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 theres 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 wellknown 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 facecentred crystal lattice
2.
Every
unit cell of the cubic facecentred crystal lattice contains four atoms or natural
units of amount of substance
3.
The
parameter of the unit cell (the edge of the facecentred 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
10^{23 }ug^{1}
The mass M of every facecentred cubic cell is:
M = 4 ee x 26,981538
u ee^{1}x(6,02214199 10^{23 }ug^{1})^{1}=
1,7922 10^{22}g
The volume V of every facecentred cubic cell is:
V = 0,40496^{3} nm^{3} = (0,40496_{ }10^{7})^{3}cm^{3 }= 6,6410 10^{23}cm^{3}
^{ }
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 v_{s} of the
aluminium at 25°C is:
v_{s}=d^{1}=0,37055 cm^{3}g^{1}
Therefore 1g of aluminium it has a volume of
0.37055 cm^{3} at the temperature of 25°C or a 1Kg of aluminium it has
a volume of 370,55 cm^{3} 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 370,55cm^{3} at 25°C and the mass of the cylinder of PtIr 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 PtIr of Sθvres.
Bologna,
mercoledμ 23 luglio 2003 (tradotto in inglese giovedμ 21 agosto 2003)