7.7) and (7.8) and the relationship /kmixHimd --/kmixGidm -+- T/kmixSimd

(7.9)

AmixHimd - - 0 .

The volume change for forming an ideal mixture can be obtained by differentiating equation (7.7) with respect to p. Since x, is independent of p, 0AmixGid/T m

(7.10)

-- Z~mix Vimd -- 0.

Op

9 /1

The change in internal energy on mixing can also be obtained from

(7.11)

/kmix Uimd - AmixHimd - PAmi x Vimd - O.

Hence, Amixgimd, Ami x Vimd,

and

AmixUimd are zero,

while

AmixGimd ----TAmixSim d =

RT ~-~xi ln xi. i a Note that the equation for calculating AmixS for an ideal liquid (or solid) mixture is the same as was derived in Chapter 2 for calculating AmixS for ideal gases.

~

t'~

r

r

Oo

11

:::r

q~

o-"

9

Oo

,~,

0

o oo

o

0

o

o

0 0

0

I

0

A i Z,i'~/(J. m o l -' )

0 9

b

7 ~ ~

b

b

-~

v'

~="

_,

.

~..,.

9

r

II

bq

r

< 9

~..

=r ~

9

o

9

c ~

~.o

~-"<

~.

~

~

c~ ~

~'~ ~

~~

_.&*. 9

r

"'d

>.

r

~_o

~"

o~ ~ ~~~~ :::::r 9

r...,.

~~0 ~ ~" ~

-.~-o

I~

~'~"

b~ ~

"~

~z

I>

~

b

o"~

~<

~

~

~"

o

The T h e r m o d y n a m i c Properties of Solutions

329

and the value of the excess function is the total enthalpy, internal energy, and volume change upon mixing. We have shown earlier [equation (7.6)] that the molar Gibbs free energy change upon mixing is given by

AmixG m

RT ~

-

(xi In xi + xi In 7R. i)

(7.6)

i

where 7R.i is the activity coefficient (Raoult's law standard state). Since Gm E-

A, mix Gm -- Amix G m id

and

AmixGid m

RT Z

--

xi In xi i

then

GEm - R T ~

xi In 7R.;.

(7.16)

i

The excess molar entropy is obtained from G

E

m -- Hm

E

-

TSm

E

or

HEm G _

_

E

m.

(7.17)

T Since these mixing processes occur at constant pressure, HEm is the heat evolved or absorbed upon mixing. It is usually measured in a mixing calorimeter. The excess Gibbs free energy, GEm, is usually obtained from phase equilibria measurements that yield the activity of each component in the mixture b and SEm is then obtained from equation (7.17). The excess volumes are usually obtained

b See section 6.5 of Chapter 6 for examples of methods for measuring the activity.

330

Chemical Thermodynamics: Principles and Applications

with a dilatometer, or from density measurements obtained from a pycnometer or a vibrating tube densimeter. The extent of deviation from ideal solution behavior and hence, the magnitude and arithmetic sign of the excess function, depend upon the nature of the interactions in the mixture. We will now give some representative examples.

Nonpolar + Nonpolar Mixtures: Figure 7.2 summarizes the behavior of HEm,

TSEmfor

E mixtures of decane with hexane ~ at 298.15 K. The small Z m demonstrates the near ideal behavior of this system. The negative G m E results from a somewhat greater lowering of the vapor fugacity of each component than is expected for an ideal mixture (small negative deviations from Raoult's law). Since HE is positive while G m E is negative, S m F must be positive, indicating an increase in disorder during the mixing process over that predicted for the ideal mixture. GEm, and

Polar+Nonpolar Mixtures: Figure 7 3 summarizes HEm G E and T S E for mixtures of acetonitrile with benzene at 318.15 K 2 The large positive H E results principally from the energy that must be added to separate the highly polar acetonitrile molecules, {which are held together by strong (dipole+dipole) 9

,

m

m

30

20

"7 o

E

10

t,qE

-10 ,

0.0

I

I

0.2

0.4

....

I

I

0.6

0.8

1.0

X2

Figure 7.2 Excess

thermodynamic

functions

at

T - 298.15 K

for

{xlCIoH22 "+-x2C6H14}, an example of a system where nonpolar chain-like molecules are mixed.

The Thermodynamic Properties of Solutions

331

800 E

Gm 600

400 "7 O

E

-q

200

%E

9

T E

-200

-400 0.0

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

Figure 7.3 Excess thermodynamic functions at T - 318.15 K for {XlC6H6-t-x2CH3CN}, an example of a system in which polar molecules are mixed with nonpolar molecules.

interactions}, as the n o n p o l a r benzene molecules are added. The large positive Gm E reflects the increased vapor fugacities (positive deviations from Raoult's law) resulting from the decreased attractive forces in the mixtures. ~ Since G iEn is larger than H E , TSEmis less than zero, and increased ordering occurs in the mixture over that expected for the ideal mixture. Mixtures

with

Hydrogen

Bonding:

Figure 7.4 summarizes H E , GEm and

TSEmfor (ethanol + water) 3 at 303.15 K and 363.15 K. The negative H E and TSEmresult principally from the formation of h y d r o g e n - b o n d e d complexes in the liquid mixture. The complexes break down as the temperature increases, and the negative contributions decrease so that H E and TSEmbecome less negative. At T - 3 6 3 . 1 5 K, H E has become s-shaped, with negative HEm at low mole C(Liquid + liquid) phase separation occurs with sufficiently large positive deviation from ideal behavior. The deviations are not large enough in (acetonitrile + benzene) to cause (liquid + liquid) equilibrium to occur. If benzene is replaced by an aliphatic hydrocarbon such as heptane, separation does occurs. In a later chapter, we will discuss (liquid + liquid) equilibrium in detail.

332 Chemical Thermodynamics: Principles and Applications

1500

1000

500

t

/ ~ _ _ _~_ _ , ~ ~ \G , ~

/

"7 9

-500

- 1000

-1500 0.0

, 0.2

I 0.4

I 0.6

9 0.8

1.0

Figure 7.4 Excess molar functions for {xlH20 + x2CzHsOH}. The solid lines represent results at T - 303.15 K and the dashed lines are for results at T - 363.15 K.

fractions of ethanol, d changing to positive H 1TI E at high mole fractions. With increasing temperature, HE becomes more positive, until at high temperatures it has a large positive value over the entire mole fraction range. 4 The ordering effect in the mixture due to complex formation is larger than the energy effect so that T S E is larger in magnitude than H E. The result is a positive G wmand an increased escaping tendency. Volume Comparison" Figure 7.5 compares VEtofor the three systems for which we have compared H E, GEm, and SnE, plus the (cyclohexane + decane) system. 5 The comparatively large negative VEtofor the (ethanol + water) system {curve (4)} can be attributed to the decrease in volume resulting from the formation of hydrogen-bonded complexes in those mixtures. The negative V m E for the ( h e x a n e + d e c a n e ) system {curve (3)} reflects an increased packing Excess

dThe stoichiometry of the complex is such that it contains considerably more water than ethanol. Since the decrease in energy due to the formation of the complex would be largest at the stoichiometric composition, the minimum in H mE o c c u r s at low mole fractions of ethanol.

