Chemical Thermodynamics of Tin - Volume 12 - OECD Nuclear ...

C.6 Procedures for data handling
497
The selected, rounded value is:
log
* ο
10
β (C.20) = − (4.9 ± 0.4).
C.6.2 Propagation of errors
Whenever data are converted or recalculated, or other algebraic manipulations are performed
that involve uncertainties, the propagation of these uncertainties has to be taken
into account in a correct way. A clear outline of the propagation of errors is given by
Bevington [1969BEV]. A simplified form of the general formula for error propagation
is given by Eq. (C.21), supposing that X is a function of Y 1 , Y 2 ,…,Y N .
2
∑ N
⎛∂X
⎞
⎜ Yi
⎟
i=
1 ∂Y
(C.21)
i
2
σ
X
= σ
⎝
⎠
Equation (C.21) can be used only if the variables, Y 1 , Y 2 ,…,Y N , are
independent or if their uncertainties are small, i.e., the covariances can be disregarded.
One of these two assumptions can almost always be made in chemical thermodynamics,
and Eq. (C.21) can thus almost universally be used in this review. Eqs. (C.22) through
(C.26) present explicit formulae for a number of frequently encountered algebraic
expressions, where c, c 1 , c 2 are constants.
X = c 1 Y 1 ± c 2 Y 2 : σ 2 2 2
X
= ( c1σ Y
) + ( c
1 2σ Y
) (C.22)
2
cY1
X = ± cY 1 Y 2 and X = ± :
Y
2
2
2 2
⎛σ
⎞ ⎛σ
⎞ ⎛
X
Y
σ ⎞
1 Y2
⎜ ⎟ = +
X ⎜ Y ⎟ ⎜ Y ⎟
⎝ ⎠ ⎝ 1 ⎠ ⎝ 2 ⎠
(C.23)
c
σ
2
X = cY ± X
σY
1
: = c2
X Y
(C.24)
cY 2
X = ce ± 1
: X c 2 Y
X
(C.25)
X =
1ln
( 2 )
σY
c c Y : σ
X
= c1
(C.26)
Y
Example C.7:
A few simple calculations illustrate how these formulae are used. The values have not
been rounded.
Eq. (C.22) : Δ
rGm
= 2[ − (277.4 ± 4.9)] kJ·mol −1 − [− (467.3 ± 6.2)] kJ·mol −1
= − (87.5 ± 11.6) kJ·mol −1 .
Eq. (C.23) :
(0.038 ± 0.002)
K = = (8.09 ± 0.96)
(0.0047 ± 0.0005)
CHEMICAL THERMODYNAMICS OF TIN, ISBN 978-92-64-99206-1, © OECD 2012