Kilogram (kg)

Definition, realization and practical mass measurement in the International System of Units

Mass is one of the most fundamental quantities in physics, engineering and chemistry. Its modern definition, and how that definition becomes a practical measurement, is less familiar. For most of its history the kilogram was tied to a physical object. It is now defined through a fundamental constant of nature.

To understand the modern unit of mass it is, as with the meter, necessary to distinguish clearly between definition, realization and practical use.

The definition of the kilogram in the SI system

Since 2019, the kilogram has been defined within the International System of Units (SI) according to the following principle:

The kilogram is the SI unit of mass. It is defined by giving the Planck constant h the exact numerical value 6.626 070 15 × 10⁻³⁴ J·s.

Since the joule can be expressed in terms of the meter, the second and the kilogram (J = kg·m²·s⁻²), this definition means that the kilogram is derived from:

the Planck constant h,
the meter, defined via the speed of light in vacuum,
the second, defined via the hyperfine transition of the cesium-133 atom.

The Planck constant thus has an exact value, and it is this fixing that defines the unit of mass, the kilogram.

Historical continuity and the 2019 SI reform

Before 2019, the kilogram was the only SI base unit still defined by a material object: the International Prototype of the Kilogram (IPK), a platinum-iridium cylinder kept at the BIPM in Paris.

Over time this artifact was found to exhibit small but measurable mass changes relative to its copies. This conflicted with the requirements of long-term stability and universal reproducibility.

With the 2019 SI reform, the definition therefore moved to one based on a constant of nature. To keep continuity, the mass of the IPK was defined to be exactly 1 kg even after the redefinition, within a very small uncertainty. No practical mass values changed at the transition, only how the unit is defined.

Fact box: Historical definitions of the kilogram

Year	Definition of the kilogram
1795	The kilogram is defined as the mass of one liter of water at its maximum density.
1889	The kilogram is defined by the International Prototype of the Kilogram (IPK), a platinum-iridium cylinder.
2019	The kilogram is defined through a fixed numerical value of the Planck constant h = 6.626 070 15 × 10⁻³⁴ J·s.

Definition, realization and use

In mass metrology, three levels must be kept apart:

The definition states what the kilogram is in principle and is entirely abstract.
The realization refers to the experimental methods by which the kilogram is made real in practice.
The use refers to mass measurement in laboratories, industry, trade and everyday life.

This distinction is essential for a correct understanding of how the kilogram works in practice.

How the kilogram is realized

The definition of the kilogram prescribes no specific experimental method. In principle, any method that leads to a mass value traceable to the Planck constant may be used. In practice, two primary realization methods have been established, both with very low measurement uncertainty.

The Kibble balance: comparing mechanical and electrical power

One method is based on the Kibble balance (formerly the watt balance). In this method, mechanical power is compared with electrical power.

In the experiment, the gravitational force on a mass is balanced against an electromagnetic force. By combining measurements of:

electrical voltage and current (linked to the Josephson and von Klitzing constants)
velocity
local gravitational acceleration

the mass can be expressed directly in terms of the Planck constant, the meter and the second.

In this way the kilogram is realized without the use of any physical mass prototype.

The XRCD method: counting atoms

The second primary method is the XRCD method (X-ray Crystal Density). Here the mass is determined by counting the number of atoms in an almost perfect crystalline object.

In practice, spheres of extremely pure silicon are used, often enriched in the isotope ²⁸Si. By:

measuring the lattice parameter of the crystal with X-ray interferometry
measuring the volume of the sphere with optical interferometry
determining the molar mass

the number of atoms in the sphere can be determined. Since the ratio between the mass of the atom and the Planck constant is known, the mass can be derived directly from the definition of the kilogram.

Primary and secondary mass standards

An object whose mass has been determined directly by a primary realization method is called a primary mass standard. These are used to calibrate secondary mass standards, which in turn are used in national laboratories, industry and trade.

The unit of mass is disseminated through an international traceability chain, in which national metrology institutes compare their realizations in organized key comparisons to ensure global equivalence.

Practical mass measurement and environmental factors

Practical mass measurement almost always uses comparative methods, for example weighing against reference weights. Although the kilogram is defined via constants of nature, practical measurements are affected by environmental factors such as:

air density (air buoyancy),
temperature,
humidity.

For precision measurement these effects must be corrected so that mass values can be related to the SI definition. This is particularly important when calibrating weights to high accuracy.

The role of the kilogram in derived units

The kilogram is a central part of many derived SI units, for example:

newton (force)
joule (energy)
pascal (pressure)
watt (power)

Table: Derived units expressed in kg

Unit	Quantity	Expression in base units
newton (N)	Force	kg · m · s⁻²
joule (J)	Energy	kg · m² · s⁻²
pascal (Pa)	Pressure	kg · m⁻¹ · s⁻²
watt (W)	Power	kg · m² · s⁻³

Summary

The kilogram has gone from being a physical artifact to being a unit defined through a fundamental constant of nature. By fixing the Planck constant, the unit of mass has become independent of material references and thus stable in the long term.

The realization of the kilogram also shows how far modern metrology has come, and how an abstract definition can be turned into practical measurements of very high precision. Together, theory, experiment and traceability are what make the kilogram a dependable basis for mass measurement in science, engineering and everyday life.