OSMOTIC PRESSURE AND MOLECULAR WEIGHT
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DAVID I. HITCHCOCK Yale University School of Medicine, New Haven, Connecticut
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calculation of molecular weight from osmotic pressure is commonly illustrated, in textbooks of physical chemistry, by an application of the van't Hoff equation to a siude determination. This equation may be written in the form P = RTC if P is the osmotic pressure in atmospheres, R is 0.0821, T the absolute temperature, and C the concentration in mols of solute per liter of solution. The equation implies that the calculation may be inaccurate unless it is known that the osmotic pressure actually is a linear function of the concentration. This equation was correctly regarded by van? Hoff as a limiting law, rigorously exact only for an ideal limiting state which solutions approach as they become more and more dilute. This implies that RT should be the limit approached by the ratio P/C as both C and P approach zero. Perhaps the most practical use of the direct measurement of osmotic pressure has been to establish the molecular weights of proteins and synthetic polymers of larw uarticle size. For solutions of such a material the osmotic pressure is rarely found t o be a linear function of the concentration. In order t o obtain the molecular weight it is necessary to make an extrapolation to
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infinite dilution before applying the limiting larx-. This procedure has been employed for some 20 years by those interested in proteins or other colloidal materials, but it. is mentioned in very few textbooks. Because a simple extrapolation makes possible the correct use of a limiting law, the method might well be taught to students beginning the study of physical chemistry. The fact that RT is the correct limitine: .-s l o ~ eof the relation between osmotic pressure and concentration can be justified empirically by a consideration of data to which the equatiou does not apply directly. Berkeley and Hartley (1) measured the osmotic pressures of concentrated sucrose solutions a t O°C., expressing the concentrations in grams of sucrose per 100 g. of water. On the basis of the accepted molecular weight of sucrose, their data have been tabulated (2) in terms of molality. In the lower part of Figure 1 the osmotic pressure (P) in atmospheres is plotted against the molality (m). The ~ o i n t sfall on a curve which bends up as the molaiity is increased. At low molalities the curve approaches the broken line, which is a graph of Morse's modification of the van't Hoff eauation. P = RTm
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SEPTEMBER. 1951
In the upper part of Figure 1 the ratios, P / m , are plotted against m. These points lie on a straight line, and its intercept a t zero molality is withm 1 per cent of 22.4, which is the value of R T for 0°C. This shows that the Morse equatiou is a correct limiting law for these data, although none of the plotted points is fitted directly by that equation. Similar plots, not reproduced here, were made from earlier data of Berkeley and Hartley (5), who then expressed the concentrations in grams of sucrose per liter of solution. After these were translated into concentrations (C) in mols per liter it was found that a plot of PIC against C was rather curved. A plot of C / P against C, while slightly curved, was straight enough to be extrapolated, and its intercept a t zero coucentration lay mithin 3 per cent of the reciprocal of 22.4. This shows that, within this margin of error, the van't Hoff equation is also valid as a limiting law for sucrose solutions a t 0%. The use of a similar extrapolation for the calculation of a molecular weight may be illustrated by means of Adair's (4) measurements of the osmotic pressures of aqueous hemoglobin solutions1 a t O°C. In the lower part of Figure 2 the osmotic pressures (P') in millimeters of mercury are plotted against the couceutra'These solutions were equilibrated against a saline buffer mixture, and the membranes used were permeable to the ions of these salts as well as to water, but not to hemoglobin. Adair showed thttt a part of the observed pressure was due to the unequal distribution of ions. For the present purpose this ion pressure difference may be disregarded, since it vanishes in tho extrapolstion.
tions (C') in grams of hemoglobin per liter of solution. It is evident that a linear equatiou does not fit the data in the range shown. In the upper part of Figure 2 the ratio Cf/P' is plotted against C'. The points are represented by a gently curved line which can be extrapolated to zero concentration; the intercept is 3.9. The van't Hoff equation, for this case, takes the form P'/760 = 22.4 C'/M if 144 is the molecular weight. If we insert the limiting value of C f / P ' and solve for M, we obtain A value uot far from this figure for hemoglobin mas first published by Adair in 1924, and later confirmed by Svedberg's measurements with the ultracentrifuge. Extrapolations of the sort described here were applied t o osmotic pressure data for other proteins by Adair and Robinson (5). Plots of P'IC' against P' were used hy others (6). LITERATURE CITED AND E. G. J. HARTLEY, PTOC. Roy. SOC. (London),A 92,483 (1916). "International Critical Tables," McGraw-Hill Book Co., h e . , Ncw York, 1928, Val. IV, p. 430.
(1) BERKELEY, EARLOF, (2)
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(3) EARLOF. AND E. G. J. HARTLEY. Phil. Trans. , . BERKELEY. Roy. SK: (Inndon); A 206,503(i906). (4) ADAIR, G. S., J. Am. Chem..Soe., 51, 702 (1929). (5) ADAIR,G. S., AND M. E. ROBINSON, Rioehem. J., 34, 1869
(1930).
(6) SCATCHARD, G., A. C. BATCHELDER, AND A. BROWN, J . Clinical Investigation, 23, 458 (1944).
