9780192806727.pdf

84 STATISTICAL PHYSICS

the ten thousand millionth of an inch' [Yl].* In 1866 Loschmidt calculated the

diameter of an air molecule and concluded that 'in the domain of atoms and mol-

ecules the appropriate measure of length is the millionth of the millimeter' [LI].

Four years later Kelvin, who regarded it 'as an established fact of science that a

gas consists of moving molecules,' found that 'the diameter of the gaseous molecule

cannot be less than 2.10~^9 of a centimeter' [Tl]. In 1873 Maxwell stated that the

diameter of a hydrogen molecule is about 6.10~^8 cm [M6]. In that same year

Johannes Diderik van der Waals reported similar results in his doctoral thesis

[W3]. By 1890 the spread in these values, and those obtained by others [B4], had

narrowed considerably. A review of the results up to the late 1880s placed the

radii of hydrogen and air molecules between 1 and 2.10~^8 cm [R2], a remarkably

sensible range. Some of the physicists just mentioned used methods that enabled

them to also determine Avogadro's number N, the number of molecules per mole.

For example, Loschmidt's calculations of 1866 imply that N « 0.5 X 1023 [LI],

and Maxwell found N « 4 X 1023 [M6]. The present best value [D3] is

* Young arrived at this estimate by a rather obscure argument relating the surface tension to the

range of the molecular forces and then equating this range with the molecular diameter. Rayleigh,

along with many others, had trouble understanding Young's reasoning [Rl].

**This relation was derived by Clausius and Maxwell. The constant is equal to I/y2 if one uses

the Maxwell velocity distribution of identical molecules. Loschmidt used Clausius's value of %, which

follows if all the gas molecules are assumed to have the same speed. References to refinements of

Loschmidt's calculations are found in [T2].

Toward the end of the nineteenth century, the spread in the various determina-
tions of N was roughly 1022 to 1024 , an admirable achievement in view of the
crudeness—stressed by all who worked on the subject—of the models and meth-
ods used.
This is not the place to deal with the sometimes obscure and often wonderful
physics contained in these papers, in which the authors strike out into unexplored
territory. However, an exception should be made for the work of Loschmidt [LI]
since it contains a characteristic element which—as we shall soon see—recurs in
the Einstein papers of 1905 on molecular radii and Avogadro's number: the use
of two simultaneous equations in which two unknowns, N, and the molecular
diameter d, are expressed in terms of physically known quantities.
The first of the equations used by Loschmidt is the relation between d, the mean
free path X, and the number n of molecules per unit volume of a hard-sphere gas:
Xnird^2 = a calculable constant.* The second relation concerns the quantity
nird^/6, the fraction of the unit volume occupied by the molecules. Assume that
in the liquid phase these particles are closely packed. Then nird/6 = p^/l. 17
Aiquid; where the p's are the densities in the respective phases and the geometric
factor 1.17 is Loschmidt's estimate for the ratio of the volume occupied by the
molecules in the liquid phase and their proper volume. Thus we have two equa-