Osmotic Pressure and the Permeability of Membranes

OSMOTIC PRESSURE AND THE PERMEABILITY OF MEMBRANES BY W. W. LEPESCHKIN*

Introduction The author’s experiments twenty years ago showed that the variation movements of plants are caused by a change of the turgor pressure in the pulvinus cells produced by t,he variation of the permeability of protoplasm to substances dissolved in the cell sap. Turgor pressure is a hydrostatic pressure caused by osmotic pressure. Thus the problem of the influence of the permeability of protoplasm upon turgor pressure is really a problem of the influence of the permeability of a membrane upon the hydrostatic pressure in the osmometer. This problem interested not only physiologists but also chemists, since no other explanation than the permeability of the membrane could be given for the discrepancy between the osmotic pressure calculated according to the van’t Hoff-Arrhenius formula and that observed in an osmometer. If it was not this explanation, one had to admit with Kahlenberg’ that osmotic pressure and gas laws are different, and although the new porous disc method of measuring osmotic pressure seems to be able to annihilate all doubt concerning the similarity of these laws, the problem of the cause sf a too low pressure in the osmometer can scarcely be considered as One explains a too low pressure in the osmometer mostly by the supposition that due to the penetration of the solute through the membrane the solvent under the membrane is gradually transformed into a more and more concentrated solution, the osmotic pressure of which brings that of the solution in the osmometer into a partial equilibrium. This supposition lies in the basis of Tammann’s considerations of the dependence of the pressure observed in an osmometer upon the permeability of membranes to solutes.s Tammann came to the conclusion that the ratio P-Pb/zP where P is the theoretical osmotic pressure calculated according to Arrhenius and van’t Hoff and Pbis the pressure observed in the osmometer, is approximately constant and proportional to the permeability of the membrane to the solute. A quite different principle lies in the basis of v. Antropoff’s calculation of the dependence of the pressure in an osmometer upon the permeability of the membrane.* Supposing that the solution formed under the membrane is continuously replaced by the solvent, v. Antropoff came to the conclusion that the pressureinanosmometer should depend not only upon the permeability of the membrane to the solute but also upon that to the solvent. Indeed,

* University of California, Berkeley, California. 1 Kahlenberg: Trans. Wieconsin Aoad. h i . , 15, 2og (1906). * R. V. Townend: J. Am. Chem. SOC.. 50,2958 (1928); F. T. Martin and L.€ Schultz: I. J. Phys. Chem., 35, 638 (1931). a G. Tammann: 2.phyaik. Chem., 9, 97 (1892). A. von Antropoff: 2.physik. Chem., 76, 721 (1911).

2626

W.W . LEPESCHKIN

the pressure in an osmometer is actually produced by the penetration of the solvent through the membrane into the osmometer and results in consequence of an increase of the volume of the liquid in the osmometer. But this volume depends not only upon the volume of the solvent coming into the osmometer but also upon the volume of the solute going out of the osmometer through the membrane. The maximum pressure in the osmometer is evidently reached when both volumes become equal. V. Antropoff gives the following formula which expresses the dependence of the maximum pressure in the osmometer P, upon the permeability of the membrane to solvent uJand that to solute u : P, = P(I - u / u ’ ) where P is the theoretical osmotic pressure. The testing of this formula was made by v. Antropoff on the basisof Cohen and Commelin’s experimental data concerning the osmotic pressure of sugar-pyridin solutions. Comparing the calculated and experimental values of the pressure in the osmometer given in v. Antropoff’s paper we may come, however, to the conclusion that the former are always greater than the latter, and sometimes the calculated pressure is more than twice as great as the observed one. V. Antropoff promised to give later his own experimental data concerning the same problem, but these data were not published during the next twenty years. The present experiments have as a purpose the explanation of the discrepancies just mentioned and show that Tammann’s results as well as those of v. Antropoff should be considered in finding a mathematic scheme of the dependence of the hydrostatic pressure in the osmometer upon the permeability of the membrane. Osmometer and Determination of the Maximum Hydrostatic Pressure The maximum hydrostatic pressure in the osmometer was determined in the present experiments by the method similar to that used by Berkeley and H a r t l e ~ that , ~ is by finding a pressure in the osmometer which just prevents the sucking of water into the osmometer. The membrane used was parchment paper obtained from the Thomas Co., Philadelphia. Preliminary experiments showed that this membrane could be used for several months without any visible change of its chemical properties, being not affected by water or bacteria, This parchment paper had an average thickness of 0.180mm. and soaked with water was thicker by twenty percent. The membrane was attached to an osmometer (0,Fig. I ) made of glass and fastened to a brass ring r to which the membrane m was stuck with a mixture of black sealing wax and venetian turpentine. The edges of the membrane were covered with a brass ring R attached to it by means of the same mixture. Both rings were pressed together by six brass clamps c which were gradually screwed on while the rings were heated and sealing wax melted. In order to fill all small cavities inside of parchment paper with water the osmometer was immersed in boiled and cooled water for some days (water surface covered with oil). The osmometer filled with a solution was then closed with a rubber stopper through which a thermometer t (divisions o . I O C ) and a glass tube a were inserted, and placed in water of a thermostat bath (Freas’, capacity 50 liters). Water was 6