The Thermodynamic Properties of Solutions 333 L ,

,,

,

,

1

0.4

J

J

/

\

/ J

\

/ 0.0

.

.

\ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

7 o

o

"~ - 0 . 4

\

/., \.

-0.8

/

/ \

/ \.. ~4. ~.. J"

-1.2 0.0

, 0.2

,. 0.4

, 0.6

, 0.8

1.0

X-,

Figure 7.5 Comparison of excess molar volumes for four mixtures as follows: Curve 1" {xICIoH22 + x2c-C6H12} at T : 313.15 K. Curve 2:{AIC6H6 + N2CH3CN} at T - 318.15 K. Curve 3" {xICIoH22 +x2C6HI4] at T - 308.15 K. Curve 4 : { x ] H 2 0 + x2C2HsOH) at T - 318.15 K.

efficiency when chain-like molecules of different lengths are mixed. The positive VEm for the (cyclohexane + decane) system {curve (1)} indicates that mixing chain-like n-alkane molecules with globular cycloalkane molecules results in less efficient packing. The smaller and s-shaped VEm curve for the (acetonitrile+ benzene) system {curve (2)} has no simple explanation. Excess volumes for mixtures of molecules with different polarities and shapes vary from large positive VEm to large negative VEto,with values in between also represented.

7.2 Calculation of the Thermodynamic Properties of Strong Electrolyte Solutes" The Debye-H~ickel Theory Solutions containing strong electrolyte solutes differ from those containing nonelectrolyte solutes in that deviations from Henry's law become important at much lower concentrations for the electrolyte solute than for the nonelectrolyte

334

Chemical Thermodynamics: Principles and Applications

solute. For example, Figure 7.6 compares the activity coefficients (solute standard state) for aqueous solutions of ethanol, a nonelectrolyte, and of hydrochloric acid, an electrolyte. The activity coefficient for ethanol changes slowly with m and is greater than 0.99 for m less than 0.1. On the other hand, the activity coefficient (,~+) for hydrochloric acid changes rapidly with m and becomes less than 0.90 for m greater than 0.01. The large non-ideality in electrolyte solutions can be explained by the distribution of charged ions in the solution as shown schematically in Figure 7.7. ~ The ions are often spherically symmetrical (or at least near-spherically symmetrical) particles, with positive or negative charges that are large in comparison to the dipolar charges present in polar nonelectrolyte solutes. Electrostatic attractions and repulsions are much stronger than other types of interactions and dominate in the solution. Repulsions between ions of the same charge and attractions between ions of a different charge cause the ions to arrange themselves in solution so that positively charged ions are preferentially surrounded by negatively charged ions, while negatively charged ions are preferentially surrounded by positively charged ions. The result is a nonrandom

9

9 ,,,

,

9

~

1.0

0.9 @

o

0.8

.,..~

<

.7 0.0

'

'

'

0.2

0.4

0.6

.

'

0.8

1.0

m/(mol.kg -1) Figure 7.6 Activity T - 303.15 K.

coefficients

of

aqueous

HCI

and

aqueous

C2HsOH

eThe ions are in thermal motion. Figure 7.7 represents the average distribution of charge.

at

The Thermodynamic Properties of Solutions 335

Figure 7.7 In a strong electrolyte solution, negative ions are preferentially surrounded by positive ions while the positive ions are surrounded by negative ions.

distribution of ions in the solution, even at low concentrations. This causes deviations from ideal behavior, since in the ideal solution, the molecules are randomly distributed.

7.2a

Derivation of the Activity Coefficient Equations

If the electrical potential in an ionic solution that results from the n o n r a n d o m distribution of charge could be calculated as a function of position of any given ion, then the activity coefficients of the ions could be calculated. Imagine a process in which the ions in their equilibrium arrangement are held in position while reducing their charges to zero. The ionic charges are then reversibly increased at constant temperature and pressure from zero to the value they have in solution, by bringing in electrical charge from infinite distance. If the electrical potential, ~b is known, the electrical work ~'~l added to the system as a result of this charging process can be calculated, since 6Wel = ~) dq where q is the charge. Integration over the charge distribution gives the work. This electrical work, in turn, equals the free energy change for the process, f which can be used to calculate the excess chemical potential and hence, the activity coefficient, g rWe remember that AG for a constant T and p process equals the non (pressure-volume) work, that occurs in the process. gAn alternate approach is to add reversibly, an ion with no charge to the solution, and calculate the work needed in bringing the ion to full charge. Both methods lead to the same result for the activity coefficient.

,.....,

,..,.

~.-,.

9

0

m,.

'-"

,--I

..q

~:=.

~"

-.~

~"~

~

0

o =s V~

~

~.~o

~"

,..... t'~

N

o_.~.

--

~:~ ,_a

H

..~ =. 7 "=r'~

-

~''~

<

=:I

~. " ~ "

II

0

N

'T"

~......, ,....,

~~. ~. _.. ~.

9

II I

" ~ "

%i~

~~

.~

~ a"

~

0

=:~'

i-,.

=

~

"

="

~= ' ~ ~ - o~

~

~

,_, =

~"

o~.~~

~.~.

~.

,I = ~

..,.

.~.=

~.~F:

-,-.

~

~

-~

o~

o

~,~

.

~

~

.

=I

o~-

~~-~

=s"

-~-

~. ~ ~ ~-o>

.

...

.~

.

~=r"

.

~

~o.~

,--.

9 ~

9 ~.~-

~

~. ' ~ ~ ~~ ~

~ . ~ > = _

~

-F_o

~-u

...,'~::I

.~

~~o ~

.

xO

,--q

+ ""

"

.,

~:

'a

_

_.

-

~-~.

~

-~

~

oo

<1

0

r

9

9

9

9

~

.H :r"

~.

9 r

::r'

ox

The Thermodynamic Properties of Solutions 337

In equation (7.21), k is the Boltzmann constant, T is the temperature, and e~is the potential energy associated with the field. It is given by

ej- zje~,j

(7.22)

where e is the electronic charge (1.6022 x 10 -19 coulombs), ZJ is the number of charges,J and ~j is the electrical potential associated with the ion. Substituting equation (7.22) into equation (7.21) gives

_cOexp/

(7.23)

The charge density in solution is obtained by multiplying the concentration of particles by the charge on each particle and summing over the number of types of particles in the solution. That is,

P- Z zjecj.

