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A Note on the Determination o~ the Avogadro Number.

~ . W. H. ADDI~K

Phil ips lgesearch Laboratories N . V. Phi t ips Gtoeilampen/abrieke~ - Ei~dl~ove,~-Nethe~'land.~

Some th i r ty years ago the dimensions of the uni t cell of crystals were pu t t o the test b y comparing the densi ty of crystals found b y weighing and the density calculated f rom the equat ion:

(1) N = ] . M / V . D ,

where

2V - - t h e Avogadro n u m b e r t aken f rom Millikan's oil-drop measurements ; ] = a factor re lated to the num ber of molecules per uni t cell; M - the molecular weight; V = the volume of the uni t cell and D = the density of the crystal.

Afterwards, as some doubt arose abou t the used wflue of N, the same equat ion was applied in order to determine the value of N.

I t was then found necessary to know the densi ty of perfect crystals as accurate ly as possible. Because of the fact t ha t D . (the densi ty found b y weigh-

ing) was found to be less t han Dr (the densi ty as de termined f rom X - r a y data), the highest valne of D for a special substance was considered to be the mos t accurately one.

In order to eomp~re measurements on different substances the value of N

is calculated according to formula (1), the lowest value being considered to be the <~ true v,%Iue ~.

Deviat ions f rom tha t value are a measure for the imperfect ion of the crystals (lattice imperfect ions, vacancies, etc.).

I n X - r a y research on crystals, in the period f rom 1920 to ~930, the di- mensions of the uni t cell, found experimental ly, were pu t to test by nume-

rous workers, who compared the crystal lographic density D~ calculated with

222 N. W . I-I. ADDINK

the aid of N (the Avogadro number) with D~ the value found experimentally (by weighing). D~ was usually found to be smaller than Dr. At that t ime the value of the Avogadro number was taken from Millikan's oil-drop mea- surements.

In the decade that followed, attention was strongly focussed on crystal imperfections (mosaic blocks), and the writer of this article was prompted to undertake an investigation of such faults [1].

I t was necessary therefore to determine the value N as accurately as pos-- sible and afterwards its deviations, depending upon the physical and chemical habit of the crystals investigated, such as impurities, non-stoechiometrie com- position, macro-holes and lattice vacancies.

In the above-mentioned paper crystals of diamond, quarte, calcite and potassium chloride (obtained from a saturated solution) were qualified as ideal. A value of N = (6.0228 =~ 0.0002).10 ~a mole -1 was found.

This value is given in the chemical scale. The following molecular weights were used: diamond 12.010 ; quartz 60.06; calcite 100.090 and potassium chloride 74.553; see [2]. The following wavelengths of the CuK~, and CuK~ lines have been taken: 1.540516 and 1.544361A resp. If Cauchois' values [9] are ap- plied (1.54051 and 1.54433 1 respectively) the value of N found by examin- ing diamond, quartz, calcite and KC1 amounts to (6.0230 ± 0.0002).102a.

In the same paper selenium, various metals and single crystals of KC1 (prepared by the Kryopoulos method) [3] were qualified as imperfect crystals. and they showed deviations from the ideal (up to about 0.05%).

The largest deviations showed yellow PbO (rhombic crystals) and various. metals. With regard to the oxide of lead it may be postulated that heat treatment at a temperature of about 700 °C causes a loss of lead (not of oxygen; cf. [4]) with the result that the density of these samples is found to be less, giving an increased value of N.

The apparent imperfection of crystals of various metals may be at t r ibuted to holes between the crystals (no mono-cryst~ls had been prepared).

These difficulties have been overcome by SH~KULA and his co-workers [5], who examined large mono-crystals. The differences between perfect crystals and those which showed the greatest deviations (T1C1 and T1Br) amounted

to 0.004%. Perfect crystals of Si, A1 and CaF2 with homopol~r, metallic and hetero-

polar links respectively showed very good agreement in regard to the cal- culated value of N: 6.023 68.10 ~ mole -1. S~AKIJLA compares the imperfections (vacancies) occurring in T1C1 and T1Br with those found in AgC1 and AgBr.

If the density of small crystals or of a crystalline powder is determined it

A N O T E ON T H E D E T E R M I N A T I O N OF TI~E A V O G A D R O N U M B E R 223

is of great impor tance t ha t the liquid, which is used as pycnome te r liquid or as immers ion liquid (floating or suspension method) , does wet t he s olid sub- stance. In m a n y cases wa te r is used as such a liquid and as i t show s a high interfacial tension the wet t ing is poor. By replacing it b y well wet t ing organic

liquids, such as paraffin oil or petroleum, the wri ter has found an i nc rea se o f

the density of lead oxide of more t han 0.5% [6]. For this r ea son dens i ty measurements of large crystals are preferable .

T h e imperfect ion of crystals of KC1, p repared according to the Kyropou los me thod at a t empera tu re near its mel t ing point, followed by a re la t ive ly rapid cooling (in some hours down to about 200 °C), remains to be explained. I t might be possible to relate the deviations found, to the larger n u m b e r of vacancies, which are formed at high tempera ture . During the re la t ive ly rapid cooling the crysta l is (( frozen >) in the s ta te of eqmlibr ium it has a t t a ined ; i t is because of this (< frozen >> equil ibrium t h a t the degree of imperfect ion has

been found to be ~ 0.02~o (el. also [7]). According to WAG?~E~ and HA~TEL= N A ~ [8], who carried out conduct iv i ty measurements a t various temperatures~

the n u m b e r of vacancies in KC1 crystMs near the mel t ing point amounts to. 1.2.10 -4, which is in good agreement wi th our results ( ~ 2 - 1 0 - %

R E F E R E N C E S

[1] N. [2] G.

[3] S. [4] N. [5] A. [6] N.

[7] J. [8] C. [9] K.

W. H. ADDINK: Recueil Tray. Chim. Pays Bas, 70, 202 (1951). P. BAXTER, )/~. GUICItARD, 0. HONIGSCttMID and R. ~VHYTLAW-GRAY. Zeits. a~wrg, allgem. Chem., 251, 430 (1943). K:fROPouLos: Zeits. anosg, alIgem. Chem., 154, 308 (1940). W. H. ADDINX: Thesis Utrecht (1933), p. 27. S~IAKULA, J. I(ALNAJS and V. Sins: Phys. l~ev., 99, 1747 (1955). W. H. ADDINK: Thesis Utrech,$ (1933), p. 61; E. ConE~ and N. W. H. ADDINK: Zcits. phys. Chem., A 168, 207 (1934). L. S~OEK: Phil. Mag., 41, 1188 (1950). WAG~ER and P. HANTELMANN: Journ. Chem. Phys., 18, 72 (1950). LONSDALE: Aeta Cryst., 3, 400 (1950).