Osmotic pressure measurements with solute-permeable membranes

JOURNAL OF POLYMER SCIENCE VOL. XXVI, PAGES 255-275 (1957)

Osmotic Pressure Measurements with Solute-Permeable Membranes

J. L. GARDON and S. G. MASON, Pulp and Paper Research Institute of Canada, and Chemistry Department, McGill University, Montreal, Quebec,

Canada

INTRODUCTION

The main difficulty in determining osmotic pressure, especially with solutions of substances having molecular weights less than 20,000, arises from leakage of the solute through the membrane. I n this paper we describe a method of computing the true osmotic pressure of a solution from the variation with time of the osmotic pressure readings for the case in which the osmometer membrane is permeable to both solute and solvent. This procedure is based upon extensions of principles due to Staverman1s2 and Laidler et and has been used to determine molecular weights of sodium lignin sulfonates having molecular weights as low as 4300.

In conventional osmometry the measurement consists of establishing the hydrostatic pressure difference equal to the osmotic pressure r where no solvent flows through the semipermeable membrane separating the solvent from the solution. When the membrane is permeable to both solvent and solute a number of corrections must be made to calculate the true osmotic pressure from the measured values. Although the concentration difference between the solutions on both side of the membrane decreases steadily, it is conceivable that a t a certain time a hydrostatic pressure difference can be applied such that solvent transfer through the membrane momentarily stops. This pressure difference, defined as experimental osmotic pressure aezp, will be shown to be smaller than the true osmotic pressure.

It is not possible to measure rezp directly but i t is, however, possible to determine experimentally the necessary conditions for which the net rate of volume change (arising from diffusion of solvent and solute in opposite direction) is momentarily zero. Under these conditions a pressure difference, defined as measurable osmotic pressure rrn must be applied; it will be shown that am < rezp.

When the osmometer is filled with a solution on one side and pure solvent on the other side of the membrane and a hydrostatic pressure difference equal to a,,, is applied between the solvent and solution, then the hydrostatic pressure differenre p , will tend to decrease with time as the concentration difference across the membrane decreases. It is shown that, after a certain time, log p , decreases linearly with time and can be extrapolated

255

256 J. L. GARDON AND S. G. MASON

to zero time to yield a value p, , where p , > rm. It will be shown how the true initial osmotic pressure ?r can be calculated from p,.

THEORETICAL PART

The Experimental Osmotic Pressure

If the activities of the solutions in the osmometer half-cells are al and a2, where al > az, the solute activities inside the membrane at the solution- membrane interfaces can be assumed t o have values of yal and ya2, respectively. The constant y is the partition coefficient of the solute with re- spect to the solutions inside and outside the membrane. Between the two membrane surfaces an activity gradient y(al - az)/d, exists, where d, is the thickness of the membrane.

Laidler and Schuler3 showed this model to be valid if the rate of desorption of the solute from the membrane surface is much faster than the rate of adsorption on and diffusion into the membrane, ie., if no solute accumu- lates in the membrane-solution interface; in this event y is the ratio of the rate constants of adsorption and desorption.

In a system consisting of a membrane permeable to the solvent which separates the solution from the pure solvent at the same hydrostatic pressure, the solute will tend to distribute itself uniformly throughout the available solvent. When y = 0 no solute can pass through the membrane and solvent will flow through the membrane into the solution under the influence of a pressure equal to the theoretical thermodynamic osmotic pressure. When y = 1 the solute can pass through the membrane without change in concentration until the solutions on both sides reach the same concentration; the solvent will then have no tendency to flow through the membrane and obviously there is no measurable osmotic pressure. In the intermediate case where 1 > y > 0, which is that with which we are con- cerned, solvent and solute diffusion will occur simultaneously.

The necessary and sufficient condition that a t any given moment no solvent should pass through the membranesolution interfaces is given by:

PI = P1,m (la>

P2 = P2.m (1b)

In these equations, and ~2 are the chemical potentials of the solvent in the osmometer cells and pl,, and pz,, at the two membrane surfaces inside the membrane.

Remembering that according to our model C L ~ , ~ = yal and U Z , ~ = yaz, the conditions satisfying the equations above can be calculated from the pressure dependence of the chemical potentials following the method6s7 of deriving the van’t Hoff equation. The derivation assumes that the solutions are very dilute, ideal, incompressible and that Raoult’s law is applicable. If the hydrostatic pressures hl, hz, and h, act on the solutions present in the two half-cells and in the membrane, where hl > h, > hz, it follows from equations (la) and (lb) that:

OSMOTIC PRESSURE MEASUREMENTS 257

where R is the gas constant and T is absolute temperature. Combining (2a) and (2b), one obtains the pressure difference defined as rezp, at which no solvent passes through the membrane:

It should be mentioned that the validity of Raoult's law in this special case implies the absence of interaction between the membrane and the solute. Interaction between the membrane and the solvent, however, does not interfere with the measurements, as it is a thermodynamic necessity that the fugacity of the pure solvent in a completely swollen membrane should be the same as that of the pure solvent outside the membrane; otherwise the membrane would either attract solvent from outside or expel solvent from inside.

From nonequilibrium thermodynamic considerations Staverman'** derived the equation:

T m p = s r (3)

where (1 - s) is d e h e d as the ratio of the concentration of the solution flowing through the membrane under the influence of a hydrostatic pressure to that of the solution outside of the membrane, when the solutions on both sides of the membrane are of identical concentration. Staverman defined s in this way because his derivation holds regardless of the model adopted to describe the mechanism of permeation. When this definition is applied to our model, it is readily seen that (1 - s) and y are identical. If a1 = a2,

the activity of the solution inside the membrane is yal, whether or not pressure is applied. It follows from this argument that equations (2c) and (3) are identical.