(7.24)

i Equation (7.23) is substituted into equation (7.24) to give the equation that relates p to

p - ~ zjec~ exp j

t

zjeOjt kT

.

(7.25)

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ~. In theory, the resulting equation can be solved to give ~ as a function of r. However, it has an exponential term in ~, that makes it impossible to solve analytically. In the Debye-Hfickel approximation, the exponential is expanded in a power series to give ,2e2cO~j

p_ ~ zjecO_ ~ "/ j

j

_3,.3 ~0~,,2

i , + Z-'~-~-J--~-~J + "'"

kT

i

(7.26)

2k 2 T 2

example, z = + 1 for H +, Na +, Ag-, .." - - +2 for Ca 2+, Mg 2-+- ,Fe-"~+ , . . _ - = - 1 forC1 , NO;,.." z = - 2 for SO~-, CO~-,...

J For

338

Chemical Thermodynamics: Principles and Applications

The first term on the right-hand side of this equation is zero, since it is simply the sum of the electrical charge in solution, which must be zero for a neutral electrolyte solution. The third term is also zero for electrolytes with equal numbers of positive and negative ions, such as NaC1 and MgSO4. It would not be zero for asymmetric electrolytes such as CaC12. However, in the DebyeHtickel approach, all terms except the second are ignored for all ionic solutions. Substitution of the resulting expression into equation (7.20) gives the linear second-order differential equation

(7.27) r 2 dr where ~2

47re2 ' ~ --

~

", o 2.'/ Cj

skT

(7 28)

.

j-

The general solution to equation (7.27) has the form exp(~cr) exp(-~r) -- A ~ + B r

(7.29)

r

where A and B are constants that can be determined from the boundary conditions of the problem. The first condition is that ~ must remain finite at large values of r. This requires that A must be zero, in which case equation (7.29) becomes - B

exp(-~cr)

.

(7.30)

The constant B is evaluated by substituting equation (7.30) into equation (7.25) to obtain an equation relating the charge density p to r. The result is applied to a central ion of charge ,'.e to give the charge density around this ion. That is, ~2e exp(-m-) & -

-B

.

47r

(7.31)

r

Electrical neutrality in the solution requires that the total charge around the ion be balanced by the charge on the ion. This condition is expressed by the

The Thermodynamic Properties of Solutions

339

relationship

(7.32)

If~ &47rr~ d r j - - z i e .

The lower limit of the integration assumes that the central ion has a finite size with a, the radius of the ion, representing a distance of closest approach of other ions to the central ion. Substituting equation (7.31) into equation (7.32) and integrating results in a value for B which, when substituted into equation (7.30), gives

/zJeexp'a't ' tExp' 'r't 9

~b/--

c(1 + eta)

(7.33)

~

r

Equation (7.33) can be used to express ~t'~l,and hence the activity coefficient, as a function of ~9z ~ c .]~~ The concentration CJ is usually converted to a quantity .I

known as the molality-scale ionic strength Ira,k where 1 l m - - - -2

' ~ z TmJ

(7.34)

In equation (7.34), mj is the molality of j type of ions. k Ionic strength can be expressed in terms of the concentration c (mol.dm-3), in which case 1Z

2

Ic ----

Z iCi. 2

i

Mole fraction can also be used, in which case

2 The form using molality [equation (7.34)] as the unit of concentration is the one most commonly encountered. It is often written as I (instead of 1,1). I For a simple 1 9 1 electrolyte (e.g. HCI, NaCI, AgNO3, etc.), Im = Im=

+m++- e m_]= 89

m.

This can be seen as follows:

340

Chemical Thermodynamics: Principles and Applications

The Debye-Hiickel final result is

In 3'+ = -

2.2+ C . ,, I ~ 2

z 2

In-),_ = -

(7.35)

1 + B~,a I ~ 2 C~I 1/2

- -

(7.36)

t

1 + B.~a I ~ 2

where 3/2

C,~ -

(27rNAPA) 112

e47rCocA k T

(7.37)

and 112

(7.38)

In equations (7.37) and (7.38), NA is Avogadro's number, PA is the density of the solvent (in kg-m-3), e0 is the permittivity of vacuum, and CA is the permittivity (dielectric constant) of the solvent. The other constants in equations (7.37) and (7.38) are as previously defined. Equations (7.35) and (7.36) can be used to calculate the activity coefficients of individual ions. However, as we discussed in Chapter 6, 7+ and 7- cannot be measured individually. Instead, 7• the mean ionic activity coefficient for the electrolyte, M , + X , _ , given by u+_ u-

7+ - (~+ 7_ )

l/u

(7.39)

is calculated, where v = u+ + u_. An expression for 7• can be obtained from equations (7.35) and (7.36). Taking the logarithm of both sides of equation (7.39) gives

In ",/• =

v+ In "7+ + u_ In 7-

.

(7.40)

0 Oo

~..

~D

0

O

m"

m"

cu

9 I'D

I=:: (u

I

o-

m-"

0

I-D

c:~

A

Cm~m

~--. I=I

I=: '-"

~m

~-...

m" -(m 0

.-I--

0

-m

o

o~

~....

E~ o -.m

+

I

I1

I',1

+

4~

---..I

II

--2 FF

1.4

..,

~"

~.._

--"

~--~

O

~

o

""

_

- ~ B

r~

==

9

~

-~-.

--'

X

~'~x

X

~

~

I

~-,.

~-

~

L-. O

~-~

~"

~

o_o

"<

~

~

~

~-~

0

--4

.

~-~

,j

.--~ r,~

~'~o~

r~

~

~-

0

-~. =" b-" o -

m

-...I ~ 9

~

J

I

I

,~

B

O0

o

X

~

,--

~

,...-,.

~r~

"--..i

-

"1 ~

~

~-,'.

0

"

~

~"

r~

0 ~

~

,-,~=

--...I

9

0

o

...B

'-tl '-< 0

.'

m. ,.,. ~"

=~~

.<

~ B ' b~ ~

1m

<~<--

=:r , - . ~

~

--" lm

9

af ~ L ~

9

I

t l

+

I

II

H-

...q

L~

...q

J

9

I

t~

~

I ~

"

I

+~.,

+

i'D

~"

~.--L.

~"

~"

~

m

0 o"

0

1m

~

~

~

I

~\I

I

+

+

m-' lm <

t'T

F

~-,. r.~

I

<

~T

~.~

'--"

m-'

,H

o

b.....

o,,_..

9

oe

,-1 ,..,.

