and with these values I calculated from the formula the following values of the coefficient of viscosity which, as before, I tabulate opposite those observed :

If we compare the numbers in these tables with each other we at once notice a certain contrast in the behaviour of the numbers as calculated by the two formulæ. For smaller values of the pressure which lie below 70 atmospheres, only the formula with variable A and B is satisfactory, the other with constant A and B being certainly not. At higher pressures, however, the formula with constant A and B corresponds with the observations very much better than the other, which gives values that deviate widely from the truth.

We must therefore conclude that carbonic acid at temperatures between 30° and 40° obeys the laws of perfect gases with sufficient accuracy so long as the pressure remains below a limit of about 70 atmospheres, which corresponds nearly to the critical pressure. But if the pressure exceeds this limit, carbonic acid behaves, at least approximately, like a liquid the density of which is scarcely altered by pressure.

Since this behaviour is confirmed also by observations of another kind, we may look on the result of our calculations as a sign that the theory of the viscosity of partially dissociated gases developed in § 89 is substantially founded on truth. We shall have to assume that the formula

n = B&1(A — d)- }

really represents the coefficient of viscosity of a partially dissociated gas of density 8, and A and B are to be looked upon as constants if the pressure is sufficiently high, but to be put

B =

bpt

for smaller values of the pressure p, a and b being constants. I might probably have found a general formula applicable to all values of the pressure if I had attempted to use as basis of my calculations one of the general laws which have been proposed by van der Waals, Clausius, and others to represent the connection between the pressure and the density. I have had to abandon doing this, as I wished to delay the appearance of this book no longer.

91. Viscosity of the Perfectly-dissociated Gas and of the Non-dissociated Gas

I have, on the other hand, sought for a more comprehensive proof of the theoretical formula by returning to the theoretical meaning of the magnitudes A and B, the values of which I have obtained from the observations, and investigating the conclusions of another kind to which they lead. According to § 90,

A = D(2 — ɛ) / (1 − e), B = n,D1(1 — ɛ)—3,

where & is determined by the ratio of the two limiting values of the coefficient of viscosity, viz. ŋ1 of the gas when completely dissociated into simple molecules CO, and ʼn of the gas when composed of only the double molecules C,O,, in accordance with the formula

D is further the density which the gas would possess under the circumstances of the experiment if it were entirely dissociated into simple molecules.

The three magnitudes n1, n2, D named above are not capable of direct measurement because we cannot know whether the limiting case in which the gas contains only molecules of one kind, single or double, has been reached in any experiment.

But from one of the laws of theoretical chemistry1 we can theoretically calculate the density D of a gas from its molecular weight. Since now, according to what we have already said, carbonic acid below the critical pressure, or at least below 70 atmospheres, may be considered as an actual gas, we are justified in extending the procedure for the calculation of D to the formulæ in which we put

Aap, B = bp.

We may therefore put these formulæ in the shape

wherein

a = ▲ (2 — ɛ) / (1 − e), b = n,A2(1 — ɛ) −a,

A = D/p

represents the density of the completely dissociated gas under the pressure of 1 atmosphere, and therefore the known magnitude which chemists call the theoretical or normal density of the gas. We have now to take into account the circumstance that we must retain for its specification the unit of density assumed by Warburg and Von Babo, so that we must not take for carbonic acid the usual value 1.5198 as referred to air, but its value as

calculated with the density of water for unit, for which we obtain at the temperature of freezing-point

A = 1.5198/773.3 = 0·0019653

and at the temperature 9

A = 0.0019653/(1+ a9),

where a 0.00367 is the coefficient of expansion.

If we put this value of ▲ in the formula for a, we obtain from the given values of a, first the corresponding values of ε = 23 (n1 / n2)*,

and thence those of the ratio which the two coefficients of viscosity bear to each other:

If we now employ the formula for b in the same way we find also the absolute values of the two coefficients of viscosity, firstly that of 7, for the gas when perfectly dissociated into the simple molecules CO,, and then that of 72 for the gas when containing only the double molecules C2H1, thus,

These numbers refer to the temperatures placed opposite them. I have therefore reduced them to 0° C. by assuming for CO, the temperature-factor

1 + ad,

which holds aproximately (§ 85) for ordinary carbonic acid, and, on the contrary, for C2O, the temperature-function

4

(1 + ad)1,

which holds for vapours (§ 87). I have thus obtained for the temperature 0° C. the following values:

A comparison of these values with those obtained for the viscosity of carbonic acid under ordinary pressures is obviously the next thing. As was mentioned before, in § 79, Graham's experiments on carbonic acid at 0° gave n = 0·000145; Puluj found the nearly equal value ŋ = 0.000143, while von Obermayer and Schumann agree in finding the rather smaller value ʼn = 0·000138; [and the mean of all these is

η

n = 0.000141].

Since this is only slightly greater than the mean value

n1 = 0·000130

now found for perfectly dissociated carbonic acid, the assumption that carbonic acid under ordinary circumstances consists almost entirely of simple molecules CO2, seems to be justified.

Nearly the same results are deducible also from the formula which contain constant values of A and B, if we make an assumption regarding the variation of the density with the pressure such as after our former explanations cannot be taken as entirely improbable. I assume that the density of gaseous carbonic acid obeys Boyle's law up to the critical pressure, which is 77 atmospheres according to Andrews; from this limit onward, however, I assume the density not to be variable with the pressure. In reality there will certainly be a continuous change from the one state to the other, but I think that we may take this assumption to be allowable as an approximation to the true behaviour.

I therefore put for the density of the intensely compressed gas which has become independent of the pressure

the value

D = 77 × 0·0019653/(1 + a9d)

=

0.15133/(1 + ad),

wherein 9 is the temperature and a the coefficient of expanOn putting this value in the formula

sion 0-00367.

A = D(2-8)/(1-ɛ)

1 Phil. Trans. 1876, clxvi. p. 421.