Using a negative pressure osmometer as a model, Barr8 also derived a relation between rezp and T. In such an osmometer the membrane is replaced by an atmosphere saturated with solvent vapor; if the solute is volatile there will be leakage. For this case, Barr found that reZp = k?r, where 0 < k < 1.

The importance of rezp is that, when the hydrostatic pressure difference between the two solutions in the osmometer is p,, the osmotic pressure reading, the driving force acting on the solvent is proportional to ( p , - rezp).

reXp cannot be directly determined as it is related only to the solvent flow. In practice only the sum of the volume changes due to solvent and solute diffusion can be observed. Thus to determine the measurable osmotic pressure, due to which no volume change occurs momentarily in the osmometer, it is necessary to derive formulas describing both solvent and solute flow.

Obviously for semipermeable membranes y = 0 and r = rezp.


I - - = -

MEMBRANE

Fig. I . Arrangement of osmometer showing solvent and solution half-cells (schematic).

Variation of the Osmotic Pressure with Time The osmotic pressure is generally measured by means of a capillary

manometer attached to either the solvent or solution half-cells. In the discussion which follows, the manometer is considered to be attached to the solvent chamber as shown in Figure 1, although the equations derived are equally applicable to the alternative case.

( I ) The capillary diameter is so small that changes in the level of the solution and of solute concentrations due to osmotic diffusion of solvent are negligible.

(2) Concentration changes in the two half-cells result solely from diffusion of solute through the membrane.

(3) The solute concentration at any time is constant throughout each half-cell, ie.) the entire concentration gradient is across the membrane.

(4) The solute is monodisperse and in ideal solution. As already shown under these conditions the concentration gradient at

any instant inside the membrane is ~ ( c 1 - cz)/d,, where c1 and c2 are the respective solute concentrations in the solution and solvent chambers, and d, is the thickness of the membrane (in further theoretical treatment the activities are replaced by concentrations). If D is the diffusion coefficient of the solute in the membrane, V1 is the volume of the solution chamber, and A is the effective membrane area, Fick’s law of diffusion takes the form :

It is assumed that:

- V i d ~ l / d t = D,47(cl - ~ z ) / d , (4) If one chamber of the cell is initially filled with a solution of concentra-

tion co and the other with solvent, we have from a material balance:

Vlco = VlCl + V2c2 = Vl(C1 + cz/q) (5) whereq = Vl/Vz.

By combining equations (4) and (5), integrating, and substituting for

OSMO1'IC PRESSIJRE MEASUREMENTS 259

(CI - c2) by means of the van't Hoff equation it follows that :

r = To exp 1 - ~ ~ t )

Kp = Y A W + q)/Vd,

(64

(6b)

where the quantity

expresses the rate constant of solute diffusion between the half-cells. In the derivation above it has been assumed that the pressure gradient

has a negligible effect on the solute diffusion; Laidler and Schuler3 have shown that this assumption is valid.

Resistance of Osmometer to Solvent Flow If both chambers of the osmometer are filled with pure solvent and the

meniscus in the capillary is initially adjusted so that there is a hydrostatic difference h(0) across the membrane, the rate of flow of solvent through the membrane may be expressed in the form:

dV,/dt z= (a/pM)(dh/dt) = -Ph (7) where u is the cross-sectional area of the capillary, pM is the density of the manometer liquid, and P is the permeability of the membrane to the solvent. Integration of equation (7) yields:

h(t) = h ( ~ ) exp 1 - ~ , t ) (8)

where the constant K, - pMP/u may be determined by means of equation (8) from the decrease in h with time.

If the capillary manometer and other parts of the osmometer provide ap- preciable resistance to solvent flow, it is readily shown that:

l/K,u = 1/PpM + V / a 2 + W (9) where v is a constant characteristic of the viscosity of the solvent and the length of the capillary tube which is calculable from the Poiseuille equation, and w is the resistance of the remaining parts of the cell.

The Measurable Osmotic Pressure (rm)

When the net volume change in the osmometer is momentarily zero the volume of the solvent flowing under the influence of the hydrostatic pressure (rezp - rm), where rm is defined as the measurable osmotic pressure, must be equal to the volume changes due to solute diffusion.

The volume flux of the solute from cell 1 to cell 2 can be expressed by the combination of the differential forms of equations (5) and (6) :

VpVI d r RT(1 + q) dt -dVp/d t = - VPV1(dCl/dt) =


where V1 dclldt is the number of molecules passing through the membrane in unit time and V p is the effective molar volume of the solute.

If one substitutes K , for P in equation (7) the volume flow of the solvent is given by:

dV,/dt = (K,a/p&f)h (11)

When h = (imp - 7rm), solvent will flow into cell 1 (if c1 > c2) at a rate corresponding to that of solute, ie.:

d V p / d t + dVs/d t = 0

The measurable osmotic pressure can be expressed from equations (2c) anh (12):

7rm = 7r(l -. y - sVp) ( 134

Variation of p7 with Time

Although 7rm can be experimentally determined by trial and error, more accurate values can be obtained if the calculation of the true osmotic pressure is based on a function which describes the variation of the osmotic pressure readings (pl) with time. This function is:

dp7ldt = (pnn/a)(dVs/dt + dVp/dt) (144

By substituting from equations (7), (1 1), and (13b), we obtain :

dp,/dt + Ksp7(t) = K,i(O)(l - y - Svp) exp { -K,t) (14b)

Integration of equation (14b) yields:

p7( t ) = B exp { -KJ) + T ( o ) ( ~ - y - sTp)[~,/(~, - KJI exp { - ~ , t )

(14c)

In practice it is advantageous to adjust the osmometer to approximately For this procedure the boundary 7rm, at the beginning of the experiment.

condition for equation (14) is:

PT(O) = T m ( 0 )

which yields for the constant of integration:

B = 7ro(l - y - sVp)K , / (K , - K,)