,-I 9

,..,..

o

m-

~P

mm-i

~J

9

0

e~

~.,io

cJ~

qm.~

e~

I

M

c~

e~

imo

e~ ~176

0

e~

q

The Thermodynamic Properties of Solutions 343

only at very low m, is given by In 7+ - -C~, I z+-

I I~ 2.

(7.45)

It is known as the limiting law expression. Note that for a simple l" 1 electrolyte with [ z+z_ I - 1 , I m - m and equation (7.45) becomes In 7+

-

-C7 ml/2.

This equation, which predicts that for a solution of a 1"1 electrolyte, lnT+ varies as m ~/2, will be especially useful in extrapolating measurements to zero concentration, n

7.2b

Comparison of the Debye-H~ickel Prediction with Experimental Values

With all the approximations involved in its derivation, one well might wonder how well the Debye-Hfickel theory works in predicting "~+. Figure 7.8 compares experimental In T+ results against the Debye-Hfickel predictions. We see from Figure 7.8(a) that the limiting law begins to work reasonably well 11/2 below-m --0.15 for HC1 and for SrCI2, when Im ~0.02. For HC1, (1"1 electrolyte), this corresponds to m = 0.02 and for SrC12 (2"1 electrolyte), m = 0.0067. We note, however, that even at this low ionic strength, "~+ for ZnSO4 deviates significantly from the Debye-Hfickel limiting law. It can be seen from Figure 7.8(b) that the curved lines predicted by the extended form of the Debye-Hfickel equation follow the experimental results to higher ionic strengths than do the limiting law expressions for the (1"1) and (2"1) electrolytes. However, for the (2"2) electrolyte, the prediction is still not very good even at the lowest measured molality. ~ Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hfickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hfickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hfickel theory is to nThe ionic strength Im = { ~2 miz] must include the concentration of all sources of ions. Thus, Imis not equal to m for a 1 : 1 electrolyte if other sources of ions are present. o Electrolytes of the 2:2 type, such as ZnSO4, apparently have significant ion pairing, even in dilute solution. This ion pairing makes an important contribution to the deviation from the limiting law for ZnSO4.

344

Chemical Thermodynamics: Principles and Applications

~':1

9 ooooOO

-1

9

9

9

9

9

9

=2:=1

X2:l

\. -2

\

.

2:} (a)

2:2

-3 I

0.0

I

0.5

I

1.0

9

1.5

2.0

l~i:/(mol.kg-') '~:

~

~

~

o 'g ~

oeoooo9

9 l:l

9

i, ~ _ ~ 1 : 1

**,.*'~

-1

. ~ 2 :1

9

9

9

9

2:1

9

===

A~2:2 A"

-2 (b)

-3

~2 i

0.0

0.5

i

1.0

I

1.5

9

2.0

W(mol.kg-';~ Figure 7.8 C o m p a r i s o n of experimental ln-7~: for 1: 1, 2 : 1 , and 2 : 2 electrolytes. The symbols indicate the experimental results, with 9 representing HC1 (z+ = 1, z_ = - 1); II representing SrC12 (z+ = 2, z_ = - 1 ) ; and A representing ZnSO4 (z+ = 2, z_ = - 2 ) . The lines are the D e b y e - H f i c k e l predictions, with the solid line giving the prediction for (z+ = 1, z_ = - 1 ) ; the dashed line for (z+ = 2, z_ = - 1); and the dashed-dotted line for ( z + - 2, z_ = - 2 ) . In (a), In%_ calculated from the limiting law [equation (7.45)] is shown graphed against llm/2. In (b), In %- calculated from the extended form [equation (7.43)] is shown graphed against Ilm/2.

provide a method for extrapolating measurements to zero ionic strength. We will see examples later in this and other chapters, p

P")/+ has not been measured to low enough concentrations for any electrolyte solution to definitively prove the square root dependence of Im on ln~~• Some other theories, in fact, predict a cube root dependence. With the absence of evidence that would contradict equations (7.44) and (7.45), the Debye-Hfickel limiting law is usually used to extrapolate results to zero ionic strength.

The Thermodynamic Properties of Solutions

7.2c

345

The Debye-H~ickel Prediction of the Osmotic Coefficient

Equation (7.45) is a limiting law expression for 7• the activity coefficient of the solute. D e b y e - H t i c k e l theory can also be used to obtain limiting-law expressions for the activity a~ of the solvent. This is usually done by expressing al in terms of the practical osmotic coefficient 0 that we described briefly in Chapter 6. For an electrolyte solute, it is defined in a general way as In al 0 - -

(7.46)

M1 ~ ~'kmk k

where uk is the number of moles of ions produced by a solute whose molality is mk, and Ml is the molecular weight of the solvent in kg-mo1-1. The sum goes over all electrolytes present in solution. For a single electrolyte, equation (7.46) reduces to q5 -

In al

.

(7.47)

Mlum For a nonelectrolyte solute, it simplifies even further to the expression In al 0 -

Mlm

(7.48)

since u - 1. The osmotic coefficient is often used as a measure of the activity of the solvent instead of al because al is nearly unity over the concentration range where 7+ is changing, and many significant figures are required to show the effect of solute concentration on al. The osmotic coefficient also becomes one at infinite dilution, but deviates more rapidly with concentration of solute than does a~. The osmotic coefficient r and activity coefficient 7• are related in a simple manner through the G i b b s - D u h e m equation. We can find the relationship by writing this equation in a form that relates a~ and a2. n l d In a l -k- n2 d In a2 - O. For a binary solution containing n2 = m moles of solute and nl = 1/M1 moles of solvent (with Ml in kg.mol-~), the G i b b s - D u h e m equation becomes

d In al + m d In a2 - O. Ml

(7.49)

::r'

