The mole and Avogadro's number

Result

The mole and Avogadro's number

T-013Home CU-103Threads stoichiometry

Statement

One mole is defined as exactly \(N_A=6.02214076\times10^{23}\) elementary entities (the fixed, exact value of the Avogadro constant, since the 2019 SI redefinition). Because the atomic mass unit \(u\) is itself defined relative to carbon-12 in a way that makes \(N_A\) atoms of mass \(1\,u\) sum to almost exactly \(1\,\text{gram}\), a substance's molar mass \(M\) in \(\text{g/mol}\) is numerically equal to its average atomic or molecular mass in \(u\) — giving the two working formulas \(n=\tfrac{m}{M}\) and \(N=nN_A\) that convert between a lab-measurable mass \(m\), moles \(n\), and a countable number of particles \(N\).

Why it matters

Every stoichiometric calculation in this and later units — limiting reagent, gas stoichiometry, solution concentration — depends on the ability to convert between a measurable macroscopic quantity (mass, on a balance) and a microscopic count (individual atoms, molecules, or ions, which cannot be counted directly). The mole is the single conversion factor that makes this possible, and its role is entirely analogous to a unit-conversion factor like "12 items per dozen," just at an astronomically larger scale appropriate to atomic-sized particles.

Hypotheses

The atomic mass unit is defined so that a carbon-12 atom has a mass of exactly \(12\,u\).This is the calibration reference against which every other element's relative atomic mass (and hence molar mass) is measured; it is a definitional choice, not a derived result, historically chosen because carbon-12 is abundant, stable, and easy to obtain in pure form for precision mass measurements. The Avogadro constant \(N_A\) is now (since 20 May 2019) defined as an exact fixed number, rather than an experimentally measured quantity with uncertainty.Before 2019, \(N_A\) was operationally defined as "the number of atoms in exactly \(12\,\text{g}\) of carbon-12" and had to be measured (historically via X-ray crystal density methods), carrying experimental uncertainty. The 2019 SI redefinition fixed \(N_A\) at the exact value \(6.02214076\times10^{23}\,\text{mol}^{-1}\) instead, and the mole is now defined directly as this many elementary entities; \(12\,\text{g}\) of carbon-12 remains extremely close to, but is no longer exactly by definition, one mole of carbon-12 atoms — the difference is far too small to matter for any ordinary laboratory calculation.

Proof

1\,u \equiv \frac{\text{mass of one }{}^{12}\text{C atom}}{12}

The atomic mass unit reference (Hypotheses); every element's tabulated relative atomic mass (as found on a periodic table) is, by construction, the average mass of its naturally occurring isotopic mixture expressed in these units. A

N_A \equiv 6.02214076\times10^{23}\,\text{mol}^{-1}\ \ (\text{exact, defined})

The fixed Avogadro constant (Hypotheses); one mole is defined as containing exactly this many elementary entities of whatever is specified (atoms, molecules, ions, electrons, formula units). A

N_A \times (1\,u\text{ in kg}) \approx 1\times10^{-3}\,\text{kg} = 1\,\text{g}

The historical calibration of \(1\,u\) (against \(\tfrac{1}{12}\) of a carbon-12 atom's mass) and of \(N_A\) (against the atom count in \(12\,\text{g}\) of carbon-12) were chosen together precisely so that this product comes out extremely close to \(1\,\text{gram}\) — to within about one part in \(10^9\) using current best measurements of \(1\,u\) in kilograms, since the two constants are now independently fixed rather than tied by a single definition. This near-exact numerical coincidence is what makes Step 4 below work. A

M\,(\text{g/mol}) \approx \text{average relative atomic/molecular mass}\,(u)

Since \(N_A\) entities of mass \(1\,u\) each sum to almost exactly \(1\,\text{g}\) (Step 3), \(N_A\) entities of mass \(x\,u\) each sum to almost exactly \(x\,\text{g}\) — so a substance's molar mass in grams per mole is numerically (not just proportionally) equal to its average particle mass in atomic mass units, read directly from summing atomic masses on the periodic table. A

n = \frac{m}{M}, \qquad N = nN_A

Moles \(n\) are obtained from a measured mass \(m\) divided by the molar mass \(M\) (Step 4); the number of individual particles \(N\) follows by multiplying moles by the defining constant \(N_A\) (Step 2). These two formulas, chained together, are the complete route from a balance reading to a particle count, or vice versa. A

Result

N_A = 6.02214076\times10^{23}\,\text{mol}^{-1}\ (\text{exact}), \qquad n=\frac{m}{M}, \qquad N=nN_A

Reading. A single defined constant, together with molar masses read straight off the periodic table (via the near-exact gram/atomic-mass-unit correspondence), converts freely between mass, moles, and particle count in either direction.

