Calculus of the density and of the specific volume of a solid substance without recourse to the kilogram of Sθvres.

 

by Cesare Curti

 

 

 

                                   Used symbols

 

 

 

Avogadro number ………………………….…………….

N

dalton  ………………………………………..................

kilodalton ……………………………………................

u

Ku

gram ……………………………………......…………..

kilogram …………………………………………………

g

Kg

centimeter ……………………………………………….

nanometer ……………………………………………….

natural unit (or elementary entity) of amount of substance ..

                         

cm

nm

ee

 

 

 

Introduction

 

1)      In  A definition of the kilogram by means of Avogadro number and dalton there’s not an explicit reference to the standard of Sθvres. Therefore the kilogram, defined as 1Kg=NKu, is formally independent from the kilogram of Sθvres.

2)      Since 1g=Nu, the Avogadro number is the equivalent in dalton of the gram.

3)      The below calculus of the density is made for the aluminium, but he is feasible for whichever solid substance of well-known composition and crystalline structure.

 

 

Calculus of the density of the aluminium

 

Only with the aim of to describe the calculus and without taking account of the relative uncertainties, we calculate the density of the aluminium at 25°C knowing that:

 

1.      The crystal lattice of the aluminium is a cubic face-centred crystal lattice

2.      Every unit cell of the cubic face-centred crystal lattice contains four atoms or natural units of amount of substance

3.      The parameter of the unit cell (the edge of the face-centred cubic cell) of the aluminium is 0,40496 nm at 25 °C

4.      The atomic mass of the aluminium is 26,981538 u ee-1

5.      N=6,02214199 1023 ug-1

 

The mass M of every  face-centred cubic cell is:

 

M = 4 ee  x  26,981538  u ee-1x(6,02214199 1023 ug-1)-1= 1,7922 10-22g

 

The volume V of every  face-centred cubic cell is:

 

V = 0,404963 nm3 = (0,40496 10-7)3cm3 = 6,6410 10-23cm3

 

The density of the aluminium at 25°C is:

 

d=MV-1=2,6987 g cm-3

 

 

Calculus of the specific volume of the aluminium

 

The specific volume vs of the aluminium at 25°C is:

 

vs=d-1=0,37055 cm3g-1

 

Therefore 1g of aluminium it has a volume of 0.37055 cm3 at the temperature of 25°C or a 1Kg of aluminium it has a volume of 370,55 cm3 at 25°C.

 

 

Conclusions

 

Known the specific volume of the aluminium, an elementary calculus allows finding the edge of a cube or the radius of a sphere of aluminium having the mass of 1Kg at 25°C.  The correspondence between the mass of a solid of aluminium having a volume of 370,55cm3 at 25°C and the mass of the cylinder of Pt-Ir  of Sθvres depends from the Avogadro number N experimentally determined using, like mass unit, the kilogram standard of Sθvres.

 

The National Physical Laboratory (NPL) in Internet to the URL http://www.npl.co.uk/mass/avogadro.html and CSIRO Telecommunications & Industrial Physics to the URL http://www.tip.csiro.au/IMP/Optical/avogadro.htm describe the Avogadro Project. This Project studies the possibility to redefine the kilogram by means of the Avogadro constant and the possibility to realize spheres of silicon that can replace the cylinder of Pt-Ir of Sθvres.  

 

 

Bologna, mercoledμ 23 luglio 2003 (tradotto in inglese giovedμ 21 agosto 2003)

 

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