ANALYTIC METHODS FOR CALCULATION OF PARTIAL MOLAR PROPERTIES

18.3 ANALYTIC METHODS FOR CALCULATION OF PARTIAL MOLAR PROPERTIES

When the value of an intensive property J can be expressed as an algebraic function of the composition, the partial molar quantities can be determined analytically.

Partial Molar Volume In the case of valine in water [5] at 298.15 K and 1 atm, for example, V (the volume of

solution in cubic decimeters for 1000 g of water) can be expressed in terms of the

following series in the molality m 2 :

V ¼ 0:9999999 þ 0:0920377 m 2 þ 0:00022207 m 2 2 (18 :40) or

V ¼ 0:9999999 þ 0:0920377 n 2 þ 0:00022207 n 2 2 (18 :41) The value of V m2 can be obtained by differentiation, because the quantity of solvent

is fixed:

@V

V m2 ¼ @n 2 n 1

¼ 0:920377 þ 0:00044414 m 2 ¼ 0:920377 þ 0:00044414 n 2 (18 :42)

where n 2 is the number of moles of solute in 1 kg of solvent. The partial molar volume of the solvent V m1 can be obtained by the integration illustrated in Equation (18.37). To evaluate dV m2 we merely need to differentiate Equation (18.42):

dV m2 ¼ 0:00044414 dn 2 (18 :43) Substituting from Equation (18.43) into Equation (18.37), we have

V ð m2

V m1 2 m1 dV m2 0 :00044414 dn 2 (18 :44)

V m2 8 0

423 As n 1 ¼ 1000/18.02 ¼ 55.51 and as V m1 †

18.4 CHANGES IN J FOR SOME PROCESSES IN SOLUTIONS

3 ¼ 18.08 cm 21 mol

55 :51 2 n 2 2 ( for 1000 g of solvent)

m 2 2 (18 :45)

Partial Molar Enthalpy Values of L m1 for the HCl solutions for which L m2 is given in Equation (18.19) also

can be obtained by analytical integration of Equation (18.19):

L m1

m 2 (432 dm 2 )

2 (cal mol )

18.4 CHANGES IN J FOR SOME PROCESSES IN SOLUTIONS We need to consider two kinds of processes involving solutions, other than chemical

changes. One kind is a transfer or differential process, and the other is a mixing or an integral process.

Transfer Process Consider the equation

(18 :47) An infinitesimal transfer of glycine from the solid phase to the solution at constant

glycine(s) ¼ glycine (m 2 ¼ 1, aq)

temperature, pressure, and composition of solution results in a corresponding change dJ in the thermodynamic property J of the system composed of crystalline glycine and a 1-molal aqueous solution of glycine. The application of Equation (9.32) leads to the expression

dJ ¼J † m2(s) dn 2(s) þJ m2(m 2 ¼1) dn 2 (18 :48)

CALCULATION OF PARTIAL MOLAR QUANTITIES AND EXCESS MOLAR QUANTITIES

As mass is conserved in the transfer,

2(s) ¼ dn 2(m 2 ¼1) ; dn

and Equation (18.48) can be written as

dJ

m2(s) dn þJ m2(m 2 ¼1) dn

(18 :49) Equation (18.49) can be integrated to obtain the change in J for the solution of

¼ [J m2(m 2 ¼1)

m2(s) † ] dn

one mole of glycine in an infinite volume of solution of molality m 2 :

ð J 2 n 2 ð þ1

dJ ¼

[J m2(m 2 ¼1)

m2(s) † ] dn (18 :50)

[J m2(m 2 ¼1)

m2(s) ] dn (18 :51)

It is characteristic of a differential process that the transfer occurs without a change in the composition of any phase. If a finite change of state is to occur without a change of composition, the aqueous solution of Equation (18.47) must have a volume suffi- ciently large that the addition of one mole of solid glycine does not change the com- position, so that J m2(m2¼1) , like J m2(s) † , is a constant. Then the integral on the right side of Equation (18.51) can be evaluated as

(18 :52) For the case in which J represents the volume of the system, we can use the data

DJ ¼ J 2 1 ¼J m2(m 2 ¼1)

† m2(s)

(Table 18.3) of Gucker et al. [1] on the partial molar volumes in aqueous glycine

TABLE 18.3. Partial Molar Volumes in Aqueous Solutions of Glycine

m /(mol kg 21 )

