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Evaluation of Measurement Uncertainty associated with the Avogadro Constant Dr. Rdiger Kessel Agenda Importance of the Avogadro Number The international approach in 2002 The Avogadro Number The task: establishing the final

SLIDE 1

Evaluation of Measurement Uncertainty associated with the Avogadro Constant

Dr. Rüdiger Kessel

SLIDE 2

Agenda

Importance of the Avogadro Number
The international approach in 2002
The Avogadro Number
The task: establishing the final value

– Sub-problem: molar volume – Removing inconsistencies – Treating correlations – Uncertainty budget

Discussion of the final approach

SLIDE 3

Importance of the Avogadro Number

Why do we need it?
Why do we need to measure it?
How do we measure it ?

– Molar mass – Volume – Mass – Lattice parameter

Why is silicon used?

SLIDE 4

The international approach in 2002

INRIM (IMGC) Lattice parameter, Density NMIJ Density,

Lattice parameter

PTB Density NMI-A Density EU/IRMM Molar Mass

SLIDE 5

The Avogadro Number

n a V m M a V m M n N

3 Si 3 Si A

⋅ = ⋅ ⋅ =

(Si: n = 8)

SLIDE 6

Mass Spectrometer at IRMM

SLIDE 7

Establishing the final value

Results and uncertainty budget have been

provided for all measurements of:

– Density – Molar mass – Lattice parameter

Two main questions:

– What is the measurement function? – How to combine different data?

SLIDE 8

Sub-problem: molar volume

28.08526 28.08533 28.0854 28.08547 28.08554 2329.024 2329.03 2329.036 2329.042 2329.048 Density ρ in kg·m-3 Molar mass in g·mol-1

Si Si m

M V ρ =

Use regression? (Data intentionally modified)

SLIDE 9

Why not use regression?

Regression leads to a complicated model

equation.

The zero point need to be included.
With the zero point the regression

degenerates.

Check of pairwise consistency is difficult.

SLIDE 10

Use of the weighted mean value

i m i Si i Si i m

V M V

, , , ,

δ ρ + =

∑ ∑

i i m i i m i m m

V u V V u V ) ( 1 ) ( 1

, 2 , , 2 m i m i

V V − =

28 / 30 , 28 / 30 28 / 29 , 28 / 29 30 28 / 30 , 28 / 30 , 29 28 / 29 , 28 / 29 28 ,

1 K r K r M K r M K r M M

i i Si i i Si i Si i Si

⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ + =

Individual molar volume Individual molar mass Weighted mean for the overall molar volume Difference between individual and overall molar volume

SLIDE 11

Removing inconsistencies

4·10-12
2·10-12

2·10-12 4·10-12 ε1.1 ε1.2 ε1.3 ε1.4 Difference of the molar volume in mol·m-3

8·10-12
4·10-12

4·10-12 8·10-12 ε1.1 ε1.2 ε1.3 ε1.4 Difference of the molar volume in mol·m-3

Before… After…

SLIDE 12

Overall data consistency

3.8·10-11
1.9·10-11

1.9·10-11 ε01 ε02 ε03 ε04 ε05 ε06 ε07 ε08 ε09 ε10 ε11 ε12 ε13 ε14 ε15 ε16 ε17 ε18 ε19 Difference of the molar volume in mol·m-3

SLIDE 13

Treating correlations

Correlations are important because we

average over large number of molar volume values

Correlations arise because the

laboratories use common quantities for different results (e.g. calibration factor for molar mass)

Results from different laboratories are

considered to be independent

SLIDE 14

Uncertainty budget

50% of the uncertainty arises from molar mass

measurements

20% from lattice parameter measurements
30% from density measurements

Quantity Value Standard Uncertainty Sensitivity Coefficient Uncertainty Contribution Index a0 543.1020880·10-12 m 16.0·10-18 m

3.3·1033
53·1015 mol-1

21.0 % K29/28Si 1.0013060 mol/mol 37.0·10-6 mol/mol 890·1018 33·1015 mol-1 8.1 % K30/28Si 0.9963150 mol/mol 58.0·10-6 mol/mol 1.3·1021 73·1015 mol-1 39.3 % M28Si 27.976926490 g/mol 220·10-9 g/mol 20·1021 4.4·1015 mol-1 0.1 % r30 0.03360280 1.00·10-6 12·1021 12.0·1015 mol-1 1.2 % δVm 0.0 5.60·10-12 2.2·1027 12.2·1015 mol-1 1.2 % ρ 2329.035464 kg/m3 900·10-6 kg/m3

70·1018
62.6·1015 mol-1

29.1 % NA 6.0221353·1023 mol-1 100·1015 mol-1

SLIDE 15

Discussion of the final approach

clear measurement

function

every lab is contributing
weighting based on

uncertainties

check for consistency