

Letters to the Editor. 
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opinions expressed by his correspondents. Neither 
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Crystal Structure of Basic Beryllium Acetate. 
Pror. G. T. Morcan recently sent me some 
well-formed crystals of basic beryllium acetate 
Be,O(C,H,O,),,. suggesting that their analysis by 
X-ray methods would, in all probability, be of con- 
siderable interest. The results show, I think, that the 
anticipation was well founded. 
The molecule is a perfect tetrahedron. The crystal 
structure is that of diamond, a molecule replacing 
each atom of carbon. The carbon atom is itself 
tetrahedral, but is very nearly a sphere. The slight 
departure from sphericity is shown by the presence 
of a very small second order in the reflection by the 
tetrahedral plane of diamond. In the acetate this 
effect is large, because the tetrahedral character is so 
much more pronounced than in the carbon atom. 
The oxygen atom must be at the centre of the 
tetrahedron. The beryllium atoms lie on the lines 
from the centre to the corners; and each (C,H;O,) 
group must be associated in a very symmetrical 
manner with one of the tetrahedron edges. 
Prof. Morgan and I hope to give, at a later date, a 
fuller description of the analysis, and to discuss the 
inferences that may be drawn from it. 
W. H. Brace. 
A Theory of the Viscosity of Liquids. 
As is well known, the viscosity of gases and its 
variation with temperature has received a satis- 
factory explanation on the basis of molecular theory. 
Little progress has, however, been made towards 
explaining the phenomena of the viscosity of con- 
densed media—that is, of liquids and solids from a 
molecular point of view. What is evidently required 
is a working hypothesis which will indicate why, 
when a substance passes from the state of vapour to 
that of liquid, its absolute viscosity is greatly increased 
but diminishes with rising temperature, while that of 
the vapour incyeases in the same circumstances. I 
propose in this note to put forward briefly the outline 
of a theory which appears to have claims to serious 
consideration, as it indicates a quantitative rela- 
tion between the viscosity of a liquid and of the 
corresponding vapour which is supported by the 
experimental data. 
The manner in which transverse stress is propagated 
through a material medium is known in the cases in 
which the substance is in the state of vapour and 
in that of a crystalline solid. In the former case, 
momentum is transferred through the diffusion of 
the molecules between parts of the medium in relative 
motion, and this is a relatively slow process. In the 
crystal, on the other hand, the stress is transmitted 
in the form of transverse elastic waves, and the latter 
process, at least for ordinary displacements, is ex- 
tremely rapid. We may conceive that in a liquid, 
momentum is transported partly by the first process 
and partly by the second, and that the effective 
viscosity depends on their relative importance. 
The ratio in which the two modes of propagation are 
operative may be determined from thermodynamical 
considerations, combined with certain simple sup- 
positions regarding the constitution of a liquid. 
NO. 2790, VOL. 111] 
NATURE 

[APRIL 21, 1923 
We shall assume that the state of aggregation of the 
molecules in a liquid is of a composite character : 
some of the molecules are quite free to move, and may 
be termed ‘‘ vapour’’ molecules; the others. are 
attached to each other somewhat as in a crystal, and 
may be termed “ crystalline ’’ molecules. In deter- 
mining the proportion of the «wo types, we shall 
consider only binary encounters between molecules. 
Let E, be the work required to separate a pair of 
molecules of the first type, and E, those of the second 
type. Then applying Boltzmann’s distribution law, 
we may, as a first approximation, take the relative 
proportion of the two types of aggregation in the 
dissociation equilibrium to be as e*:/8T to ef2/kt, where 
R is the gas-constant and T the absolute temperature. 
The next step is to determine the rate of transport of 
momentum through the medium. In the “ vapour ”’ 
part of the aggregation, the transport occurs by 
bodily movements. In the “‘ crystalline ”’ part, the 
rate of transport may be considered to be practically 
infinite. The effective rate of transport in the liquid 
is therefore greater than in the vapour at the same 
temperature and pressure in the ratio e®:/RT/e®:/RT, 
The viscosity of the liquid is therefore given by the 
formula Miquia =vapour €'=2— =1/8T, Since E,>E, it follows 
that the viscosity of the liquid will diminish with 
rising temperature. 
The next step is to determine the absolute magni- 
tudes of the energy constants E, and E,. As was first 
pointed out by Sutherland, in the cases of gases and 
vapours the attractive forces between the molecules 
tend to increase the frequency of collisions and thus 
diminish the viscosity. The matter has been further 
examined by Chapman, who has shown that Suther- 
land’s constant is one-sixth of the mutual potential 
energy of the molecules when in contact. It is con- 
venient to use an amended form of Sutherland’s 
formula and write 
Nvapour & Ty eo Fake, 
where E, is another energy-constant. From Chap- 
man’s work it would appear that E,=6E;, and we 
may also take E,=E;. Hence, finally, we have 
Niiquid = vapour eSEa/RT, 
E, may be found from the data for the viscosity of 
vapour at different temperatures, and the formula 
thus enables the viscosity of the liquid to be calcu- 
lated a priori. : 
To illustrate the matter, it will suffice to take the 
case of benzene as an example. The table shows the 
viscosity of liquid benzene at different temperatures 
as determined by Thorpe and Rodgers, and also as 
calculated from an empirical equation of the type 
n= Aesit, 
Viscosity OF BENZENE LigQuID. 


A =0-0000951. B=1237; 
Temperature. eee Mews _ Difference. 
767° 0-00781 0:00789 +8 
13°46 0-00714 0-00717 +3 
19°39 0-00654 0-00654 ° 
25°96 0°00595 0°00595 7 
32°07 0:00549 0°00547 a 
38°47 000504 000502 -2 
45°35 0-00464 0-00461 =3 
51°66 0-00429 0-00429 te) 
57°37 000403 000402 -1 
63°29 0:00377 0:00377 o 
69°41 0:00353 0:00354 +1 
73°36 0°00332 0-00333 +1 



Viscosity of benzene vapour at 100° C, =0°0000930. 
5E, calculated from the value at 212°5°C. is 1300. 