V 3 m2 3 /(cm mol 21 ) V m1 /(cm mol 21 ) 0 43.20 18.07

1 44.88 18.05 Pure solid

425 solutions. Then we calculate DV for the change of state in Equation (18.47) as

18.4 CHANGES IN J FOR SOME PROCESSES IN SOLUTIONS

DV ¼ V m2(m2 ¼1)

m2(s) †

¼ 44:88 cm 3 mol

3 mol

(18 :53) Thus, the volume change for Equation (18.47) is the sum of the volume change

3 mol

for the disappearance of one mole of solid glycine, 2V m2(s) † , and the volume change for the addition of one mole of solid glycine to a large volume of solution

with m 2 ¼ 1, V m2 .

A process analogous to that of Equation (18.47) is

H 2 O(1) ¼H 2 O(solution, m 2 of glycine ¼ 1) (18 :54) for which

DJ ¼ J m1(m 2 ¼1)

m1 †

or, for the volume, DV ¼ V m1(m2 ¼1)

¼ 18:05 cm 3 mol

Integral Process More typical of common experience is a mixing process such as

2 55 3 :51 H 2 O; glycine(s) þ 55:51 H 2 O(l) ¼ solution 4 1 glycine; 5 (18 :56)

m 2 ¼1 for which

DJ ¼ J final

initial

But the expression for J final and J initial includes terms for all components of the initial and final phases, because the composition of the phases changes during the mixing process. Thus,

DJ ¼ n 1 J m1 þn 2 J m2

1 J † m1

2 J † m2(s) (18 :57)

CALCULATION OF PARTIAL MOLAR QUANTITIES AND EXCESS MOLAR QUANTITIES

which for Equation (18.56) becomes DJ ¼ 55:51(J m1

m1 ) þJ m2

m2(s)

18.5 EXCESS PROPERTIES: VOLUME AND ENTHALPY Excess Volume

As with other excess thermodynamic properties (Section 16.7), the excess volume is defined as

(18 :59) because DV I mix , the volume change on mixing for an ideal solution, is equal to zero.

V E M I ¼ DV mix mix ¼ DV mix

The values of V E M can be measured directly with a dilatometer, or they can be calcu- lated from density measurements of pure components and solutions. For mixtures of components A and B

E [XM A B ] XM A (1

M ¼ (18 :60)

where M A and M B are the molar masses of the components, r A and r B are the corre- sponding densities, X is the mole fraction of A, and r is the density of the solution. As with the excess Gibbs function, analytical expressions for excess volumes can be obtained by fitting experimental data to a Redlich – Kister expression of the form

where X is the mole fraction of component A and N is determined by fitting poly- nomials with successively larger number of terms until an additional term does not improve the sum of squares of deviations between experimental and calculated values, which is a quantity provided by the fitting program (see Section A.1).

Excess Enthalpy From Equation (16.58),

H M E ¼ DH mix

427 so experimental values of enthalpy of mixing give the excess enthalpies directly.

EXERCISES

As with excess Gibbs function and excess volumes, the results can be fitted to a Redlich – Kister expression of the form

Enthalpy of dilution data can be used to calculate excess enthalpies by a procedure analogous to that we used to calculate relative molar enthalpy from enthalpy of dilution. We can describe the dilution of an aqueous solution of n 1 moles of water and n 2 moles of solute S with (n 1f 2n 1i ) moles of water by the equation

[n 1i H 2 O þn 2 S] þ (n 1f 1i )H 2 O ¼ [n 1f H 2 O þn 2 S] (18 :63) For this process, the enthalpy of dilution is given by

1i H m1i þn 2 H m2i þ (n 1f 1i )H m1 8 ] (18 :64) Exercise (18.20) leads to the result

DH dil ¼n 1f H m1f þn 2 H m2f

(18 :65) For solutions for which the molality is a convenient measure of composition, it has

DH m,dil ¼H E mf E mi

been suggested [6] that H E m can be expressed as a polynomial in m

H E ¼h m þh m m 2 1 2 (18 :66) Consequently, DH m,dil ¼h 1 (m f i ) þh 2 (m f i ) 2 (18 :67)

The constants needed to obtain a value of H E m as a function of m can be obtained by fitting the enthalpy of dilution data to Equation (18.67) by a nonlinear least-squares method. (see Section A.1).