Proportionality factor in the context of Coherent derived unit

SI derived units are units of measurement derived from theseven SI base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate power of exponentiation (see: Buckingham π theorem). Some are dimensionless, as when the units cancel out in ratios of like quantities.SI coherent derived units involve only a trivial proportionality factor, not requiring conversion factors.

The SI has special names for 22 of these coherent derived units (for example, hertz, the SI unit of measurement of frequency), but the rest merely reflect their derivation: for example, the square metre (m), the SI derived unit of area; and the kilogram per cubic metre (kg/m or kg⋅m), the SI derived unit of density.

A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. It is a system in which every quantity has a unique unit, or one that does not use conversion factors.

A coherent derived unit is a derived unit that, for a given system of quantities and for a chosen set of base units, is a product of powers of base units, with the proportionality factor being one.

The Boltzmann constant (k_B or k) is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the molar gas constant, in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann.

As part of the 2019 revision of the SI, the Boltzmann constant is one of the seven "defining constants" that have been defined so as to have exact finite decimal values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380649×10 joules per kelvin, with the effect of defining the SI unit kelvin.

In chemistry, the Avogadro constant, commonly denoted N_A, is a conversion constant or ratio between an amount of substance and the number of particles that it contains. The particles in question are any designated elementary entity, such as molecules, atoms, ions, or ion pairs. It is an SI defining constant with the exact value 6.02214076×10 mol (reciprocal mole). The numerical value of this constant when expressed in terms of the mole is known as the Avogadro number, commonly denoted N₀. The Avogadro number is an exact number equal to the number of constituent particles in one mole of any substance (by definition of the mole), historically derived from the experimental determination of the number of atoms in 12 grams of carbon-12 (C) before the 2019 revision of the SI, i.e. the gram-to-dalton ratio, g/Da. Both the constant and the number are named after the Italian physicist and chemist Amedeo Avogadro.

The Avogadro constant is used as a proportionality factor to define the amount of substance n(X), in a sample of a substance X, in terms of the number of elementary entities N(X) in that sample: