Let A B or A B' be the true curve showing the relation of equivalent depression (6) to ionization coefficient (a), for a given electrolyte. Such curves sometimes bend in the one way sometimes in the other. If AC have the length unity, the point A will indicate extreme dilution. With the true values of k and l, applicable at extreme dilution, and with true values of the ionization coefficients, the line k (1-a) +l a will be A E, the tangent to A B or A B' at A. Since for a = 1, δ = 1. the line A D must have the length ; and since for a = 0, 6=k, the line OE must have the length k. With observations ranging in dilution from P to Qor P' to Q' and affected by errors, the line

a

=

A A A"

C

determined will be say A' E' or A" E" instead of A E. The value we obtain for k will thus be the length of O E' or O E", and for the length A' D or A" D. Obviously if P Q or P'Q' is not far from A, I will in general be determined with a much closer approximation than k, and the error in its determination will be smaller the more dilute the solutions on which observations are made and the more accurate the observations. Unfortunately owing to the wide limit of error in measuring the depression in very dilute solutions, it is not possible to use observations corresponding to parts of the curve in the immediate neighbourhood of A. The fact also that in drawing the experimental curves, we must use electrically determined ionization coefficients may possibly introduce another source of error of appreciable magnitude; but at the dilutions at which it is possible to make trustworthy depression observations the error due to the difference between electrically determined coefficients and actual coefficients will probably be small.

While therefore the determination of even the depression of the free ions, per gramme-ion, from the value found for l, may be affected by a considerable error, it is worth while to find what it would be in the

case of the electrolytes examined. The following table contains the values of k and I for the electrolytes considered above, and the depression constants obtained from these values, for the undissociated molecules and the free ions respectively. In the case of the sulphates, as the solutions are fairly dilute, the assumption is made that they dissociate into three free ions, although the positive differences in the case of the sulphates in Table I would seem to make it probable that the dissociation is in part into two free ions.

As seen above, the values of the depression produced by molecules may be expected to be wide of the mark, while those for the free ions. may be expected to be much closer. One sees at a glance that the latter have very nearly the value which they are expected from a theoretical stand-point to be found to have when accurately determined. It is interesting to note that the values obtained for potassium chloride and sodium chloride are in close agreement with Raoult's results, obtained by an entirely different method which makes no use of ionization coefficients. This distinguished experimenter made a series of observations of depression on dilute solutions of these electrolytes, and having plotted his molecular depressions against the depressions themselves, he obtained curves exhibiting rapid curvature in the region of great dilution. By producing these curves beyond the range of observation until they cut the axis of molecular depression, he found the molecular depression for infinite dilution to be about 36 8 for potassium chloride and 37.2 for sodium chloride.

It is usually assumed, by analogy from the results of observations on non-electrolytes of different kind and constitution, that the depression

produced in dilute solutions by all molecules, whatever the substance and whether they are undissociated molecules or free ions, is the same. On the assumption that the depression constants for undissociated molecules and for free ions are the same for all electrolytes, we may use the above determinations to get still closer values of them. For some of the errors involved in the determination of the k's and the l's are quite as likely to be positive as negative. The straight lines drawn in to represent the experimental curves in their most dilute portions are as likely to be too high as too low and to have too great an inclination to the depression axis as to be too slightly inclined to it. Errors due to the impurity of the water, the ionization of the water itself, and the use of electrically determined ionization coefficients may or may not be thus variable in sign. But as the former sources of error are probably the main ones, the average values of the respective depression constants of the above table will give a closer approximation to the true common values, if there are common values, than the individual determinations. Taking the averages we find the depression constant for undissociated molecules to be 1 895 and that for free ions to be 1.850.

On the additional assumption that the depression constants for undissociated molecules and for free ions are the same, the common value would be that found for the latter, as being affected by the smaller error. That the depression constant, assumed the same for undissociated molecules and for free ions, as well as for all electrolytes, should thus be found to have the value 1.85 is of interest, because it is expected to be found to have approximately this value both on theoretical grounds and because of the results of observations on non-electrolytes. It is of interest, however, rather as indicating the value of the method used above than because of the value of the result. If thoroughly treated the above observations would give a somewhat different result; for in finding the average value of the constants we have regarded all the individual determinations as having the same weight. Obviously they are of different weights. It is hardly worth while to endeavour to estimate their relative trustworthiness, because in an investigation intended to give a close determination of the depression constant by the method used above, it would be necessary to take into account the whole body of our knowledge of the depression of the freezing-point by electrolytes.

The result is of interest, however, as showing that the method used above gives us a means of handling the bewildering multiplicity of observations which have been accumulated in this department of experimental investigation. It has been found practically impossible to get at values of the depression constants for different electrolytes, and consequently to determine what its value is on the assumption that it has a common value, by carrying observations to extreme dilution, because of the percentage error to which such observations are

subject. The above method gives promise of enabling us to reach an approximate solution of the question by means of observations on solutions at dilutions at which determinations of depression can be accurately made.