Thus the time-dependence of p r is described by:

pr(t> = ~ ( 0 ) ( 1 - Y - ~ v p ) ( ~ X P { - ~ p t ) + W,/K - K,)I[~XP 1 - K J ) - exp f - K ~ ) I ) (15)

It follows from equation (15) that [ p ( t ) - ~ ( t ) ] reaches a maximum at the time :

tmaz = 2.303 [log (Ks/Kp)l/(Ks - Kp) (16)

If K, >> K , (as was found experimentally) it follows that, at sufficiently great values of 2, equation (15) may be approximated to:

pr(O = ~rn(0) (1 + ~ p / ~ s ) ~ X P { - K J ] (17)

i.e., after a certain time a plot of In p , ( t ) against t yields a straight line of slope -K,. If the linear portion of this plot is extrapolated to zero time to yield a value p, , it follows from equation (17) that:

pc = (1 + K p / K s ) T m ( O )

40) = Pe/[ l ( l + KP/KS)(l - Y - SV,)l

whence the true osmotic pressure is given by

(18)

Pe +,ax hrs.

119.0 -

101.0 -

15.00 69.3

13.80 55 0

12.10 40.0 11.45 32.5 la50 15.7 10.00 0.00

I I 1 I I I I 100 140 160 180 200

Fig. 2. Log p , ueTsus t plots calculated from equation (15) corresponding to a,(O) = 10 cm. and K , = 0.01 hr.-l, and various values of K,.


In Figure 2, a series of values of pr ( t ) calculated by means of equation (15) is shown plotted logarithmically against t. In these plots ~ ~ ( 0 ) and K , have been assigned constant values of 10 cm. solvent and 0.01 hr.-l, respectively, and K , varies SO that K J K , covers the range of values found experimentally. From a number of such plots it was established that the plots are linear after t,,, when K,/KJ < 0.3 and that the extrapolated value p e agrees within 5% of the value given by equation (18).

In practice it is impossible to adjust the osmometer so that p,(O) = ~ ~ ( 0 ) since ~ ~ ( 0 ) is not known, but as will be described later it is possible to adjust the osmometer a t a certain time t o after filling so that:

pr(t0) - ~ r n ( t o > = A

where A is the error of adjustment at to, and then to make subsequent measurements of p l ( t ) without further adjustment.

It is readily shown that under these circumstances integration of equation (14b) leads to (15) with the additional term A exp { - K,(t - to)) added to the right-hand side. Since K , >> K , and A can be kept much smaller than rm(0), the adjustment error soon becomes negligible and does not interfere with the extrapolation.

In equation (18), ~ ( 0 ) and v, are unknown while the other quantities can be calculated or determined experimentally. The equation may be rearranged to the form:

a = l/T(O) + vp/3

P = (1 + K,/Ks)(Vpe)

(19) where : = r(1 - 7)/pei(1 + K,IKJ and :

A plot of values of Q versus /3 at constant V , and T ( O ) , which can be made by changing the membrane and/or the capillary manometer so as to vary 7, K,, and/or K,, should therefore yield a straight line of slope V, and having an Q intercept of l/a(O).

The foregoing treatment leading to equation (14c) is in many respects similar to that of Laidler and S c h ~ l e r . ~ , ~ By assuming that the driving force acting on the solvent is ( p , - T), these authors did not take into account the membrane coefficient y. As demonstrated above and by Staverman12 the driving force should be ( p , - By setting y = 0 equation (14c) becomes identical with the corresponding equation presented by Laidler and Schuler.

Laidler and Schuler solved (14c) for the boundary condition p,(O) = 0 . 4 In their experiment^,^ the authors calculated the osmotic pressure from the concentration and determined the constants which correspond to K,, K,, and 6 from a single osmotic experiment. The experiments were conducted using sugar solutions in a soft membrane osmometer and thus the volume changes due to diffusion may have been masked by movement of the membrane. Nevertheless, they obtained good qualitative agreement with the theoretical predictions.


The treatment presented in this paper has the advantage that the constants used are so defined that it is possible to determine them by means of independent experiments. The use of the boundary condition p,(O) = ~ ~ ( 0 ) yields an expression which allows interpretation of data obtained by the techniques of conventional osmometry and makes possible the determination of osmotic pressures, of number-average molecular weights, and in principle of the effective model volumes.

Suggestions to correct osmotic pressure measurements for membrane leakage have also been presented by M e n ~ i k , ~ Dialer and Elias, lo and Donnet and R0th.l' These authors, however, have not made all of the corrections which are considered necessary.

The Effect of Polydispersity and Concentration-Dependence

In deriving equation (15), the solute was assumed to be homogeneous and the solution ideal. Deviation of these conditions may have a strong influence on the interpretation of the experimental results.

If the solute is polydisperse, number average y, K,, and V p values have to be used to calculate T. Number average K, and P, can be determined from the osmotic experiments and the number average partition coefficient yn can be separately determined according to a procedure presented below.

For K , and V, only the polydispersity of the fraction which can permeate through the membrane has to be considered. In practical osmometry this fraction is usually small and hence does not represent a wide molecular weight range. If, however, the fraction permeating through the membrane is a large portion of the dissolved material and highly polydisperse, the molecular weight distribution of the solute in the osmometer cells will change appreciably with time and neither K , nor V , can be considered constant.

It should also be noted that F, is essentially the molar volume of that fraction of the solute which diffuses through the membrane. It can be expected that, when membranes of different selectivities are used for polydisperse solutes, the measured V, values will differ from membrane to membrane. The C U , ~ values obtained with different membranes should be treated separately.

If the reduced osmotic pressure and the diffusion coefficient vary with concentration, it is necessary to carry out experiments at different concentrations and to extrapolate to zero concentration. If the concentration- dependence of the diffusion coefficient is strong, the observed values of K, may vary as the concentration difference between the osmometer cells decreases. Generally it is preferable to use poor solvents to minimize the concentration effects.

