that we must not see in this any ground for doubt; for Graham's measures were made at quite a different temperature from those of the observers at Tübingen. On the contrary I should sooner think that Graham's observations are of small value because they were made with a short tube, the section of which was neither circular nor regular. If, however, Lothar Meyer's observations also were to be affected by a constant error, that would be without effect on the mutual agreement of the numbers; and the conclusion remains that the entrance of chlorine or bromine or iodine into a chemical combination substantially increases the viscosity of the vapour.

Within each group, however, the value of the viscosity. is the same for all substances. In this lies a confirmation of the proposition stated in § 79, that the viscosity-coefficients of gases whose molecules are made up of a considerable number of atoms are of nearly the same magnitude. They are certainly not so different from each other as the coefficients of viscosity of bodies in which fewer atoms are bound together to form the molecule; this is seen also from a comparison of the numbers just given, both with each other and with other numbers tabulated earlier.

If, as this shows, vapours obey the laws of viscosity in many respects like gases, there still remain essential differences to take into account, and these we have now to consider more in detail.

87. Dependence on the Temperature

The first of these differences concerns the mode in which the viscosity of vapours depends on the temperature. Vapours exhibit a much more marked variation with temperature than gases. Hence Sutherland's formula, which is in excellent agreement with the behaviour of gases, is only imperfectly satisfied by many vapours.

We can at once see that the validity of the formula may be limited; for it is not possible by a determination of the value of C to represent every possible ratio in which the viscosity may increase with the temperature.

'Capillary tube K, Phil. Trans. 1849, pp. 353, 357.

This constant C serves as a measure of the cohesion of the molecules of the vapour in comparison with the energy of their motion. With vapours we must expect that C will assume a larger value, which is to be taken the larger the more easily the vapour can be condensed to the liquid state. Considering then that C increases, the factor

1/(1 + C/O) = 0/(C + 0),

which comes into the formula, takes approximately the simpler form

/C,

and is thus simply proportional to the absolute temperature when C is so large that the value of is small compared with it. The length of the free path in such vapours will thus increase in nearly the same ratio as the absolute temperature, or will be proportional to

1 + ad

This limiting case cannot, however, be exceeded, so that on this theory, as has been already indicated in § 71, the free path can only increase as rapidly as 1 + ad at the most when the temperature rises. The coefficient of viscosity, which, in accordance with the formula

ʼn = 0·30967 pNL,

contains, in addition to L, the second factor

that is variable

with the temperature, cannot therefore, from Sutherland's formula, increase with the temperature more rapidly than in proportion to

(1 + a9)3.

But Synesius Koch' has shown by experiments which embrace a range of more than 100 degrees of temperature that the viscosity of mercury vapour increases with the temperature proportionally to

(1 + a9)1,

in which the coefficient of expansion a is taken equal to

1 Wied. Ann. 1883, xix. p. 857.

0.003665, as for gases. Mercury vapour therefore alters its viscosity in a rather larger ratio than can be explained by Sutherland's theory.1

We meet with similar difficulties in regard to the observations made by O. Schumann' on the viscosity of the vapour of benzol and of different esters; for the function of the temperature that represents the behaviour of benzol is

(1+0.001859)2(1+0.004 9),

in which each term increases more rapidly with the temperature than the theory can explain. For many of the esters, certainly, the law of alteration with temperature that was found lies between the theoretical limits; but the function

(1 + 0·00164 9)2(1+0.004 9),

which Schumann has deduced as the most probable mean of all his observations on esters, increases nearly as rapidly as the extreme limiting case admissible under Sutherland's theory.

Let us now examine whether Schumann's observations satisfy this limiting value, and for this let us express the coefficient of viscosity by

n = n。 (1 + ad) 3,

where n is its value at 0° C., and a is taken for all vapours no equal to 0.00367; or more simply let us put

η = ΗΘ'.

where H is a constant and the absolute temperature; we then find a tolerably good agreement between the theoretical formula and the results of experiment. The following tables contain the mean values of the magnitudes measured by Schumann and the values of H calculated from them.

Compare § 92.

2 O. Schumann, Tübinger Habilitationsschrift, 1884; Wied. Ann. 1884,

xxiii. p. 353.

Q

From the mean values of these numbers found for H I have calculated the value of the coefficient of viscosity 7 at 0°, and thence the values of n at the temperatures of the experiments in order to see if the formula

n = n。(1 + a9)*,

with the value of a, viz. 0-00367, which is satisfied by gases, reproduces the observed values of the viscosity within the limits of possible errors of observation. I have thus obtained the following values of 10 for the vapour of benzol, which I have tabulated along with those observed by Schumann and those also which are calculated by his formula.

The new formula therefore agrees with the observations not essentially worse than Schumann's, which contains a disposable constant. For the esters investigated by Schumann I have likewise calculated the following values, which I tabulate opposite the observed values:

The calculated numbers fit in well as a rule with the course of those observed. We may therefore assert with certainty that Sutherland's theory corresponds pretty exactly to reality even in the limiting case now considered, in which the cohesive forces of the vapour are taken extremely large. The deviations of the theoretical formula from the results of observation are always only a few hundredths of the magnitudes measured. We might therefore defend the view that the differences may be due only to errors of observation, arising partly in the measurement of