Earl of Berkeley and E. G. J. Hartley: Phil. Trans., 206A,481 (1906); a09,I77 (1909).

OSMOTIC PRESSURE AND PERMEABILITY OF MEMBRANES

2627

continuously stirred in the latter. The temperature in the osmometer was always adjusted to 2 5 O C rt o.02~C. Water used for the bath as well for the solutions in the osmometer was spring water (laboratory tap water). It contained 0.029 percent dissolved substances (CaO 0.004%~ MgO O . O O ~ % , KzO and NazO b.003%, C1 o.ogz%) SO8 0.004%~ Si02 0 . 0 0 3 ~C02 , o.008%) and had pH = 7.3. The tube a (Fig. I ) was connected (thick wall rubber tube) with the other part of the instrument in which the volume of water filtered through the membrane or sucked into the osmometer was measured. This part consisted

U

U

FIQ.I Osmometer. Explanation is given in the text. The black line on the scale is thicker at the nght than at the left.

of an exactly horizontal capillary tube b provided with a three channel glass cock d, the upper branch of which was connected with a funnel and served for filling the instrument with water or with the same solution with which the osmometer was filled. Air bubbles were carefully driven out. The lower branch of the cock was united with the osmometer by a thick wall rubber tube. The capillary tube had a short branch e provided with a rubber ball which could be squeezed by means of a clamp and served for filling the capillary tube with the liquid when it was disconnected from the funnel but connected with the osmometer. I n this case the filtration of water through the membrane produced a movement of the meniscus in the capillary tube to the right, the sucking in of water its movement to the left. Every movement of the meniscus in the capillary tube by one division of the millieterscale corresponded to 0.002015 cc of water passing through the membrane. The height of the capillary tube above the water level in the thermostat bath represented the pressure expressed in cm of the column of water or the solution used. In tables below the pressure is expressed in cm of water column. When the pressure used was greater than zoo om it was produced by mercury. For this purpose a U-shape tube was inserted between the osmometer and the capillary tube, and the latter as well as a half of the U-shape tube was filled with mercury.

2628

W. W. LEPESCHKIN

The same osmometer was used for the filtration of water and solutions to determine the permeability of the membrane to water, and for the determination of the osmosis rate of the solute through the membrane (permeability to solute). The diameter of the lower opening of the osmometer covered by the membrane, and therefore that of the part of the membrane through which the filtration or osmosis took place was 7.4 cm. The area of this part after the osmometer was filled with water or solutions and the membrane stretched was about 55.6 sq. cm. The volume of the osmometer was 289 cc. Every change of temperature in the osmometer by o . I O C produced therefore a movement of the meniscus in the capillary tube by 3.7 divisions. This was considered in all experiments. In a few experiments (indicated below) the osmometer was immersed in a cement trough of a sink (cubic form, capacity 45 liters), filled with water. In this case the temperature varied more, and the correction was necessary. Preliminary experiments showed that by the stretching of the membrane an increase of the pressure in the osometer was completed in two to three hours when this increase was greater than 100 cm of water column. A smaller change of the pressure correspondingly required less time for the settlement of the equilibrium. About two hours were necessary for the temperature equilibrium. Accordingly, the observation of the meniscus always began three hours after the immersion of the osmometer in the thermostat bath and its connection with the capillary tube. The pressure was changed several times until a pressure was found which did not produce any filtration or sucking in of water through the membrane during a certain period of time (one to four hours). This pressure was considered as equal to the maximum pressure which can be reached in the osmometer. The concentration of the solution in the latter was determined immediately after the determination of this pressure. Sodium chloride (Merck, blue label) was used in these experiments as solute. Its concentration was determined by titration with silver nitrate (potassium chromate as indicator). From the amount of chlorine found in the solution its amount contained in water was substracted.

Is the Accumulation of Solute under the Osmometer responsible for a too Low Pressure? I t is well known that the maximum hydrostatic pressure in the osmometer with parchment paper as a membrane is many times smaller than the osmotic pressure calculated according to the van't Hoff-Arrhenius formula. Is the accumulation of solute in water under the osmometer responsible for this low pressure? Even if the solute passing through the membrane should form a solution under the membrane which falls down a t the bottom of the vessel, one could think that a layer of the solution always remains under the lower surface of the membrane balancing by its osmotic pressure the most part of the osmotic pressure in the osmometer. I t is very easy to prove, however, that a continuous stirring of the liquid under the osmometer does not increase the hydrostatic pressure in the osmometer. The following experiments prove


2629

that even the removal of the very thin layer of the solution from the lower surface of the membrane does not increase the maximum hydrostatic pressure in the osmometer noticeably. Experiment 1.-The osmometer was filled with a solution of sodium chloride, 0.1425 g-mol in a liter, and immersed in a sink trough filled with water. No stirrer was used this time. The temperature in the osmometer varied from 26OC to 26.6'C. The maximum hydrostatic pressure was found to be 1 2 2 cm of water column. The final concentration of sodium chloride in the osmometer was found t o be 0.1298 g-mol. The theoretical osmotic pressure of this solution (calculated according to the formula P = iRCT) is 5795 cm of water column. Experiment %-The above experiment was repeated but modified. The osmometer was filled with the solution of sodium chloride of the same concentration, and the maximum hydrostatic pressure found was I I O cm. Then a strong current of water from a glass tube (diameter 1.5 cm) was directed on the membrane of the osmometer from below. This current gave 1 2 liters of water in one minute. The distance between the tube opening and the membrane was 4 om. When directed toward water surface the current gave a water jet of about one meter above the surface. The temperature in the osmometer varied from 2 5 . 5 to 26.1'C. I n about 50 minutes the maximum hydrostatic pressure in the osmometer increased to 116 cm. This increase might result from a decrease of the stretching of the membrane. After the water current was stopped the maximum hydrostatic pressure decreased again, and it was found to be 109 cm. The final concentration of sodium chloride was found to be 0.1250 g-mol. The corresponding theoretical osmotic pressure is 5580 cm. Experiment &-The concentration of the solution of sodium chloride in the osmometer was 0.0669 g-mol in a liter. The maximum hydrostatic pressure found was 53 cm. After a water current of 180 liters an hour was conducted into the trough a t a distance of 2 0 cm from the osmometer the maximum hydrostatic pressure in the latter increased to 54 cm. Then the water current was directed on the membrane from below. In one hour the pressure was found to be 59 cm. The final concentration of the solution in the osmometer was 0.0645 g-mol. The theoretic osmotic pressure corresponding to this concentration is 3173 cm of water column. Temperature variations in the osmometer did not surpass 0.4'C. From the experiments cited we may conclude that the removal of solute from the lower surface of the membrane does not affect noticeably the maximum hydrostatic pressure in the osmometer. Thus, the accumulation of the solute under the membrane is not responsible for the low pressure in the osmometer. It is evident that the penetration of the solute through the membrane affects this pressure in some other way. We may try now to elucidate this way.

