Avogadro’s Number — Value, Coulometry, Electron Mass & Metrology

A compact, student-friendly guide to Avogadro’s number (definition and exact SI value), historical and modern measurement routes (coulometry and silicon-sphere / Kibble-balance approaches), how it relates to the Faraday constant and the electron mass, worked examples and FAQs.

Table of contents

• Definition & SI value
• Coulometric measurement & relation to Faraday constant
• Electron mass from molar quantities
• Silicon-sphere method & Kibble balance (metrology)
• Worked numerical examples
• FAQs

Definition & SI value

Avogadro’s number (Avogadro constant, \(N_{\!A}\)) is the number of constituent particles (atoms, molecules, ions, electrons, etc.) contained in one mole of a substance. In modern SI the Avogadro constant has an exact defined value:

\[ \boxed{N_{\!A} = 6.02214076\times 10^{23}\ \text{mol}^{-1}} \quad\text{(exact, SI definition)} \]

This exact value is part of the 2019 SI redefinition, which fixed several fundamental constants so that units are realized from invariant quantities of nature.

Coulometric measurement & relation to the Faraday constant

A classical laboratory route to determine \(N_{\!A}\) uses high-precision electrolysis (coulometry). The two useful relations are:

\[ F = N_{\!A}\,e \] where \(F\) is the Faraday constant (charge per mole of electrons) and \(e\) is the elementary charge.

So, if you can determine \(F\) by careful coulometric measurements and you know \(e\), then

\[ N_{\!A} = \frac{F}{e}. \]

In a typical silver coulometry experiment an anodic silver sample is dissolved under a precisely measured current \(I\) for time \(t\). Measuring the mass of silver removed, and knowing silver’s molar mass, allows an experimental value for \(F\) to be deduced. Early and mid-20th century high-precision experiments used this approach; modern metrology relies on additional independent methods as well.

Electron mass from molar quantities

The mass of a single particle is related to its relative atomic mass and Avogadro constant. Define:

• \(\mathrm{A_r}(X)\) = relative atomic mass (dimensionless) of species \(X\).

Then the mass of one particle is

\[ m_X = \frac{M}{N_{\!A}} = \frac{\mathrm{A_r}(X)\,M_u}{N_{\!A}}. \]

For the electron specifically:

\[ m_e = \frac{\mathrm{A_r}(e)\,M_u}{N_{\!A}}. \]

Using CODATA values for \(\mathrm{A_r}(e)\) and the exact constants above yields the standard electron rest mass (about \(9.109\times10^{-31}\ \mathrm{kg}\)).

Silicon-sphere method & Kibble balance (modern metrology)

Modern determinations of \(N_{\!A}\) have used two complementary high-precision approaches:

1. Silicon-sphere (XRCD) method — count the number of atoms in a nearly perfect silicon single-crystal sphere by measuring its mass, volume and lattice spacing (X-ray/optical interferometry). Knowing the crystal’s unit-cell volume and the number of atoms per cell lets you compute atoms per sphere and hence \(N_{\!A}\).
2. Kibble balance (formerly watt balance) — measures mass in terms of electrical quantities tied to the Planck constant \(h\). Since the 2019 SI redefinition fixed \(h\) (and also fixed \(e\)), these electrical measurements and silicon-sphere results are interrelated; modern metrology cross-checks both routes for consistency.

Early silicon-sphere campaigns used natural silicon; later improvements used spheres enriched in \(^{28}\)Si to reduce isotopic-composition uncertainties. Metrologists worked to reconcile small disagreements between methods — improvements in surface characterization (oxide layers) and lattice-parameter measurement removed remaining inconsistencies.

Worked numerical examples

Example 1 — Compute \(N_{\!A}\) from \(F\) and \(e\).
Use the relations \(F = N_{\!A} e\). With the exact CODATA SI values:

• elementary charge \(e = 1.602176634\times10^{-19}\ \mathrm{C}\) (exact),
• Faraday constant (derived) \(F \approx 96485.332123\ \mathrm{C\;mol^{-1}}\) (derived from the exact \(N_{\!A}\) and \(e\)).

Thus

\[ N_{\!A} = \frac{F}{e} \approx \frac{96485.332123\ \mathrm{C\;mol^{-1}}}{1.602176634\times10^{-19}\ \mathrm{C}} \approx 6.02214076\times10^{23}\ \mathrm{mol^{-1}}. \]

Example 2 — Compute the electron mass from \(\mathrm{A_r}(e)\).
Use \(m_e = \dfrac{\mathrm{A_r}(e)\,M_u}{N_{\!A}}\). Taking the CODATA relative atomic mass (example value shown below) \(\mathrm{A_r}(e) \approx 5.48579909070\times10^{-4}\), \[ m_e \approx \frac{5.48579909070\times10^{-4}\times 1.0\times10^{-3}\ \mathrm{kg\;mol^{-1}}}{6.02214076\times10^{23}\ \mathrm{mol^{-1}}} \approx 9.10938370\times10^{-31}\ \mathrm{kg}. \] This reproduces the standard electron rest mass to the displayed precision.

FAQs

Q: Is Avogadro’s number measured anymore, or is it defined?
A: Since the 2019 SI redefinition, the Avogadro constant has an exactly defined numerical value \(N_{\!A} = 6.02214076\times10^{23}\ \mathrm{mol^{-1}}\). Modern measurements (silicon sphere, Kibble balance, coulometry) are still critically important because they test and validate consistency between fundamental constants and our realization of units.

Q: What is the Faraday constant and how does it relate to \(N_{\!A}\)?
A: The Faraday constant \(F\) is the magnitude of charge per mole of electrons: \(F = N_{\!A} e\). Historically \(F\) (or precise electrochemical methods) were used to infer \(N_{\!A}\) by measuring transferred mass and charge in electrolysis experiments.

Q: Why did metrologists use silicon spheres to count atoms?
A: A silicon single crystal has a well-known lattice structure and a simple cubic-based unit cell, so measuring lattice spacing and sphere volume with exceptional precision lets you count the number of unit cells (and atoms) in the sphere — a direct atom-count method with very small uncertainties after careful surface and isotopic-composition corrections.