Scope. Applies to any pure substance with a well-defined chemical formula (element, compound, or ion); "particles" means whatever discrete entity the formula specifies — atoms for an element, molecules for a molecular compound, formula units for an ionic compound.

Corollaries & converses

The mass of a single particle (one atom or molecule) is obtained from \(M/N_A\) — an extraordinarily small number in grams (of order \(10^{-22}\) to \(10^{-24}\,\text{g}\) for ordinary atoms and small molecules), directly illustrating why \(N_A\) must be as large as it is for molar quantities to be lab-scale.
The mole concept extends beyond mass entirely: at fixed temperature and pressure, equal numbers of moles of any ideal gas occupy equal volumes (Avogadro's original 1811 hypothesis, later formalised as the ideal-gas-law result), and molar quantities of charge, volume, and heat capacity are all defined analogously.
Converse: given any two of \(\{m,\ n,\ N\}\) for a known substance, the third follows immediately from Step 5 — the three quantities are never independent once the substance's identity (and hence \(M\)) is fixed.

Fails without

Drop the shared calibration between \(1\,u\) and \(N_A\) (Step 3): if the two constants had been defined independently with no numerical relationship, \(M\) in \(\text{g/mol}\) would not equal the periodic-table atomic mass in \(u\) at all, and every molar mass would require a separate, otherwise unnecessary conversion factor — the entire convenience of reading molar masses straight off the periodic table depends on this historical calibration choice.
Use an average atomic mass that ignores natural isotopic abundance (e.g. using only the most common isotope's exact mass): for elements with significant multi-isotope abundance (chlorine, at roughly 76% \(^{35}\text{Cl}\) and 24% \(^{37}\text{Cl}\), giving an average atomic mass of \(35.45\,u\), matching neither isotope individually), this would introduce a small but systematic error into every molar-mass-based calculation for that element.

Common errors

Confusing "moles of atoms" with "moles of molecules" for compounds — e.g. one mole of \(\text{CO}_2\) contains one mole of carbon atoms but two moles of oxygen atoms, a distinction that matters whenever a problem asks for a specific atom's count rather than the molecule's.
Forgetting to multiply by a formula subscript when computing molar mass (e.g. treating \(\text{H}_2\text{O}\)'s molar mass as the sum of one H, one O, and one more H mass without doubling the hydrogen contribution correctly, or more generally miscounting any repeated atom in a formula).
Applying \(n=m/M\) with \(M\) in the wrong units (e.g. \(\text{kg/mol}\) instead of \(\text{g/mol}\)) without correspondingly converting the mass \(m\), producing an answer off by a factor of 1000.

Discussion

The mole's conceptual roots trace to Amedeo Avogadro's 1811 hypothesis that equal volumes of gas at the same temperature and pressure contain equal numbers of particles — a hypothesis made without any means of counting or even estimating that number, which was only determined much later (from the late 19th century onward, via methods including Brownian motion analysis, X-ray crystallography, and electrochemistry) and named in his honour once established. The 2019 redefinition of the mole and \(N_A\), part of a broader overhaul of the SI base units (which also redefined the kilogram, ampere, and kelvin in terms of fixed fundamental constants rather than physical artefacts or reference states), completed a decades-long shift toward defining measurement units by exact numbers rather than by physical reference objects — the same underlying philosophy, at the level of the whole SI system, that motivated fixing \(N_A\) itself.

A frequently repeated but slightly imprecise statement is that "\(12\,\text{g}\) of carbon-12 contains exactly \(N_A\) atoms" — this was true by definition before 2019, but since the redefinition, \(N_A\) is fixed independently and the mass of \(12\,\text{g}\) worth of carbon-12 atoms is now an experimentally measured quantity (extremely close to, but not defined to be exactly, \(12\,\text{g}\)); the practical numerical difference is negligible for essentially all laboratory purposes, but the logical direction of the definition has reversed.

Common misconception: that Avogadro's number is some special property of carbon or of chemistry specifically, rather than an arbitrary (if historically motivated) choice of scale factor, analogous to why a dozen is 12 and not some other number — \(N_A\)'s specific value is a product of the historical choice to define the gram and the atomic mass unit as closely as possible, not a fundamental constant of nature in the way the speed of light or Planck's constant are.