Theoretical Aspect of the Problem It should be emphasized that the problem of the dependence of the maximum hydrostatic pressure in the osmometer upon the permeability of the

2630

W.W. LEPESCHKIN

membrane should be distinguished from that of the dependence of osmotic pressure upon this permeability. We may consider first the latter problem. This problem seems a t first incomprehensible because one defines osmotic pressure as a thermodynamic value independent of any membrane. If we define it, however, as a force necessary t o separate solvent from solute, as Nernst does6 and if we try to separate the former from the latter by means of a membrane, we will find that this force is greater in the case of a complete impermeability of the membrane to solute than in the case of its permeability. Indeed, a membrane which is permeable to solute can not separate the solvent from the solute completely. Also from the standpoint of the kinetic theory of osmotic pressure is the influence of the permeability of a membrane to solute on osmotic pressure comprehensible. Indeed, the osmotic pressure which can be considered from this standpoint as a sum of strikes on the membrane produced by molecules of solute is evidently greater if all molecules moving toward the membrane strike it than is the case when some of these molecules do not strike the membrane but penetrate through it, and this independent of whether the penetrating substance accumulates on the other side of the membrane or not. Should we define osmotic pressure as a force which causes the solvent to penetrate into the solution through the membrane or as a force preventing this penetration, we will find that this force is smaller in the case of a permeable than of an impermeable membrane because this force results from the difference in concentration of the solute on two boundaries of the membrane and this difference is smaller when the solute penetrates the membrane. Even if we continuously replace the solution accumulating under the membrane by pure solvent, the force causing the solvent to pass into the solution through the membrane will be smaller because the layer of the solution adjacent to the membrane would always have a smaller concentration than that of the whole solution. Another question is whether such an incomplete force could be called osmotic pressure. We may propose to call osmotic pressure, as it is usually done, a force necessary for a complete separation of the solvent from the solute, or resulting from a molecular bombardment of full strength, or necessary to prevent the passing of the solvent through the membrane when the latter is completely impermeable to the solute. We will call, however, the reduced force resulting from a certain permeability of the membrane to the solute suction force of the solution. We may try now to find the dependence of this force upon the permeability of the membrane to the solute. Let us suppose we have a cylindric vessel closed a t both ends in which a piston made of a semipermeable membrane separates a solvent from a solution of a substance in the same solvent. We now move the piston in the direction of the solution so that the volume of the latter is diminished by a very small value A v while the solvent passes from the solution through the piston on its &her side. We produce a work equal to P A v where P is the osmotic pressure of the solution. If the 8

Kernst, W.: “Theoretical Chemistry,” English transl. by L. Codd, 136 (1923).


263 I

piston is permeable not only to the solvent but also to the solute, this work is smaller than P A v because not the whole solute contained in the volume A v is separated from the solvent. If the solute penetrates during the movement of the piston in such an amount through the latter that it would form a solution in the volume A v of the solvent which has an osmotic pressure p, the work we produce by the movement of the piston is equal to P A v - p A v. As the rate of diffusion of the solute through the piston can be supposed to be proportional to the osmotic pressure P and to the permeability of the membrane to the solute, the osmotic pressure p is proportional to both. If p is a number proportional to the permeability of the membrane to the solute, we can conclude that the work done is equal to P A v - p P A v or P(I - p)Av. The force with which this work is done is equal to P(I - p), and this force is evidently the suction force of the solution a t the membrane of which the piston is made. The work done by the movement of the piston is evidently independent of the permeability of the latter to water, because we can move the piston with a desirable speed, and the work done remains the same. We may write therefore Po = P(I - p) where Po is the suction force of the solution, P is the osmotic pressure of the solution and p is a number proportional to the permeability of the membrane to the solute. A similar formula expressing the dependence of the suction force of the solution can be deduced from Tammann’s formula mentioned in the introduction. Indeed, if M is a number proportional to the permeability, we may write: (P - Pb)/zP = M or Pb = P(I - 2M). This formula expresses, however, the hydrostatic pressure in the osmometer only in the case when the permeability of the membrane to solvent is so great that the penetration of the solute through the membrane producing a change of the volume of the liquid in the osmometer can be disregarded. In order to get an exact idea about the maximum hydrostatic pressure in the osmometer one should consider with v. Antropoff that this pressure results from the entering of the solvent into the osmometer and that the increase of the volume of the liquid in the latter is diminished by the exosmosis of the solute. It is evident, however, that v. Antropoff’s “treibende Kraft der Osmose” is really the suction force of the solution. Thus P in his formula is equal to P(I - p) where P is the osmotic pressure of the solution, and the formula expressing the dependence of the maximum hydrostatic pressure P, in the osmometer should be written as follows: P m = P(I - Y)(I - CL/U) I n this formula p is a number proportional to the permeability of the membrane to the solute, u is proportional to that to the solvent, and P is the osmotic pressure of the solution in the osmometer. We may call p and u permeability factors for solute and solvent. It should be emphasized that this formula concerns the case when the solvent under the membrane into which the solute diffuses from the osmometer, is continuously renewed. If the membrane is both sides in touch with a solution, the formula should be written as follows: Pm= [Pa(l - p/u) - P ~ ] ( I- p) where P, is the osmotic

2632

W. W. LEPESCHXIN

pressureof the solutioninside, and P b is that of the solution outside the osmometer. We are now going to test this formula on parchment paper. But in its application to the experiments described below one can simplify it. Namely, the volume of water into which the osmosis of the solute took place was in these experiments about 170 times greater than that of the solution in the osmometer, and the concentration of the solute in the outer liquid was at the end of the experiment in most cases more than 500 times smaller than that in the osmometer. Thus, without committing an error which surpasses the error unavoidable in the determination of the maximum hydrostatic pressure in the osmometer, we may omit Pb in the formula and test the formula: P, = P (1

- P)(I - dd.

Dependence of the Maximum Hydrostatic Pressure upon Osmotic Pressure From the above formula it follows that the ratio P,/P should be constant if the permeability of parchment paper to water and sodium chloride is independent of the concentration of the lattcr. We may try whether this ratio is really constant. The osmometer filled with a solution of sodium chloride (the Concentration is given in Table I) was suspended above the water surface of the thermostat bath in a water vapor saturated atmosphere for three hours while the pressure in the osmometer was raised to a height corresponding to the maximum hydrostatic pressure found in a preliminary experiment. One could avoid in this way a too long osmosis of the solute and at the same time produce the necessary stretching of the membrane. After the indicated period of time the osmometer was immersed in water, and the determination of the maximum hydrostatic pressure could begin nt once. The results are given in Table I, where C is the concentration of sodium chloride immediately after the determination of the maximum pressure expressed in g-mol in a liter, P is osmotic pressure corresponding to this concentration and P, is the maximum hydrostatic pressure in the osmometer, both expressed in cm. of water column.

TABLEI Maximum Hydrostatic Pressure P, as compared to Osmotic Pressure. Sodium Chloride. Parchment Paper C P, P PmIP

0.475

0.290

0.

I77

0,135

0.088

502

20910 0.024

367 13050

233 6240

I44 4110

0.028

278 8120 0.034

0.036

0.035

C P, P P,/P

0.067

0.050

0.029

0.026

0.016

I22

97

59

3150 0.035

2370 0 : 040

62 I390

I250

39 770

0,045

0.047

0.050


2633

From the results cited above we may conclude that the ratio P,/P is approximately constant at the concentrations ranging from about 0.06 to 0.3 g-mol, but increases with a further decrease of the concentration. This may show that the permeability of parchment paper is constant at middle concentrations, but changes at the extremes. I n the experiments just described the membrane of parchment paper used was an old one, that is, it was used for filtration of water and salt solutions for three knonths. Such a filtration increases the permeability of parchment paper to water as well as to sodium chloride. The data concerning a fresh membrane of the same parchment paper will be revealed later.