~

~c~"

~~~~

~

"-~

o,

--.

"a ~

"" 0

"~"

>~~

~~

~ ~ "- .

4:~

~

~

~

~ - ~= ~"

~ ~ "

~

~-

o~

~.~

=

~. 0

~= ,---, ~ ~ ~

~

~

~

c~ ~ ~

~

""

gQ

~-=

~

~

~ ' ~ ~

~

~

~

~

~

I

I

-

=

" o ~ ~

t.~

~

~"

o-~

~ ~

0

~

~ ' ~

~

~

o,

"~

-

-0-

-~ H-

=

O"

~-.

"~ FF

E" -II

~-~_

-,-

~-,

~

~

::s II

o

=.~

_.

~.

"0 - ~

~'~

~-

_~

~o

L~

+

II

I

9

ox~ ,..~o <

o'~

L~

~

0

0

L~

+

FF

~

"-

~

""

~

-'"

~

-F

+~

+

~"

~'~"

,_.

FF

~

~

+~

~ 0

0

L~

+

I

II

""

~.,

~r

0

I

II

cr

0

oo

o

o

~....

0

~..io

oo

0

,...

>

,..,.

_~.

..

~....

0

H

~r

b~

II

--.-I ~ ~ ~

t~ +

~..,o

9

9

~..,o

~-'"

t'~

.

0

I"~1

-~

=

o

II

I

I-J

+ ~,~

L,|

II

I

0

~

L-I

I tl

b~

L,u I I ~

+J

I

I

+

+

11

I

+

+

r

bJ

t.l rJ

Q

r~

I

+

I

t')

+

+

t~

II

I

9

l.J

l~J

., m,.

"--.3

=,

~ ,.~

-.

--J

=

i.-+.

c/] = G'

I

I

t~

L~

I

+

+ t-~

t

II

-4

'-'"

-..

~

r

,~

"+

L~

I

,~

+

II

t.m

"-..3

9

II

~

I

9

t.m

---.1

e

~:

~

:~"

~

~.--,

i,-,1

I~

l:::::l

~

~..,.

~

0

0

0

i- t +-+. i-,.

i- I 0 "t::l

~3

~..,..

9

i- t

-]

348 Chemical Thermodynamics: Principles and Applications

As with 7+, an extended form of equation (7.59)can be written

1 - c h - Co ]z+z_ ]

~/2 m i1/2 ' 1 +-m

(7.60)

with the extended form applying over a larger molality range than the limiting law expression.

7.2d

The Debye-HQckel Prediction of Thermal and Volumetric Properties of the Solute

From the Debye-Hfickel expressions for ln'~• one can derive equations to calculate other thermodynamic properties. For example L2, the relative partial molar enthalpy, q and ~'2, the partial molar volume are related to 7+ by the equations /

\

L2 - H2 - H~ - - v ' R T 2 ( 0 In 7+ \ OT

v2

-

(7.61)

(7.62)

Op

where H~_ and ~'~. are the partial molar enthalpy and the partial molar volume in the standard state. Starting with the limiting law expression equation (7.45) and differentiating gives

L2 - - [ z + z _

V2 - V 2

I I ~ 2 2R

T2/C1

2]z+z_ ]Ilm/2 2 R T

(7.63)

(7.64)

q In Section 5.3 of Chapter 5, we introduced the concept of relative partial molar properties. In the next section of this chapter we will describe these properties in detail.

The Thermodynamic Properties of Solutions 349

Differentiating the equation for C~, and substituting into equations (7.63) and (7.64) gives

(7.65)

and

/

(7.66)

where

Ctt - - 3 R T 2C.~ T +

OT

(7.67)

+ 3

and

Cv-

RTC.~

3

/0'nl t0'nm1 Op

-

~

Op

,

(7.68)

with c~ and Vm as the coefficient of expansion and molar volume of the solvent. Differentiation of equation (7.65) with respect to temperature gives an equation for J2, the relative partial molar heat capacity, given by

-

J2 -

-

c~.~ -

-o

c~_

-

t0E2/, -0-7

"

The result is

,2

(7.69)

where Cj is an expression involving multiple derivatives of c and Vm with respect to temperature. Values for the coefficients CH, Cv, and Cj as a function

350 Chemical Thermodynamics: Principles and Applications

~)f temperature for aqueous solutions can be found in Table 7.1 from which the thermal and volumetric properties can be calculated.

7.3

Relative Partial Molar and Apparent Relative Partial Molar Thermal Properties

In Chapter 5, we defined the partial molar property Z; and described how it could be used to determine the total thermodynamic property through the equation

Z - Z niZi. i m

w

D

We described methods for obtaining values for V;, Cp.~, and S;, but did not apply the methods to H i and Gi (or #;), since absolute values of Gm a n d Hm cannot be obtained. We did describe a procedure for obtaining the volume difference P ' i - ~'* using equations (5.40), (5.41) and (5.42), r where ~'* is the volume of the pure substance, and indicated that equations of the same form can be used to obtain H ; - H* We will return to this method later in this chapter as we describe ways for measuring relative partial molar enthalpies.

7.3a

Relative Partial Molar Enthalpies

~0 The quantity H; - H; is called the relative partial molar enthalpy and given the symbol L;. It is the difference between the partial molar enthalpy in the solution and the partial molar enthalpy in the standard state. That is,

Zi-

Hi-

(7.70)

H i9

For a Raoult's law standard state, H , ? - H* and L i - H i - H* These are the differences described in Chapter 5. For a Henry's law standard state, H- oi is the enthalpy in a hypothetical m - 1 (or x2 - 1 or c - 1) solution that obeys Henry's law. To help in understanding the nature of these standard state enthalpies, we will show that Hi(hyp m - 1)

-

Hi(m

- 0).

r The difference G i - G*(or # i - #*)is obtained from the activity through the relationship #i - #*= R T lnai

where ai is the activity with a Raoult's law standard state.

,.-..

~.

0

~

0

,-,.,.

~. ,--,

,...,. r.~

~'~

,--. r.~

t~

o"

0 ,-1

O ~

0

~ ~

.

~

,.--.

-.. E. I>

0

:::I

r

"u

~

0

=," ,..q

0

0

~

~

~"

,--,.

~

-.-..o ,-.

~

,.--,

o_. ~-. =

~

r

,--:

~ . ~

~oq

8Pro

,.~

"~"

o_.

o

0

~'~'~

l~

'-1

I'D

~

~ ~-

~

nr'

o

~D

~.

,-,

go

~

O~

go

0

~.~"

~o " - -

0

~

~1

t'D

~,

0

,-~ ;-.,.

0

C~

~-'"

,~_.~"

~"

9

<

,~

~ r ~~

0

"~. o

~

~ ~1

o-.o

II

-.

"

~-~1

o

~1

l~

~

~

~ ~

,-~

0

0

~,-.* 9

o

o

~

oE " ~

"--"

~.~

~

:~.~

~

~

I

II

I

~1

~

~

II

~z.

"~

~. -"

= 0o ~D

~

o,-,.

~

~

:=I

~

,q

c~

tl

II

I

c~

~

~

~-..

"I::

<

=

~-"

~

~

~

~

..

I

"

~

-o

"I:::

-

0"

o" 0<

II

~

o~

oo"

~

-

~

'-~

II

~o

-.o

"~

-

0

~"

0"

c~

-,.

4~

~-~.

o~

~

0 ,-~ ::r

~

~r'

o

<.~

~-'*"

~

0

0

0

~.-.~

0

0

9

0

,..,,. I'D

0

9

r

I'D

-q

352 Chemical Thermodynamics: Principles and Applications

write n

LI -- Hi - H*I L2 - H2 - H 2 ( m -

0), 0

m

and in the infinitely dilute solution, L ~ - L 2 - 0.

7.3b

Calculation of ~ H from Relative Partial Molar Enthalpies