In this case extrapolation to zero time may not be successful.

EXPERIMENTAL PART

In order to apply the theory presented above it is necessary to determine To make an a,P plot at least three sets of values of p,, K,, K,, and -yn.

osmotic pressure measurements were made.

2a J. L. GARDON AND S. G. MASON

Materials

The substances used in the experiments were sodium lignin sulfonate fraction (Nos. 1 to 6) having various molecular weights. A detailed account of their preparation is given elsewhere.12 To suppress ionization and thus avoid Donnan membrane and other effects, 0.5 N sodium chloride from a large stock was used as solvent.

Osmometers

The osmometers employed had the characteristics assumed in the theoretical part and have been described in detail in a separate communica- tion. l3 The connecting capillaries between the toluene manometer and the cell were made at least 5 cm. long to insure that solute concentration changes were confined to the cell section.

Some characteristic data of the three osmometers used are given in Table I. From the weight-average partition coefficients (rut) obtained (Fig. 3), it can be concluded that the denitrated nitrocellulose membrane had the smallest and the 300 P.T. cellophane the largest pores. Various capillaries having diameters ranging from 0.3 to 2 mm. were employed with these units. The membranes and stainless steel reinforcers were supported by rubber discs in osmometer A, and by glass rods in B and C. All measurements were carried out in a water thermostat at 30°C.

TABLE I Osmometer Data

Cell

A . . . . . .

B . . . . . .

c. . . . . .

Capil- lary

a , . . . . . b . . . . . .

d . . . . . . . e . .. . . ,

c . . . . . . .

Membrane

300 P.T. cellophane

450 P.T. cellophane

Denitrated nitrocellulose

Diam., cm.

0.0324 0.0418 0.0702 0.0864 0.113

f . . . . . . . . g . . . . . . . .

0.144 0.191

Vol. soh. chamber, v1, cc.

13.5

8

21

Cross-section U, cm.2

8.20 x 10-4 1.38 x 10-3 3.75 x 10-3 5.85 x 10-3 1.01 x 10-2 1.63 X 2.85 X

VOl. solv. chamber, vz, cc.

13.5

8

24.3

Cell A

0.275 0.223 0.193 0.154 0.112 - -

Effective membrane

area, cm.

30

80

200

K., hr.-1 Cell B

0.725 0.623 0.371 0.267 0.165 0,099 0.061

K;m, hr. -1

0.97

2.03

2.85

Cell C

0.375 0.346 0.303 0.283 0.196 0.140 0.112

Osmotic Pressure Measurements

The osmometer readings were corrected for the “zero pressure,’’ i.e., for the finite pressure reading observed when solvent was placed on both

OSMOTJC PRESSURE MEASUREMENTS 265

.?

t z w 0 t $ 2

8 w

z 0 t a t

2 .I

5 10 I5

MO LECU LA R w E I G HT x I O - ~

Fig. 3. Variation of yw with molecular weight of sodium lignin sulfonate fractions in 300 P.T. cello hane (curve A), 450 P.T. cellophane (curve B), and denitrated nitrocellulose (curve 8). From the experimental data of Tables I1 to VI.

sides of the membrane. The zero pressure tended to change, presumably from sorption of solute on the membrane, but the original value was re- established by periodically rinsing the membrane on both sides with solvent for periods ranging from 12 to 24 hours. The zero pressures for denitrated nitrocellulose and 450 P.T. cellophane were 0.8 and 0.5 cm. toluene, respectively. No zero pressure could be measured with the osmometer equipped with the 300 P.T. cellophane.

The osmometers were filled in such a way that the pressure difference between the solution and the solvent was within 3 cm. of the expected osmotic pressure. This pressure was maintained in the initial 0.5 to 2 hours needed to establish temperature equilibrium. The purpose of this adjustment was to avoid dilution effects due to solvent diffusion during this period.

After establishing thermal equilibrium at the time to , the measuring capillary was put into operation. The value ~ ~ ( 2 0 ) was then established within 0.5 cm. by a trial and error method in which the level in the capillary was adjusted by using the leveling device and observing whether the level tended to rise or fall. This procedure was repeated until two con- secutive levels were obtained, one rising and the other falling, not differing by more than 0.5 cm. The ornometer was adjusted between these levels and the recorded pressure values p, ( t ) were followed until a t least 2t,,,, i.e., for 24 to 48 hours.

Determination of K,,

K , was determined from the slope of the log p,(t) versus 2 plot, using equa- In addition, it was computed from the change in solute concen- tion (17).


trations in the two half-cells using the following equation, which is readily obtained by combining (4) and ( 6 ) :

K6 = (2.303/t) log C~/ (CI - C Z ) (20)

At the end of a series of measurements of p,, the solution and solvent chambers were emptied and the lignin sulfonate concentrations c1 and cz were determined from the optical densities a t 280 mp in a Beckmann spec- trophotometer.

Kb values calculated from (20) are different from K , values calculated from (17). For monodisperse solutes they should be the same; for hetero- disperse solutes K , is a number-average and Kg is a weight-average with K , > Ka.

Determination of K,

Ka was calculated using equation (20).

The K, of each osmometer fitted with capillary manometers of various sizes was determined. Both cell chambers were filled with distilled water, toluene was placed in the capillary manometer, and the unit was put into the thermostat. Plots of log h against time yielded straight lines from which K, was calculated by means of equation (8). Typical plots are shown in Figure 4. The solvent used in the osmotic pressure measurements was 0.5 N sodium chloride solution, but this could not be used for the determination of K, values. The semilogarithmic plots with sodium

10 20 30 40 50 60 70 80 90 100 110 120 130 MINUTES

Fig. 4. Experimental log h versus t plots used for the evaluation of K, of osmometer B and designated capillaries ( e f . Table I).


chloride solution showed breaks. known.