Worked examples

\text{Mass of one }\text{H}_2\text{O molecule: } M(\text{H}_2\text{O})=18.015\,\text{g/mol} \Rightarrow \frac{18.015}{6.02214076\times10^{23}}\approx2.991\times10^{-23}\,\text{g}

Dividing molar mass by \(N_A\) (the reverse of Step 5's \(N=nN_A\), applied to exactly one particle) gives the mass of a single water molecule — a vivid illustration of just how many molecules are packed into an ordinary, lab-scale sample (a single drop of water, roughly \(0.05\,\text{g}\), contains on the order of \(10^{21}\) molecules). A

10.0\,\text{g NaCl}\ (M=58.44\,\text{g/mol}):\ n=\frac{10.0}{58.44}\approx0.1711\,\text{mol}, \quad N=0.1711\times6.02214076\times10^{23}\approx1.030\times10^{23}\,\text{formula units}

A routine two-step application of \(n=m/M\) then \(N=nN_A\); "formula units" (not "molecules") is the correct term for an ionic compound like \(\text{NaCl}\), which does not exist as discrete molecules in the solid state. A

m(\text{one }\text{H}_2\text{O})\approx2.99\times10^{-23}\,\text{g}; \qquad 10.0\,\text{g NaCl}\approx0.171\,\text{mol}\approx1.03\times10^{23}\,\text{formula units}

Reading. The same pair of formulas runs in both directions: from a molar mass alone (single-particle mass) or from a measured sample mass together with molar mass (moles, then particle count).

Scope. Identical arithmetic applies to any pure substance, molecular or ionic, given only its formula (to compute \(M\)) and either a sample mass or a target particle count.

Problems

A sample of carbon dioxide, \(\text{CO}_2\) (\(M=44.01\,\text{g/mol}\)), has a mass of \(22.0\,\text{g}\). Find the number of \(\text{CO}_2\) molecules and the total number of oxygen atoms present.
Solution
\(n=\dfrac{22.0}{44.01}\approx0.4999\,\text{mol}\). Molecules: \(N=0.4999\times6.02214076\times10^{23}\approx3.010\times10^{23}\). Each \(\text{CO}_2\) molecule contains 2 oxygen atoms, so total oxygen atoms \(=2\times3.010\times10^{23}\approx6.02\times10^{23}\) — coincidentally very close to \(N_A\) itself, since the sample happens to be almost exactly half a mole of \(\text{CO}_2\), giving almost exactly one mole of oxygen atoms.
Glucose, \(\text{C}_6\text{H}_{12}\text{O}_6\), has molar mass \(M=6(12.011)+12(1.008)+6(15.999)=180.156\,\text{g/mol}\). Find the number of moles and the number of glucose molecules in a \(5.00\,\text{g}\) sample.
Solution
\(n=\dfrac{5.00}{180.156}\approx0.02775\,\text{mol}\). \(N=0.02775\times6.02214076\times10^{23}\approx1.671\times10^{22}\) molecules.
A sample of iron (\(M=55.845\,\text{g/mol}\)) is found to contain \(3.011\times10^{23}\) atoms. Find the mass of the sample in grams.
Solution
First find moles: \(n=\dfrac{N}{N_A}=\dfrac{3.011\times10^{23}}{6.02214076\times10^{23}}\approx0.5000\,\text{mol}\) (the given particle count is almost exactly half of \(N_A\)). Mass: \(m=nM=0.5000\times55.845\approx27.92\,\text{g}\).
Explain, without doing any further arithmetic, why "one mole of \(\text{H}_2\text{O}\) molecules" and "one mole of \(\text{H}_2\)O formula units" would, if the terms were used, mean exactly the same physical quantity, whereas "one mole of \(\text{NaCl}\) molecules" is not a meaningful description of solid sodium chloride.
Solution
Water genuinely exists as discrete, individually identifiable \(\text{H}_2\text{O}\) molecules (in gas, liquid, and solid ice phases alike), so "molecule" and "formula unit" refer to the identical physical entity for water — the two terms are interchangeable for a molecular compound. Solid \(\text{NaCl}\), by contrast, is an extended ionic lattice with no discrete, individually bonded \(\text{Na}^+\text{Cl}^-\) pairs to call a "molecule" — each \(\text{Na}^+\) ion is simultaneously bonded (electrostatically) to six surrounding \(\text{Cl}^-\) ions and vice versa, so "formula unit" (the smallest whole-number ratio of ions, \(\text{NaCl}\)) is the correct term, while "molecule" misleadingly implies a discrete, bonded pair that does not actually exist as a separate physical unit in the solid.