Relative partial molar enthalpies can be used to calculate A H for various processes involving the mixing of solute, solvent, and solution. For example, Table 7.2 gives values for Ll and L2 for aqueous sulfuric acid solutions 7 as a function of molality at 298.15 K. Also tabulated is A, the ratio of moles H20 to moles H 2 8 0 4 .t We note from the table that L l - L 2 - 0 in the infinitely dilute solution. Thus, a Raoult's law standard state has been chosen for H20 and a Henry's law standard state is used for H2SO4. The value L * - 95,281 J.mol -l is the extrapolated relative partial molar enthalpy of pure H2SO4. It is the value for H~ - H- o2. The results given in Table 11.1 can be used to calculate A H for various mixing processes. Example 7.1- Calculate A H at 298.15 K for the process H2SO4 + 2 5 H 2 0 - Solution(HzSO4 + 25H20). The enthalpy change for this process, in which we mix pure liquids, is known as an integral enthalpy of solution. m

Solution: A H -- nl H i + n2H2 - nl H T - n ~ H * with nl - 2 5 and n 2 - 1.

Rearranging gives AH-

n l ( H l - H O + n2(H2 - H ~ . m

m

o

With the choice of standard states used in Table 7.2, H] ~- H~. Making this - o2 gives substitution and adding and subtracting n 2 H AH-

nl(Hl -

H ~ ) + n2(H2 - # * +

H~ - H2)

t Also tabulated is 4~L, the apparent partial molar enthalpy. We will define this quantity and

describe its application later.

m

--..I

I

II

t~

I

§

I

t.m

II

t>

--..I

m

I

§

II

t>

<

9

.,<

0

o

I

.,:-.,

:.,~ 9

I

t . , ~ . X-

I

~0

t~

t~

+

nO

I

,.._,

II

I>

,,_,......

~ ~B"

0

0~"

...

"

~~

~ ~

0

~

~t

~ z~

---~ ZH m

t~

9

,,_,.

~

~:~~

4~ ~

~,....

~'~.

I

~-...~,,o-...~

I

L~

w,

~

~

I

i~L~

i

o

9

9

N~

o

I

I

-~.~"

9

~-~_.

~-o

9

----

::Z:: ~ "a

9

9

k,--.

II " ]

,..]

9

9 ,,,,_.

0

~t

~t 9

,..,.

0

B

,,-t

-]

-]

354 Chemical Thermodynamics: Principles and Applications

An easier way to solve this problem is to define a quantity L that we call the total relative enthalpy. It is given by L-

--

H-

--O

H ~ - nlHl + n2H2 - n l H l

--Q

-

n2H 2.

Collecting terms gives L-

(7.72)

niL1 + n2L2.

A useful application of L comes from the fact that for any solution process,

AM-- ~ prod

prod

tliH i -- Z

"i~]`

react

react

or

(7.73)

AH--AL.

In our example A L - L ( s o l u t i o n ) - L ( H 2 S O 4 ) - L(H20 ) L(solution) L(H 2 S O 4 ) - -

L1 +

-

nl

-

I/2 L - -

n2L2

- - o

L ( H 2 0 ) - n i L e - - n l ( H ~ - H l) - - 0 A H - - niL1 +

n2L2

-

n2L2

-

niL1.

Substituting values from Table 7.2 gives the same answer as before.

Example 7.2: Calculate A H for the mixing process 25H20 + solution(1H2SO4 + 25H20) - solution'(1H2SO4 + 50H20). In this process, the original solution is diluted by the addition of pure solvent, and hence, the enthalpy change is called an integral enthaipy of dilution.

The Thermodynamic Properties of Solutions

Solution: nl - 2 5 , n2 F r o m Table 7.2"

1, n '1- 5 0 ,

n;-

1

A = 25"

L1 -- - 9 9 . 2

L2 - 26.166

A = 50:

L ' 1 -- - 2 5 . 5

L '2 - 23.785

AH-

355

AL = (1)(23,785) + ( 5 0 ) ( - 2 5 . 5 ) - -1176

(1)(26,166)- (25)(-99.2) -(25)(0)

J. Q

T h e last t e r m in the calculation is L ]~, the relative partial m o l a r e n t h a l p y for p u r e water, which is zero. E x a m p l e 7.3: Calculate A H w h e n one mole of H2SO4 is a d d e d to a large v o l u m e of the solution (H2SO4 + 10H20). Solution: In this example, it is a s s u m e d that we add a solute to a large e n o u g h

v o l u m e of solution so that the c o m p o s i t i o n of the mixture does not change. T h e e n t h a l p y c h a n g e for this process is referred to as a differential enthalpy of solution. We can represent this process by n 2rH2804 + solution (nlH,O_ + n~H_2 S 0 4 )

-

solution [ n l H 2 0 + (n2 %- n~)H2SO4] AH-

nlLl + (n~_ + n'2)L2 - - n i L-1 -- n 2 L-2 - n 2' -L* . m

Since the c o n c e n t r a t i o n s are a s s u m e d to stay the same, L~ and L2 have the same values in b o t h terms of the expression, they cancel, and we get A H - - n '2E 2 - n '2E *-. F r o m T a b l e 7.2 A - 10: Since L ~ - 9 5 , 2 8 1

L, - 38.525. and n ' ~ - 1

A H - - 38,525 - 95,281 - - 5 6 . 7 5 6 J. A similar calculation involves the a d d i t i o n of a mole of H : O (instead of H2SO4) to a large a m o u n t of solution. W h e n we add a mole of H 2 0 (n' l - 1) to a

La

_

~~

~I

="

-~

~a

""

r

o

-.

oo

~

~'I

~

,~

O

~

~-

~o"

F "~

~

o

=--

~

~

~

9

~,1

o

~ ~ ~ ~

~

=

_

--"

o o~

~'~

~a

~ ~I

O

~

O

~o ~

-

~

~

~

lh~

H~

o ~ o ~

o

m

o

~-~

= ~ ~.~_,~

"

=~o ~ ~ o ~~

~a

II

O"

~.~.

o"

o

"0

o

0

oo

m

>

N

--.

r.#)

o

0

~ ~

_m

r

~0

-~

m ~

i'D

II

n

9

oo

II

I>

-x

il

o

II

"--X-

~o

O0

I

II

o

=r

9

0

oo

9

r

>

r ,....

r

0

The Thermodynamic Properties of Solutions 357

OI"

n2qbL- L.

(7.78)

Equation (7.78) allows us to use OL to calculate A H for a solution process since AH-

AL-

nx&OL.

(7.79)

Values of 4~L are tabulated and can be used to calculate A H for mixing processes. For example, for the integral enthalpy of solution process given in Example 7.1, we used L1 and L2 values to show that for the process H2804 + 2 5 H 2 0 - solution(H2SO4 + 25H20)

A H -- -71,595 J. We could get this same value by writing AH-

n2qSL - n2OL*

where 4~L - 23,686 J is the relative apparent molar enthalpy for the solution (see Table 7.2) and O L * - 9 5 , 2 8 1 is the relative apparent molar enthalpy of pure H2804. Thus ~ H - (1)(23,686) - (1)(95.281) = -71,595 J. From the nature of apparent properties, we note that the apparent molar enthalpy assigns all of the enthalpy change in forming a mixture to the solute. The result, as shown in equation (7.79), is that all we need to do to calculate A H for a solution process is find the difference in oL between the products and reactants. Thus, to solve Example 7.2 using apparent molar enthalpies, we would write 25H20 + solution(1 H2SO4 + 25H20) - solution'( 1H2SO4 + 50H20) ~H

-

n2(q~L'

-

r

From Table 7.2 A H - - (1)(22,510- 2 3 . 6 8 6 ) - -1176 J which agrees with the value obtained from L l and L2.

9

Oo

I

9

-I~

(:~

I

I

DO

bO

I

I

--

~+

~--o+

J~

~"

+~+

_

9

I

I

--

--

b~

oo

=+~.+

++

,,.,

=+

==

0

3

I>

V

3

D.

,....

r~

,--1 ,.....

9

~ ~=~

9

~.~

or~

= ~ ~'~"

Or)

--

~

"

,-~

~

o ~ _

<

~

~

~

~

~

E~

,-,. ~

=

" ~

r~ =

'E ~

~~ ~~~

~0~.:.

~"

('I> ~

~

0"~

~ ~~;~ -"

9

~

=

"-'~-"~

~

~o

~

--~ ~ "~ ~~ ~

~

Or;

.

~~=~~~

~ .

9

~

~

0

~~

=

g.,,-,.