The reasons for this behavior are not

Determination of y

The partition coefficient y was, in effect, determined by comparing the dialysis and diffusion coefficients of the solute. The procedure consisted of the following steps:

(1) The diffusion coefficient was determined by the porous plate tech- nique14Js using potassium chloride as the calibrating liquid. In this method two half-cells are separated by a porous plate in which y = 1 for the solute. The half-cells are initially filled with solution and solvent, respectively, and diffusion occurs in accordance with equations (4), (6) , and (20). This is repeated with potassium chloride and from the two sets of measurements Kg and K;3,KC1 are obtained for the porous plate cell. Since the diffusion coefficient D K C ~ of the potassium chloride is known, the diffusion coefficient D of the solute can be calculated from:

DKC1/D = [K~ ,KC1/K~lporous plate (21)

(2) The value of Ka ,~c l for potassium chloride diffusing through the It is assumed that in this

It follows from equation (6b) that : membrane in the osmometer cell is determined. case the partition coefficient is unity.

DKCl/D = Yzo[Kb,KCI!Kblmembrane (22)

where yur is the weight-average partition coefficient of the solute. As the calibrating potassium chloride is monodisperse, K,,KC1 = Kb,KC1, the ap- proximate number-average partition coefficient Y~ can be determined similarly :

DKCIID = ~n[I(~,KC1/KpIrnernbrane (23)

The true number-average partition coefficient could be determined only if the number-average diffusion coefficient were used instead of D, the weight- average value. However, in certain cases discussed later, the use of equation (23) only introduces a small error. In calculating a and p, the yn values were used.

The values of Kb,KCI of the membranes were determined using 0.1 N potassium chloride and distilled water in the osmometer cell, the other parts of the osmometer being closed off. These experiments were carried out a t the prevailing room temperature rather than a t 30°C. because the cell could not be brought t o temperature equilibrium quickly enough; a typical KC1 diffusion measurement took about 0.5 hr. The potassium chloride concentrations were determined by the Mohr method. Kb,Kcl values were corrected to the 30°C. values from the temperature coefficient of diff usion.16

To determine the diffusion coefficients of the sodium lignin sulfonates, medium-grade sintered glass discs 30 cm.2 in area were used. These discs


- I .

were substituted for the membrane in the smallest osmometer cell (A, Table I). The four inlets to the cell were closed by stopcocks. During the diffusion measurements the cell was in a horizontal position. The experimental procedure was the same as suggested by other author^.'^^^^ The diffusion coefficient of 0.1 N potassium chloride was taken as 0.00742 cm.2 hr.-l by converting Gordon's value at 30°C.16317

FRACTION No. 5 CELL-A=--A--- CELL6 -X-

FRACTION No.6 CELL-C- CELLC -m-*o-*-

RESULTS AND DISCUSSION

A typical set of log p , 2)s. t plots is shown in Figure 5. These curves were obtained using 0.2 g./100 cc. of fraction 5 in osmometer B equipped with various capillaries so as to vary K, at fixed K,. It will be observed that the points fall on a curve similar to those in Figure 2, and that the linear portions are parallel t o one another as required by equation (17). It may be concluded from this that the rate of solute diffusion through the mem-

151 1

CAPILLARY: a b c d e f SYMBOL : 0 A 0 e A 0

H O U R S

Fig. 5. Experimental log p , versus t plots for fraction 5 in osmometer B (cf. Table 111).

I I I I I I I I

10 15 20 25 30 35 40 45 50

lo7 cm:' 9:'

Fig. 6. a,@ plots for fractions 5 and 6. The straight lines were calculated by the method of least squares (cf. Tables I1 and 111).


TABLE I1 Osmotic Pressure Measurements for Fraction 6

D = 9.05 X cm.2 hr.-l. co = 0.2 g./lOO cc.

Unit C: K p = 0.029 hr.-l, T~ = 0.078, yn = 0.087. K i = 0.026 hr.-l,

Capil- Pa, ff, B lary K P / K ~ g. cm.+ cm.2 g.-1 cm. -l g. -l

a . . . . . . . . . . . . . 0.077 9.88 0.0992 40.00 X lo-' b . . . . . . . . . . . . . 0.084 10.4 0.0946 24.75 x lo-' c . . . . . . . . . . . . . 0.095 10.4 0.0956 10.4 x 10-7 d . . . . . . . . . . . . . 0.102 11.95 0.0842 e . . . . . . . . . . . . . 0.148 12.48 0.0838 f . . . . . . . . . . . . . 0.207 12.8 0.0850

r0 = 11.92 g./cm.*. rp = 3950 cc. M = 4350.

TABLE I11 Osmotic Pressure Measurements for Fraction 5

D = 8.34 X cm.2 hr.-l. co = 0.2 g./lOO cc.

Capil- P. 7 ff,

lary KP/K* g. cm.-2 cm.2 g.-I

6.28 x 10-7 5.22 x 10-7 4.62 x 10-7

8, cm.-' g.-1

Unit A: K, = 0.0304 hr.-1, yn = 0.279, yu = 0..238. K; = 0.0259 hr.-1,

a . . . . . . . . . . . . . 0.110 7.56 0.106 51.2 x 10-7 b . . . . . . . . . . . . . 0.136 8 . 0 0.102 50.2 x 10-7

8.52 0.0935 13.3 x 10-7

9.78 0.0937 8.02 x 10-7 d . . . . . . . . . . . . . 0.197 8.67 0.0935 10.95 X

~ ( 0 ) = 10.92 g./cm.-2. M = 4700. v, = 2530 cc.

