Guide for the Use of the International System of Units (SI)

 
8.6.9 Specific volume 
 
Quantity symbol: υ.
SI unit: cubic meter per kilogram (m
3
/kg). 
Definition: volume of a substance divided by its mass: υ = V / m. 
Note: Specific volume is the reciprocal of mass density (see Sec. 8.6.7): υ = 1 / ρ. 
8.6.10 Mass fraction of B 
 
Quantity symbol: w
B
.
SI unit: one (1) (mass fraction is a quantity of dimension one). 
Definition: mass of substance B divided by the mass of the mixture: w
B
= m
B
/ m. 
8.7 Logarithmic quantities and units: level, neper, bel 
 
This section briefly introduces logarithmic quantities and units. It is based on Ref. [5: IEC 60027-3], 
which should be consulted for further details. Two of the most common logarithmic quantities are level-of-
a-field-quantity, symbol L
F
, and level-of-a-power-quantity, symbol L
P
; and two of the most common 
logarithmic units are the units in which the values of these quantities are expressed: the neper, symbol Np, 
or the bel, symbol B, and decimal multiples and submultiples of the neper and bel formed by attaching SI 
 
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Guide for the Use of the International System of Units (SI) 
prefixes to them, such as the millineper, symbol mNp (1 mNp = 0.001 Np), and the decibel, symbol dB 
(1 dB = 0.1 B). 
Level-of-a-field-quantity is defined by the relation L
F
= ln(F/F
0
), where F/F
0
is the ratio of two 
amplitudes of the same kind, F
0
being a reference amplitude. Level-of-a-power-quantity is defined by the 
relation L
P
 = (1/2) ln(P/P
0
), where P/P
0
is the ratio of two powers, P
0
being a reference power. (Note that if 
P/P
0
= (F/F
0
)
2
, then L
P
= L
F
.) Similar names, symbols, and definitions apply to levels based on other 
quantities which are linear or quadratic functions of the amplitudes, respectively. In practice, the name of 
the field quantity forms the name of L
F
 and the symbol F is replaced by the symbol of the field quantity. 
For example, if the field quantity in question is electric field strength, symbol E, the name of the quantity is 
“level-of-electric-field-strength” and it is defined by the relation L
E
 = ln(E/E
0
). 
The difference between two levels-of-a-field-quantity (called “field-level difference”) having the 
same reference amplitude F
0
is ΔL
F
 = 
2
1
F
F
L
L
−
= ln(F
1
/F
0
) − ln(F
2
/F
0
) = ln(F
1
/F
2
), and is independent of 
F
0
. This is also the case for the difference between two levels-of-a-power-quantity (called “power-level 
difference”) having the same reference power P
0
: ΔL
P
 = 
2
1
P
P
L
L
−
= ln(P
1
/P
0
) −
ln(P
2
/P
0
) = ln(P
1
/P
2
). 
It is clear from their definitions that both L
F
 and L
P
 are quantities of dimension one and thus have as 
their units the unit one, symbol 1. However, in this case, which recalls the case of plane angle and the 
radian (and solid angle and the steradian), it is convenient to give the unit one the special name “neper” or 
“bel” and to define these so-called dimensionless units as follows:
One neper (1 Np) is the level-of-a-field-quantity when F/F
0
= e, that is, when ln(F/F
0
) = 1. 
Equivalently, 1 Np is the level-of-a-power-quantity when P/P
0
= e
2
, that is, when (1/2) ln(P/P
0
) = 1. These 
definitions imply that the numerical value of L
F
 when L
F
 is expressed in the unit neper is {L
F
}
Np
= ln(F/F
0
), 
and that the numerical value of L
P
 when L
P
 is expressed in the unit neper is {L
P
}
Np
= (1/2) ln(P/P
0
); that is 
L
F
 = ln(F/F
0
) Np
L
P
 = (1/2) ln(P/P
0
) Np. 
One bel (1 B) is the level-of-a-field-quantity when F/F
0
=
10
, that is, when 2 lg(F/F
0
) = 1 (note that 
lg x = log
10
x – see Sec. 10.1.2). Equivalently, 1 B is the level- of-a-power-quantity when P/P
0
= 10, that is, 
when lg(P/P
0
) = 1. These definitions imply that the numerical value of L
F
 when L
F
 is expressed in the unit 
bel is {L
F
}
B
= 2 lg(F/F
0
) and that the numerical value of L
P
when L
P
is expressed in the unit bel is 
{L
P
}
B 
= lg(P/P
0
); that is
L
F
 = 2 lg(F/F
0
) B = 20 lg(F/F
0
) dB 
L
P
 = lg(P/P
0
) B = 10 lg(P/P
0
) dB. 
Since the value of L
F
 (or L
P
) is independent of the unit used to express that value, one may equate L
F
 
in the above expressions to obtain ln(F/F
0
) Np = 2 lg(F/F
0
) B, which implies 
1 B
2
10
ln
=
Np exactly 
≈ 1.151 293 Np 
1 dB ≈ 0.115 129 3 Np. 
When reporting values of L
F
 and L
P
, one must always give the reference level. According to Ref. [5: 
IEC 60027-3], this may be done in one of two ways: L
x
 (re x
ref
) or L
x / xref
where x is the quantity symbol for