~~~

--

~ ~-.~-o

II

II

~

'<

.---

~._.-, ,~ ~ g ~

~,_~

~ o ~~ ~~ ~ ~' - . -~~ _ m ~"

CD

~ ~.>~.

,~,

9

,~

I

II

~

~

~

~

o

~-

= =T'

o~

o

~

~..,o

:~

::::r

9

o

-+-

o

~

_-

:s

E"

~D

"-

,--'

r

o

~

:~

~

('1)

~

o

~

-~=,-

~.

(I)

~:~

~

..~

0

~.

~-~

~

m

~. 0

~~.~

0

~ ~ ~"~ ,'-I

o

~g..~

e-~

~

~-o~ = _

o _~.

~

=~ ~ ~

=

0

~

=

~"

0

,...

>

,.._.

,,.,,.,.

m.

'-u

,_.. tb ..

0

,-t

,....

The Thermodynamic Properties of Solutions

359

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The Thermodynamic Properties of Solutions 367

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Conversions similar to those used to obtain equations (7.86), (7.87), (7.89), and (7.90) can be used to obtain equations involving nz (or m 1/2) instead of n2.

7.4

The Osmotic Pressure

The osmotic pressure is a property that has proven to be especially valuable in the study of solutions of macromolecules, including those of biologic and polymeric interest. The apparatus for measuring this quantity is shown schematically in Figure 7.10. Two compartments are separated by a m e m b r a n e that will allow the flow of liquid solvent between the two chambers. If solvent is added, flow will occur until the liquid level on the two sides of the membrane is the same. We now add a solute to the c o m p a r t m e n t on the left. The solute is confined to this compartment, since we have chosen a membrane that will allow the flow of solvent, but not the flow of solute. Such a device is called a semipermeable membrane.

Figure 7.10 Schematic representation of the apparatus for measuring osmotic pressure. The flow of solvent through the semipermeable membrane is followed by observing the movement of the meniscus of the flow indicator. The osmotic pressure II is the pressure that must be applied to the solution to prevent the flow.

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The Thermodynamic Properties of Solutions

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Also, the freezing point of this solution would be lowered 0.2 K below that of pure water. The large H makes osmotic pressure measurements useful for determining molecular weights of high molecular weight solutes, such as polymers, since a very dilute solution still gives a substantial effect. The method is often used in biochemistry to determine the molecular weight of solutes such as polypeptides. For example, for a polypeptide with a molecular weight of 10 kg.mo1-1, a solution obtained by dissolving 10 g of this substance in 1 kg of water gives an osmotic pressure of 2.5 kPa, a value that is easily measured. In contrast, a freezing point lowering of 0.002 K would need to be accurately measured if one tried to determine the molecular weight of the polypeptide from the freezing point lowering. As a caution, however, one must carefully purify the sample since small amounts of low molecular weight impurities can contribute significantly to H, even though their mass percentage in the sample is small.

7.4a

Osmosis

If the pressure on the piston in the left cell in Figure 7.10 is less than H, solvent will flow from the pure solvent on the right to the solution on the left. This process is called osmosis. The process can be reversed by increasing the pressure on the left to a value greater than H. The result will be a flow from left to right. This process is known as reverse o s m o s i s and is used to purify water in desalination plants. The key to making this process work is to find membranes that will allow the passage of water at a reasonable rate, that are strong enough to withstand the pressure difference, and are substantially impermeable to salt that is present as a solute in the impure water solute. Membranes of cellulose acetate or hollow fibers with very small pores are often used. Osmosis across cell membranes is very important in biology, but the process is more complicated than we have pictured it in Figure 7.10. Cell membranes are permeable to water, and also to other substances such as CO2, O~, N2, and low molecular weight organic compounds such as amino acids and simple sugars. Large polymer molecules such as proteins and polysaccharides, however, will not pass through the membrane. Intermediate to these two extremes are inorganic ions (such as NaC1) and disaccharides (such as sucrose) that pass slowly through the membrane. In a living organism, the cells are bathed in body fluids that contain various solutes while the fluid within the cell contains other solutes. Thus solutes are present on both sides of the membrane, and a flow of fluid in or out of the cell depends upon the difference in chemical potential across the cell membrane, which in turn depends upon the relative concentration of solutes on the two sides. If the activity (approximately concentration) of solutes in the fluid surrounding a cell is higher than in the cell fluid, then the cell will lose water by osmosis. When this occurs, the surrounding fluid is said to be hypertonic. For the reverse process when fluid flows into the

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378 Chemical Thermodynamics: Principles and Applications

(a) What are the standard states chosen for K and Na in generating the data? Justify your answer. (b) Calculate A H and AG (in Joules) at 384 K for the process: K(4 moles) + Na(1 mole) = solution. E7.12 The following table gives the osmotic pressures as a function of concentration for the polymer polyisobutylene dissolved in benzene at 298.15 K.

c/(g.dm -3)

H/(Pa)

20.0 15.0 10.0 5.0

210.3 150.4 100.5 49.5

Extrapolate the data to infinite dilution to obtain a value for the molecular weight of the polymer. (Note that an average molecular weight is obtained since the polymer consists of a mixture of molecules of different chain lengths.) E7.13 Twenty mg of a protein are dissolved in 10 g of water. The osmotic pressure at 298.15 K is determined to be 0.040 kPa. Estimate the molecular weight of the protein. E7.14 Estimate the vapor pressure lowering and the osmotic pressure at 293.15 K for an aqueous solution containing 50.0 g of sucrose (M2=0.3423 kg.mol -~) in l kg of water. At this temperature, the density of pure water is 0.99729 g.cm -3 and the vapor pressure is 2.33474 kPa. Compare your results with those given in Table 7.3.

