Classical-Mechanics · Unit 1 · Video 4 · Interactive Practice

Four Units, Four Constants: Defining Reality with Frozen Numbers

IKey Formulas

Relationship	Meaning
Choose constant → fix value as exact → invert to get the unit	The three-step recipe for every defining constant
$1\,\text{A} = 6.789687\times10^{8}\; e\,\Delta\nu_{Cs}$	Ampere from the elementary charge $e$
$1\,\text{K} = 2.266665\; \dfrac{h\,\Delta\nu_{Cs}}{k}$	Kelvin from the Boltzmann constant $k$
$1\,\text{mol} = 6.02214076\times10^{23}\; \dfrac{1}{N_A}$	Mole from the Avogadro constant $N_A$
$\Delta E = k\,\Delta T,\quad k = 1.380649\times10^{-23}\,\text{J/K}$	Temperature is energy per degree

IIVisualization 1 — Which Constants Define Each Unit

Some 2019 units trace back to a single constant; others secretly drag in $h$ and $\Delta\nu_{Cs}$ — which, and why?

Base unit

💡 A unit whose frozen constant carries a mechanical unit — the joule for $k$ , the watt for $K_{cd}$ — must trace mass, length, and time back to $h$ , $c$ , and $\Delta\nu_{Cs}$ .

IIIVisualization 2 — Temperature Is Energy per Degree

The Boltzmann constant $k$ turns temperature into thermal energy on a line whose slope is frozen forever.

Temperature T (K) 300

IVVisualization 3 — The Mole Counts, It Doesn't Weigh

A mole is a fixed count of $N_A$ entities — identical for every substance, no matter how much each one weighs.

Amount n (mol) 1

💡 Since 2019 the mole is defined by fixing $N_A$ alone — no kilogram, no reference artifact, just a number.

VQuiz Questions

Question 1

In the 2019 redefinition, which constant of nature is frozen to an exact value in order to define the ampere, the unit of electric current?

✅ Correct! The ampere is anchored to the elementary charge $e$ .

❌ Not quite. That constant defines a different unit. Current is about charge per second.

Show solution

Solution:

The ampere is defined by fixing the elementary charge $e = 1.602176634\times10^{-19}$ C. Since a coulomb is an ampere-second, inverting the relationship gives:

$1\,\text{A} = 6.789687\times10^{8}\; e\,\Delta\nu_{Cs}$

The Boltzmann constant defines the kelvin, the Avogadro constant defines the mole, and the luminous efficacy defines the candela.

Question 2

True or False: A mole is a unit of mass.

✅ Correct! A mole counts entities, not grams.

❌ Not quite. A mole counts how many things, independent of their mass.

Show solution

Solution:

False. A mole is not a unit of mass — it counts entities (atoms, molecules, ions, electrons). Since 2019 it has been defined purely by fixing the Avogadro constant:

$1\,\text{mol} = 6.02214076\times10^{23}\;\text{entities}$

The mole was deliberately divorced from the kilogram. It is pure counting, with no kilograms or seconds involved.

Question 3

The kelvin and the candela both end up depending on the Planck constant $h$ and the cesium frequency $\Delta\nu_{Cs}$ , but the mole and the ampere do not. What best explains this difference?

✅ Correct! Carrying the joule and watt forces a trace back to mechanical base units.

❌ Not quite. Think about which constants carry energy or power in their units.

Show solution

Solution:

The kelvin fixes the Boltzmann constant $k$ , whose units include the joule (energy), and the candela fixes the luminous efficacy $K_{cd}$ , whose units include the watt (power). Energy and power are mechanical quantities — they unpack into $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}$ and so on. Tracing mass back to $h$ , length back to $c$ , and time back to $\Delta\nu_{Cs}$ is exactly what pulls in those extra constants.

By contrast, the ampere fixes $e$ (charge, no mechanical units) and the mole fixes $N_A$ (a pure count), so they stay clean.

Question 4

True or False: The 2019 redefinition changed the physical size of one kelvin and one ampere, so old measurements no longer match new ones.

✅ Correct! The definitions changed; the magnitudes were preserved within uncertainty.

❌ Not quite. Recall: they changed who holds the ruler, not the size of the ruler.

Show solution

Solution:

False. The numerical values of the defining constants were deliberately chosen so that the new units match the old ones to within measurement uncertainty. The definitions changed — we moved from physical artifacts to frozen constants — but the magnitudes of the units did not. A kelvin is still a kelvin and an ampere is still an ampere; nothing in the lab actually shifted.

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