Unit B: K , = 0.0231 hr.-l, 7% = 0.1018, yw = 0.0849. KL = 0.0193 hr.?,

a , . . . . . . . . . . . . 0.032 8.52 0.109 6.37 I ) . . . . . . . . . . . . . 0.037 8.67 0.108 4.30

9.12 0.104 2.58 d . . . . . . . . . . . . . 0.085 9.37 0.104 2.26 e . . . . . . . . . . . . . 0.14 10.65 0.096 1.97 f . , , . . . . . . . . . . 0.23 11.70 0.094 2.03

~ ( 0 ) = 10.15 g./cm.-2. M = 5050. v, 1800 cc.

Unit C: K p = 0.0188 hr.-l, yn = 0.0582, yw = 0.0427. K i = 0.0137 hr.-l,

a , . . . . . . . . . . . . 0.052 9.18 0.108 27.2 c . . . . . . . . . . . . . 0.061 9 . 3 0.107 7.16 d . . . . . . . . . . . . . 0.067 9 .7 0.103 4.86 e . . . . . . . . . . . . . 0.096 10.75 0.096 3.78

. . . . . _ . . . . 0.134 11.6 0.092 3.13 g . . . . . . . . . . . . . 0.167 12.45 0.088 2.14

~ ( 0 ) = 10.70 g./cm.-2. M = 4770. vp = 6850 cc.


TABLE IV Osmotic Pressure Measurements for Fraction 4

D = 6.30 X cm.2 hr.-l. co = 0.5 g./100 cc.

Capillary a C d e

Unit A: Kp = 0.010 hr.-l, yn = 0.092, Kh = 0.0087 hr.+, yw = 0.08.

K,/K,. . . . . . . . . . . . 0.036 0.052 0.065 0.089 p c ................ 13.4 13.15 12.9 13.7 ~ ( 0 ) . . . . . . . . . . . . . . 14.65 14.4 14.1 15.2

Unit B: Kp = 0.008 hr.-l, Y~ = 0.035, yw = 0.031. K; = O.O07hr.-',

K,/K,. . . . . . . . . . . . 0.011 0.021 0.03 0.061 p , . . . . . . . . . . . . . . . . 14.35 14.35 14.0 14.6 ~ ( 0 ) . . . . . . . . . . . . . . 14.8 14.8 14.4 15.0

Unit C: Kp = 0.006 hr.-1, yn = 0.0194, K; = 0.005 hr.-l, yw = 0.016.

K,/K,. . . . . . . . . . . . 0.016 0 019 0.021 0.043 p , ................ 15.7 15 7 14.95 15.1 ~ ( 0 ) . ............. 16.05 16 0 15.3 15.4

r(0) = 15.00 g./cm.2. M = 8450 f 5%.

brane was not influenced by the simultaneous flow of solvent through the membrane, as was assumed in deriving equations (15) and (17).

The effects attributed to V p and Y~ expressed in (18) were a t least quali- tatively demonstrated by the a,P plots shown in Figure 6. Because of imperfections in the method discussed later, the points showed consider- able scatter. Nevertheless, LY clearly tended to increase with increasing p, indicating that the volume of the solute diffusing through the membrane effectively contributed to changes in the capillary meniscus level.

The molecular weights shown in Tables I1 and I11 were calculated from the osmotic pressure values obtained from the treatment of the a and p values by the method of least squares using equation (19).

Because of the high degree of scatter of the a,P plots and the limited range of values of a, the data were of limited use in computing vp by means of (19). Values of vp obtained from the mean least square plots and given in Tables I1 and I11 represent orders of magnitude only.

Results for fractions 2, 3, and 4 are given in Tables IV, V, and VI. Here the values of K , / K s were less than the estimated error in measuring pe and corrections were made for yn alone using the relation (cf. Eq. (18)) :

It is significant that the values of ~ ( 0 ) so obtained are independent of yn.


TABLE V Osmotic Pressure Measurements for Fraction 3

D = 5.26 X o m 2 hr.-l. ~0 = 0.8 g./100 cc.

Capillary a d e

Unit A: K p = K i = 0.0038 hr.-l, y = 0.0632.

p. ..................... 11.7 11.9 12.2 ~ ( 0 ) . . . . . . . . . . . . . . . . . . . 14.45 12.8 13.0

Unit B: K p = KL = 0.0052 hr.-l, y = 0.0292.

....................... 12.3 12.1 12.65 ~ ( 0 ) . . . . . . . . . . . . . . . . . . . 12.65 14.45 13.15

Unit C: Kp = KL = 0.00077 hr.-*, y = 0.0027.

p , = T(0) . . . . . . . . . . . . . . 12.9 13.25 13.25

~ ( 0 ) = 13.32 g./cm.2. M = 15,500 + 3.7%.

TABLE VI Osmotic Pressure Measurements for Fraction 2

D1 = 3.07 X cm.2 hr.-I, co = 1 g./100 cc.

Capillary a d e

Unit A: K , = K i = 0.0028 hr.-I, y = 0.0542.

T c . . . . . . . . . . . . . . . . . . . . . 12.45 12.7 12.65 ~ ( 0 ) . . . . . . . . . . . . . . . . . . . 13.15 13.4 13.3

Unit B: K , = KL = 0.0035 hr.-l, y = 0.0321.

Ac . . . . . . . . . . . . . . . . . . . . . 12.45 12.3 12.8 r(0). . _ _ _ _ _ . . _ _ . _ _ . _ . _ . 12.85 12.65 13.25

Unit, C: K , = K i = 0.0004 hr.-l, y = 0.0025.

x , . . . . . . . . . . . . . . . . . . . . . 13.4 13.25 13.5

~ ( 0 ) = 13.35 g./cm.2, I I f = 19,200 f 3%.

It was found that within the experimental error pe/cO was constant over a range of concentrations for fractions 1, 3, and 5 (Table VII). Thus the molecular weights could be calculated from measurements a t a single concentration without the necessity of extrapolating to zero concentration. Osmotic pressures for fraction 1 for which y = 0 for all three membranes are also shown in Table VII. The calculated molecular weight of this fraction was 58,000.

As shown by Staverman,2 s, and thus y, are dependent on concentration. When the reduced osmotic pressure T / C changes appreciably with concentration, the ideal values of both y and the reduced osmotic pressure can be evaluated by extrapolation to zero concentration.


TABLE VII Concentration-Dependence of Reduced Osmotic Pressure of Various Fractions

Frac- tion

1..

3 . .

5 . .

. . .

...

c,

100 cc. g. /

. 0 . 3 0 . 5 0 . 8 1 .0 2 . 0

. 0 . 4 0 . 8 1 . 2

0 . 2 0 . 3

. 0.15

Capil- Unit lary

B a B a A a C b B a A a A a A a B a B a B a

TJ

g./ T / C

1.27 4.25 2.19 4.38 3.65 4.56 4.40 4.40 8.72 4.36 -- - - - __ - __ - __ - -- -

- - 5.95 14.8

11.7 14.6 18.05 15.1 6.27 41.8 8.52 43.3

12.9 43.0

Stavermanls assumes that in a homologous series the relation between the reflection coefficient s, ie., (1 - y), and the molecular size is given by:

s(y) = 0.5 [I + tanh r(y - l)] (24)

where y is the ratio of the molar volumes of the solute to a volume characteristic for the membrane and r is a parameter characteristic of the homologous series and the membrane. The data presented in Figure 3 do not cover a range large enough to give experimental proof for this equation.

According to equation (24), y changes very little with the molecular weight in the molecular weight range giving y values either close to unity or close to zero. On the other hand the value of y is very sensitive to the molecular weight near the inflection point of the s-y curve corresponding to equation (24)) i.e., in the range of yn = 0.5. If the solute is polydisperse and the membrane is so chosen that the y values of the individual components are low enough not to fall into the range near the inflection point, the effective value of y will change very little as a result of diffusion. In the present experiments K , remains reasonably constant and extrapolation t o zero time is therefore possible. On the other hand if the initial yn is in the range of 0.5, the effective value of y will change considerably during the experiments, K , will vary as the diffusion of the solute proceeds, and the extrapolation will not yield consistent results. This effect was observed when parchment membranes, which are more porous than cellophane, were tried for fractions 1-6 and when the cellophane and denitrated nitrocellulose membranes were used for lower fractions.

The method of determining y is based on the assumption that a very low molecular weight reference substance, such as potassium chloride, is capa- ble of diffusing freely through the membrane, ie., with y = 1. Probably this assumption is only approximately correct, but nevertheless, it will not lead to a large error in the determined molecular weight of the polymer as long as the y value of the polymer itself is low. If, for example, the par-


tition coefficient of potassium chloride were 0.8 instead of 1, and if the partition coefficient of the polymer is in the range of 0.1, it follows from the procedure adopted that the error in assuming y = 1 for potassium chloride leads only to 2% error in the calculated true osmotic pressure and molecular weight.

The procedure involves the additional assumption that the ratio of the true diffusion coefficients (D in Eq. (6b)) of the reference substance to that of the polymer is the same inside the membrane and in the solution outside the membrane. This assumption is probably justified for lignin sulfonates which are in form of tight water-impermeable molecular coils in sodium chloride solution.20 Such coils are inelastic and hence lignin sulfonates would hardly change their shape, and thus their diffusion coefficient, inside the membrane. The lowered diffusion rate of the polymer relative to the reference substance inside the membrane is accordingly not caused by slower diffusion of the individual molecules but by the lower solubility of the polymer in the membrane. This lowered solubility is expressed by the partition coefficient.

In the present experiments the y n values were determined with the aid of the weight-average diffusion coefficient D (cf. Eq. (23 ) ) . If D' is the number-average diffusion coefficient the true number-average partition coefficient y i is given by:

Y: = ynD'/D For a given molecular weight distribution the ratio D'/D increases with the exponent b in the equation:

D = const X M - b describing the dependence of the diffusion coefficient on the molecular weight. For flexible polymers b is in the rangeI9 of 0.5 to 1 and for lignin sulfonates in the range15 of 0.5 to 0.6. When b < 1, the relative difference between number- and weight-average diffusion coefficients is necessarily smaller than between weight- and number-average molecular weights which is favorable for the approximation applied in equation (23) .

Obviously a high degree of polydispersity may also cause D'/D to depart considerably from unity. The molecular weight distribution of the unfractionated lignin sulfonate12 may indicate that the polydispersity of the middle fractions 2-6 is not excessive. Thus the error involved in using equation (23) to evaluate y n is in all probability small.

It should be noted that, as indicated by Staverman,2 y can also be determined by ultrafiltration. In a previous paperz1 we described an experimental method of doing this. The advantage of the ultrafiltration technique is that it does not involve assumptions concerning a reference substance and leads to well-defined y values. I t is, however, suitable only for the determination of the weight-average partition coefficient while for osmotic corrections the number-average values are to be used. Furthermore the ultrafiltration technique is applicable only if large sheets of very uni-


form membrane material is available, as it is experimentally not feasible to use the same piece of membrane for both osmotic and ultrafiltration experiments. Ultrafiltration may, however, prove to be a valuable tool to check the assumptions which are involved in the application of the diffusion technique.

An additional source of experimental error is believed to be the concentration gradients in both half-cells caused by the movement of both solute and solvent. Concentration gradients caused by solvent diffusion may be expected to be confined to the immediate vicinity of the membrane. As solvent flows through the membrane, the concentration of the solution at the membrane increases on the solution chamber side and decreases on the other side as a result of the filtering action of the membrane on the solute. The effect of solvent flow is thus opposite to that of solute diffusion in that it tends to increase the measured osmotic pressure.

The need for membrane supports for permeable membranes is greater than for semipermeable membranes. Ballooning of semipermeable membranes has an adverse effect on the speed of the measurements but not on their accuracy. The slightest membrane ballooning in membranes permeable to the solute has an adverse effect on the accuracy of K,, V1, and q. When the membrane moves, one part of the solvent diffusing through the membrane causes changes in the hydrostatic pressure, and the other part is needed to compensate for the volume changes in the half-cells. In the medium-sized osmometer used (unit B) the membrane support was prac- tically perfect. The reproducibility of the K , determinations was about 1%; furthermore, a plot of l/K,cr against l / u 2 using the data given in Table I yields a straight line, as predicted by equation (9). In the other two units the K , determinations were reproducible only to f 3%.

It is concluded that the general principles of the theory and method have been established. It is fully applicable to monodisperse solutes and when K,/Ks < 0.3. From the present experiments the method appears to be applicable to polydisperse solutes if vn is smaller than about 0.2. A con- siderable improvement in the method may be expected from the use of an osmometer with rigidly supported membranes and built-in stirrers to elim- inate concentration gradients in the half-cells.

References 1. Staverman, A. J., Rec. trav. chim., 70,344 (1951). 2. Staverman, A. J., ibid., 71,623 (1952). 3. Laidler, K. J., and Schuler, K. E., J . Chem. Phys. , 17,851 (1949). 4. Laidler, K. J., and Schuler, K. E., ihid., 17, 856 (1949). 5. Laidler, K. J., Dames, C. A,, and Schuler, K. E., ibid., 17, 860 (1949). 6. Glasstone, S., Textbook o j Physical chemistry , 2nd ed., Van Nostrand, New York,

7. Hildebrand, J. H., Science, 121, 117 (1955). 8. Barr, T. A., Am. J . Phys. , 23,436 (1955). 9. Mencik, Z., J. Polymer Sci., 18, 425 (1955).

1946, pp. 668-670.

10. Dialer, K., and Elias, A. G., J . Polymer Bci., 18, 427 (1955).


11. Donnet, J. B., and Roth, B., Bull. soc. chim. France, 49, 1257 (1954). 12. Gardon, J. L., and Mason, S. G., Can. J . Chem., 33, 1477 (1955). 13. Gardon, J. L., and Mason, S. G., ibid., 33, 1453 (1955). 14. McBain, J. W., and Liu, T. H., J . Am. Chem. Soc., 53,59 (1931). 15. Northop, J. H., and Anson, M. L., J . Gen. Physiol., 12, 543 (1929). 16. International Criticat Tables, Vol. V, McGraw-Hill, New York, p. 68. 17. Gordon, A. R., Ann. N . Y . Acad. Sci., 46,785 (1945). 18. Staverman, A. J., Ind. Chim. belge, 18,235 (1953). 19. Hengstenberg, J., Die Physik der Hochpolymeren, H. A. Stuart, ed., Vol. 11,

20. Gardon, J. L., and Mason, S. G., Can. J . Chem., 33, 1491 (1955). 21. Gardon, J. L., and Mason, S. G., Can. J . Chem., 33, 1625 (1955).

Springer, Berlin, 1953, pp. 483-484.

Synopsis A method of determining the osmotic pressure of a solute which can diffuse through

the membrane is described. It is shown theoretically that after a certain time interval a plot of the logarithm of the indicated osmotic pressure against time yields a straight line which can be extrapolated to zero time. The true value of the initial osmotic pressure is calculated by correcting the extrapolated value for the rate of transfer of solute and solvent through the membrane and for the partition coefficient of the solute between the membrane and the solvent. The various correction factors are calculated from experimentally determined quantities. By the use of cellophane membranes, osmotic pressures of lignin sulfonates with molecular weights as low as 4,300 have been evaluated.

Resume Une methode permettant de determiner la pression osmotique d h n e substance dis-

soute qui peut diffuser h travers la membrane est d6crite. On montre sur des bases theoriques qu’aprbs un certain intervalle de temps le diagramme du logarithme de la pression osmotique en fonction du temps fournit une ligne droite qui peut &re extrapolbe au temps z6ro. La valeur exacte de la pression osmotique initiale est calculBe en cor- rigeant la valeur extrapolbe en tenant compte de la vitesse de passage du solute e t du solvant h travers la membrane et du coefficient de partage du solute entre la membrane et le solvant. Ces diffbrentes corrections sont calculees au depart de valeurs expbri- mentales. E n utilisant des membranes de cellophane, les pressions osmotiques des sulfonates de lignine de poids mol6culaires aussi bas que 4,300 ont 6tB Bvalues.

Zusammenfassung Eine Methode zur Bestimmung des osmotischen Druckes einer gelosten Substanz,

welche durch die Membrane diffundieren kann, wird beschrieben. Es wird theoretisch gezeigt, dass nach einem bestimmten Zeitinterval cine Darstellung des Logarithmus des angezeigten osmotischen Druckes gegen die Zeit eine gerade Linie ergibt, die auf Zeit Null extrapoliert werden kann. Der wahre Wert des anfanglichen osmotischen Druckes wird durch Korrektion des extrapolierten Wertes fur die tfbergangsgeschwindigkeit der gelosten Substanz und des Losungsmittels durch die Membrane und fur den Verteilungs- koeffizienten der gelosten Substanz zwischen der Membrane und dem Losungsmittel berechnet. Die verschiedenen Korrektionsfaktoren werden aus experimentell bestimmten Mengen berechnet. Durch Verwendung von Zellophanmembranen werden os- motische Drucke von Ligninsulfonaten mit Molekulargewichten bis auf 4,300 herab abgeschiitz t .

Received January 15, 1957

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Osmotic pressure measurements with solute-permeable membranes