Problems P7.1

Vapor pressure data for ethanol (l) + 1,4-dioxane (2) at T = 323.15 K is given in the following table, where xl is the mole fraction in the liquid phase and yl is the mole fraction in the vapor phase. (a) Graph the data to construct the (vapor + liquid) phase diagram and determine the composition of the minimum boiling azeotrope. (b) Use the vapor pressure data to calculate GEm at T - 323.15 K at each of the mole fractions.

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The Thermodynamic Properties of Solutions 381

Thermodynamic Properties of xi H20 + x2(1,4-C4H802) D

X2

L1/(J.mol-J)

L2/(J.mol -l)

al

a2

0.000 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6OOO 0.650O 0.7000 0.7500 0.8000 0.8500 0.9000 1.0000

0 -314 -690 -858 -941 -962 -962 -962 -962 -962 -962 -900 -586 -126 167 649 1234 2092 ...

... -2469 21 774 1088 1130 1130 1130 1130 1130 1130 1088 920 690 565 439 314 209 0

1.0000 0.9151 0.8770 0.8476 0.8236 0.8068 0.7963 0.7843 0.7723 0.7638 0.7544 0.7408 0.7242 0.7004 0.6711 0.6264 0.5631 0.4764 0

0 0.4002 0.5450 0.6368 0.6966 0.7375 0.7585 0.7778 0.7946 0.8044 0.8134 0.8249 0.8366 0.8502 0.8647 0.8826 0.9031 0.9261 1.0000

P7.5

The osmotic coefficients of aqueous CaCl2 solutions at 298.15 K are as follows:

m/(mol.kg -1)

~5

m/(mol'kg -1)

0

0.0001 0.0005 0.001 0.005 0.01 0.02 0.04 0.05 0.1

0.9869 0.9719 0.9615 0.9250 0.9048 0.884 0.867 0.861 0.854

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.862 0.876 0.894 0.917 0.940 0.963 0.988 1.017 1.046

(a) M a k e a graph of In 0 against m for the experimental results given in the table. Also show on the graph the predictions of In (~ against m as predicted by the D e b y e - H f i c k e l limiting equation and the extended form of the D e b y e - H f i c k e l equation.

382

Chemical Thermodynamics" Principles and Applications

(b) Use the osmotic coefficients in the table to calculate 7+ of a 0.001, 0.01, 0.1, and 1.0 molal CaCI2 solution and compare your answers with those obtained from both the limiting and extended forms of the Debye-Htickel equation.

References l.

3.

4.

5.

6.

7.

K . N . Marsh, J. B. Ott, and M. J. Costigan, "'Excess Enthalpies, Excess Volumes, and Excess Gibbs Free Energies for (n-Hexane+n-Decane) at 298.15 K and 308.15 K", J. Chem. Thermodyn., 12, 343- 348 (1980). D. A. Palmer and B. D. Smith, "Thermodynamic Excess Property Measurements for Acetonitrile-Benzene-n-Heptane System at 45 ~C", J. Chem. Eng. Data, 17, 71-76 (1972). R.C. Pemberton and C. J. Mash, "'Thermodynamic Properties of Aqueous Non-Electrolyte Mixture I]. Vapour Pressures and Excess Gibbs Energies for Water + Ethanol at 303.15 to 363.15 K Determined by an Accurate Static Method'", J. Chem. Thermodvn., 10, 867-888 (1978). J.B. Ott, C. E. Stouffer, G. V. Cornett, B. F. Woodfield, C. Guanquan and J. J. Christensen, "Excess Enthalpies for (Ethanol + Water) at 398.15 K, 423.15, 448.15, and 473.15 K and at Pressures of 5 and 15 MPa. Recommendations for Choosing (Ethanol + Water) as an HEm Reference Mixture", J. Chem. Thermodvn., 19, 337-348 (1987). The VEtoresults for (cyclohexane + decane) are obtained from J. R. Goates, J. R. Ott, and R. B. Grigg, "Excess Volumes of Cyclohexane+n-Hexane, +n-Heptane, +n-Octane, +n-Nonane, and +n-Decane'" J. Chem. Thermodvn., 11, 497-506 (1979). Excess volumes for the (ethanol+water) system were obtained from K. N. Marsh and A. E. Richards, "Excess Volumes for Ethanol + Water Mixtures at 10-K intervals from 278.15 to 338.15 K", Aust. J. Chem., 33, 2121-2132 (1980). Excess volumes for the (acetonitrile + benzene) and the (hexane +decane) systems were obtained from the same source as the H mE * G I"11 E and S mE results referenced earlier. We refer those who are interested in the details of the Debye-Htickel derivation to the following sources: R. A. Robinson and R. H. Stokes, "'Electrolyte Solutions", Academic Press, Inc., New York (1955). The Robinson/Stokes reference does an especially good job of summarizing and evaluating the assumptions made in the derivation" H. S. Harned and B. B. Owen, "The Physical Chemistry o/Electroh'tic Solutions," Reinhold Publishing Corporation, New York (1958); K. S. Pitzer, ""Thermodynamics," Third Edition, McGraw Hill, Inc., New York (1995). Values taken from S. Glasstone, Thermodrnamics[br Chemists, D. Van Nostrand Company Inc., Toronto, p. 443 (1947), The values tabulated in this reference were taken from D. N. Craig and G. W. Vinal, J. Res. Natl. Bur. Stami., "'Thermodynamic Properties of Sulfuric Acid Solutions and Their Relation to the Electromotive Force and Heat of Reaction of the Lead Storage Battery", 24, 475-490 (1940). More recent values at the higher molality can be found in W. F. Giauque, E. W. Hornung, J. E. Kunzler and T. R. Rubin, "'The Thermodynamic Properties of Aqueous Sulfuric Acid Solutions and Hydrates from 15 to 300 ~ K", J. Am. Chem. Sot,, 82, 62-70 (1960).