Determination of Permeability Factors I n order to determine the permeability factors p and u the maximum hydrostatic pressure was determined for an ordinary membrane of parchment paper and for a double layer membrane of the same paper. The latter was made of two sheets of parchment paper stuck together with potato starch paste. The membranes were used for a filtration of water only for some days. After the maximum hydrostatic pressure in the osmometer had been determined, the same osmometer was used for the determination of the filtration rate of water and for the estimation of constants of the diffusion of sodium chloride through the membranes. The former, divided by the pressure used for the filtration gives the permeability of the membranes to water, while the diffusion constant gives the permeability of the latter to sodium chloride. After the maximum hydrostatic pressure was determined, using one of the above membranes, the osmometer was filled with the solution of sodium chloride of a similar concentration but it was not connected with the capillary tube: it was closed with a rubber stopper through one of whose two openings a stirrer was introduced into the osmometer, while through the second opening a graduated glass tube was inserted bent horizontally in its outer part. This tube served for measuring of the volume of water sucked into the osmometer during the experiment. After a period of time indicated in tables during which the osmometer was immersed in the water of the bath the concentration of the solution in the osmometer was determined again. The concentration changes not only because sodium chloride diffuses through the membrane but also because water is sucked into the osmometer, and the concentration found by titration should be therefore corrected. The volume of the osmometer was 289 cc. If n cc of water were sucked into the osmometer, the corrected n / d g times greater than the found one. The diffusion of concentration is I sodium chloride through the membrane proceeded directly in the water of the thermostat bath. The volume of the latter was I 7 0 times greater than that of the osmometer, the water in both was continuously stirred, and the concentration of sodium chloride in the bath a t the end of experiments was not greater than 0.0005 g-mol in a liter. Thus, this concentration was simply substracted from the final concentration of sodium chloride in the osmometer. The diffusion constant K was calculated according to Fick’s formula: dS/dt = - KQ dC/dt, which in our case takes the form: VdC/dt = - KQC, where V is the

+

2634

W. W. LEPESCHKIN

volume of the osmometer, C the changable concentration of sodium chloride in the latter, t is time period, and Q is the area of the membrane. After the integration we have: K = (log C, - logC)/Qt where C, is the initial concentration, C the final concentration of sodium chloride in the osmometer. K is expressed in tables in g-mol passing through one sq. cm. of the membrane in one hour. The maximum hydrostatic pressure in the osmometer was determined for each membrane for three concentrations of sodium chloride, while the diffusion constant was found for two middle concentrations. The results are given in Tables I1 and 111. Concerning the permeability of both membranes to water, it was determined in the following manner. The osmometer was washed, filled with water and connected with the capillary tube as it was done in the case of the determination of the maximum hydrostatic pressure, but the capillary tube was filled with water. Then the osmometer was immersed in the water of the thermostat bath and the pressure in it was raised to I O O cm of water column. After three days of the filtration of water through the membrane under this pressure the rate of filtration was determined. It was found that this rate expressed in cc of water passing through one sq. cm. of the membrane during one hour is for the ordinary membrane equal to 2 I S X IO-^ and for the double layer membrane to 11j X IO-^, This result is in accord with the supposition that the filtration rate is inversely proportional to the membrane thickness (Manegold and Hofmann.' The permeability of the membranes to water is d X IO-'. accordinglyz~gX ~ o - ~ a n115

TABLE I1 Maximum hydrostatic pressure P, (cm of water column) as compared to osmotic pressure P. I for the ordinary membrane, I1 for the double layer membrane. C is the final concentration of sodium chloride in the osmometer (g-mol in a liter). I. Ordinary membrane 0.124 0.035 0 071 I11 39 60 5750 3340 1670 0.018 o 019 o 023

C P, P P,/P

11. Double laler membrane

o

122

18 5650 0.039

0.072

0.033

I20

81

3400 0 035

1580

2

0.0jI

TABLE I11 Constant K of the diffusion of sodium chloride through the ordinary and double layer membrane. C, is the initial concentration, C is the corrected final concentration of sodium chloride. T is the time period during which the diffusion took place, in hours. CO

C T K 7

I. Ordinary membrane 0.1355 o 0629 0.1271 0.0589 2 2.03 0.0724 0.0720

11. Double layer membrane 0.1360 0.0705 0.1276 0.0636 4 6 73 0.0360 0,0345

E. Manegold and R. Hofmann: Kolloid-Z., 49, 372 (1929); 50,

22

(1930).

