Avogadro's law and Avogadro's number N_A = 6.022 × 10²³: Foundations

This page assumes nothing. Before you can read the parent note Avogadro's law and $N_A$, you must own every symbol it throws at you. We build them one at a time, each one earning its place before the next arrives.

0. What is an atom, a molecule, a particle?

Look at the figure: the same box can be described as "3 molecules" or "9 atoms" — same stuff, different count. This is why the parent note keeps warning "moles of WHAT?".

Figure — Avogadro's law and Avogadro's number N_A = 6.022 × 10²³

Picture: three water molecules floating in a box.
Why the topic needs it: Avogadro's number counts entities, and the answer changes depending on which entity you name.

1. The counting number: $N$

Picture: a bag with a number written on it.
Why we need it: the whole point of chemistry-by-weighing is to find out $N$ without counting by hand.

The trouble is that real $N$ for anything you can hold is astronomically large — like a hundred sextillion. Writing all those zeros is a nightmare, which forces us to the next tool.

2. Scientific notation and powers of ten: $1 0^{23}$

Picture (below): a number line drawn in exponents, so $1 0^{- 24}$ and $1 0^{23}$ both fit on one page.
Why the topic needs it: $N_{A} = 6.022 \times 1 0^{23}$ and $1 u = 1.66 \times 1 0^{- 24} g$ are unreadable any other way.

3. The mole: $mol$ and the number $N_{A}$

The bridge between the raw count $N$ and the packet count $n$ is one multiplication:

Picture (below): identical boxes, each labelled " $N_{A}$ particles," stacked to show $n$ boxes = $n N_{A}$ particles.
Why the topic needs it: this is literally the definition line of the parent note.

4. Mass, molar mass, and the atomic mass unit

Watch the units cancel — this is why the formula is trustworthy: $= mol \frac{g}{g/mol} \times N_{A} \frac{1}{mol} = plain count$

5. Pressure, volume, temperature: $P$ , $V$ , $T$ (for the gas half)

Picture (below): a piston box with $P$ (arrows on walls), $V$ (the boxed space), $T$ (a thermometer), and $n$ (particle count) all labelled together.
Why the topic needs it: Avogadro's law is the statement " $V / n$ is constant when $T$ and $P$ are held fixed" — you cannot even state it without these three.

6. Ratios and "solving for" a symbol

7. How the foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer aloud.

What does the symbol

n

mean, and how is it different from

N

n

= number of moles (packets);

N

= number of individual particles. They differ by the factor

N_{A}

Write

6.022 \times 1 0^{23}

as an ordinary number.

602,200,000,000,000,000,000,000 (slide the decimal 23 places right).

What does

1 0^{- 24}

mean?

One divided by

1 0^{24}

; a decimal point followed by 23 zeros and a 1 — a very tiny number.

What is a mole, in one sentence?

A fixed-size packet holding exactly

N_{A} = 6.022 \times 1 0^{23}

entities.

What is

N_{A}

numerically and in words?

6.022 \times 1 0^{23}

per mole — Avogadro's number, the count of entities in one mole.

How do you get moles

n

from mass

m

Divide by molar mass:

n = m / M

What is molar mass

M

and its unit?

The mass of one mole of a substance, in grams per mole (g/mol).

What is 1 u equal to in grams, and why?

1.66 \times 1 0^{- 24}

= 1/ N_{A}

g, because

N_{A}

was chosen so 1 mol of C-12 weighs exactly 12 g.

Why must temperature be in kelvin for gas laws?

Kelvin has a true zero (no motion); Celsius's zero is arbitrary, so ratios of temperature only make sense in K.

What does

V \propto n

mean, and under what conditions?

Volume is proportional to moles (double

n

, double

V

) — only when

T

and

P

are held fixed.

Rearrange

P V = n R T

to isolate

V / n

V / n = R T / P

"3 molecules of water" is how many atoms?

9 atoms (each H₂O has 3 atoms), showing you must always say "of WHAT".

Recall Feynman check: could you teach this?

If you can explain to a friend (a) why we need a "counting box," (b) why $1 0^{23}$ isn't the number twenty-three, and (c) why weighing something in grams secretly tells you how many atoms are inside — then you are ready for the parent note. Every later formula is just these six symbols ( $N, n, N_{A}, m, M$ , and for gases $P, V, T$ ) shuffled around.

Connections

Parent topic: Avogadro's law and $N_A$ — this page builds every symbol it uses.
The mole concept — the mole defined and used in depth.
Molar mass and atomic mass unit — where $M$ and $u$ come from.
Ideal gas law PV=nRT — the home of $P, V, T, R$ .
Boyle's law and Charles's law — the parts that combine into the gas law.