OSMOTIC PRESSURE AND PERMEABILITY O F MEMBRANES

2635

From these results we may conclude that the decrease of the permeability of the membrane used to water and sodium chloride approximately two times brings about the increase of the hydrostatic pressure in the osmometer two (I--;)(I-G) P 2P times. We may write, therefore:

2=

(1

- Pa)

or p = 0.66. Coa-

sidering now that the ratio Pm/Pin the case of the ordinary membrane is equal at an average to 0.02 we may conclude that: 0 . 0 2 = 0.33 (I - o.67/u), o r u = 0.71. Test of the Formula Comparing the data given in Tables I and I1 we may conclude that the ratio P,/P in the case of the membrane used for the data of Table I is greater than that of the membrane used for the data of Table 11. As it was mentioned before, the first membrane was an old one through which water and salt solutions were filtered for three months while the second membrane was fresh and used for filtration only for some days. The difference in the ratio P,/P is evidently explained by the fact mentioned before that prolonged filtration increases the permeability of parchment paper to water and sodium chloride. It was interesting to find both permeabilities for the membrane of Table I and to calculate according to the so obtained data the ratio Pm/P which should be similar to the same ratio found by the experiments if the formula expressing the dependence of the maximum hydrostatic pressure upon the permeability is correct. The data concerning the diffusion constant of sodium chloride are given in Table IV. TABLE IV Constant K of the diffusion of sodium chloride through the membrane of parchment paper through which water had been filtered during three months. C, the initial concentration, C the final concentration (g-mol). T is time period during which the diffusion took place. CO C T K

0.1525 0. I431 1.5 0.0930

0.0970 0.0895 2

0.0910

0.0490 0.0443 2.48 0.0908

0.0258 0.0222

3.17 0.0801

We see therefore that the diffusion constant for the membrane of Table I is at an average 0.0887, while for the membrane of Table I1 this constant is 0.0722. I n other words, a prolonged filtration increased the permeability of the membrane for sodium chloride 1.23 times. The permeability to water was found to be also increased. For the membrane of Table I it was found to be 310 X IO-' while this permeability for the membrane of Table 11is, as it was mentioned before, 215 X IO-', that is, 1.44times smaller. The ratio Pm/Pfor

2636

W. W. LEPESCHKIN

the membrane of Table I may be calculated therefore from the following equation:

P,/P = ( I - 1 . 2 3 ~ ) (I--

I*" ') ,where p = 0.67 and u = 0.71 or P,/P = 0.034. 1.44 The found ratio is very near the ratio found by the experiments for the membrane of Table I. Namely, it is a t an average 0.037 (see Table I). We may conclude therefore that the formula expressing the dependence of maximum hydrostatic pressure upon the permeability of a membrane found theoretically is confirmed by these experiments. In order to test the formula for different concentrations of sodium chloride a t which the ratio P,/P was found to be different the membrane of Table I was used not only for the determination of the diffusion constant but also of the permeability to water a t different concentrations of sodium chloride, Brukner8 observed an increase of the filtration rate by addition of sodium chloride to water while the increase produced by weaker concentrations was found to be greater than that produced by strong concentrations. In the present experiments the osmometer was placed in a metal box covered inside with wet filter paper to prevent evaporation from the surface of the membrane, and after it was filled with a solution of sodium chloride the filtration rate was determined. It was found that, indeed, the filtration rate was greater than that in the case when water was filtered through the membrane, but between 0.04 and 0.15 g-mol it does not vary distinctly. We may assume therefore that it is constant at this concentration range. At concentrations which were lower than 0.04 g-mol the permeability decreased, however by about ten percent. In Table V are given the results. In this table C is the concentration of sodium chloride (g-mol), P is osmotic pressure in cm. of water column, K is diffusion constant, P, is the maximum hydrostatic pressure experimentally found, P,' the same pressure but calculated according to the formula P, = P (I - p ) ( ~- p/u) where p is supposed to be proportional t o K, and u is proportional to the permeability of the membrane to water at different concentrations. As it was found above at concentrations ranging from 0.07 to 0.08 g-mol p i s equal to 0.82 and u is equal to 1.02.

TABLE V Maximum Hydrostatic Pressure experimentally found and calculated C P K CL P, Pm' (exp. f.)

0 .'59

0.106 0.071

73 10 4930 3320

0,093 0.092 0.091 0.091

0.84 0.83 0.82

254 I79 119 95

(calc.)

206 I55 117

0.053 2520 0.82 89 0.082 0.74 62 52 0.029 7390 Considering the fact that p and in the above formula were calculated approximately we may conclude from the results cited that this formula is confirmed by the experiments in a satisfactory degree. 8

Brukner: 2. Ver. deutsch. Zuckerind., 72, 3

(1926).


263’1

Outlook on the Further Application of the Formula In the experiments presented in the preceding chapters one and the same sort of parchment paper was used for the membrane. One can not doubt that it will be found correct also for other membranes. It should be emphasized, however, that p and u in the formula expressing the dependence of the hydrostatic pressure in the osmometer upon the permeability of the membrane should be determined for every membrane separately because the coefficients of proportionality between them and the permeability can be expected to be different. Some experiments made on two different sorts of parchment paper showed, for instance, that the diffusion coefficient of sodium chloride can be three to four times greater than in the case of parchment used in the experiments described above. It is evident that p for these sorts of parchment paper can not be three or four times greater than it was found in these experiments. Otherwise the osmotic force of the membranes would be a negative value which is impossiple. On the other hand, the ratio p / a was near one for the membrane of parchment paper used in these experiments, because the exosmosis of sodium chloride from the osmometer into water expressed in grams through I sq. cm. was about one to one and a half times less than the endosmosis of water into the osmometer. If this ratio decreases, the hydrostatic pressure in the osmometer increases, and if the permeability of a membrane to water is very great in comparison with that to solute, the formula is practically transformed to: P, = P(I - p ) . Such a case is evidently observed on the precipitation membrane of copper ferricyanide. Indeed, Tammann’s experiments showed that the values p calculated from Pfeffer’s experiments in supposition that P, = P(I - p ) is proportional to the constant of the diffusion through this membrane.9 The same case is also observed on living protoplasm which moreover possess a very small permeability factor p so that the osmotic pressure which develops inside of the cells is near the theoretic value of osmotic pressure. summary

A too low hydrostatic pressure observed in an osmometer in which parchment paper is used as a membrane is not due to an accumulation of the I.

solute (sodium chloride) under the membrane. The pressure which is about thirty times smaller than the theoretic osmotic pressure corresponding to the concentration of sodium chloride in the osmometer is increased very little when a water current giving I Z liters in one minute is directed on the membrane from below. 2. The force with which a membrane sucks water into an osmometer depends upon the permeability of the membrane to solute (sodium chloride). This dependence can be expressed by the formula Po = P(I - p ) where Pois the suction force of the solution, P is its osmotic pressure and p is the permeability factor proportional to the permeability of the membrane to solute. 0

Here p is substituted for y/nv of his formula.

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W. W. LEPESCHKIN

3. The maximum hydrostatic pressure in the osmometer depends in general not only upon the permeability of a membrane to solute but also to its permeability to water. This dependence can be expressed by the formula: P, = P( I - p ) ( I - p / a ) where P, is the maximum hydrostatic pressure, and (r is the permeability factor proportional to the permeability of the membrane to water. I n the case of a copper-ferricyanide membrane the formula of paragraph 2 is practically valid for the maximum hydrostatic pressure in the osmometer because the permeability to water is much greater in this case than the permeability to the solute (Tammann). 4. The factor of permeability as calculated from the experiments described in this paper for one kind of parchment paper and sodium chloride are equal: p = 0.67 and (r = 0.71. If the same parchment paper was used for filtration of water and salt solutions during three months, these factors became p = 0.83 and u = 1.02.

Osmotic Pressure and the Permeability of Membranes

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