Avogadro Constant (Particle Number)

Multiply the amount of substance by the Avogadro constant: $N = n\cdot{}N_{A}$ with $N_{A} = 6.022\times{}10^{23} mol^{-1}$. Example: 0.25 mol of water contain $N = 0.25\cdot{}6.022\times{}10^{23} \approx{} 1.51\times{}10^{23}$ molecules. If a mass is given instead of an amount, proceed in two steps: first $n =\frac{ m}{M}$, then $N = n\cdot{}N_{A}$. Thus 9.0 g of water $(n = 0.50 mol)$ contain about $3.0\times{}10^{23}$ molecules. The result is a pure count without a unit. Always check the order of magnitude: everyday quantities of a substance almost always lie between $10^{22}$ and $10^{25}$ particles; a result like $10^{12}$ would be a sure sign of a calculation error.

The value was historically chosen so that particle mass and molar mass share the same numerical value: one carbon-12 atom weighs 12 u, one mole of carbon-12 weighs practically 12 g. For that, the number of particles per mole must be exactly the reciprocal of the atomic mass unit in grams. Originally the mole was therefore defined via 12 g of carbon-12, and N_A was measured, among others by Jean Perrin via Brownian motion and later with high precision using nearly perfect silicon single-crystal spheres. Since the 2019 SI reform the tables have turned: $N_{A} = 6.02214076\times{}10^{23} mol^{-1}$ is fixed exactly and in turn defines the mole.

Divide the molar mass by the Avogadro constant: $m_{T} =\frac{ M}{N_{A}}$. A water molecule accordingly weighs $\frac{18.02 g}{mol}$ divided by $6.022\times{}10^{23} mol^{-1}$, i.e. about $2.99\times{}10^{-23} g$. A carbon atom comes to $\frac{12.01}{6.022\times{}10^{23}} \approx{} 1.99\times{}10^{-23} g$. Such numbers show why particles are not weighed individually but in mole portions: only $10^{23}$ particles add up to weighable gram quantities. It works the same way in reverse: from a particle mass in u you directly get the molar mass in $\frac{g}{mol}$, because the numerical values are equal. This conversion likes to appear hidden in exams, for instance in mass-spectrometry tasks or nuclear physics.

You decide that yourself, and that is exactly why you must always state the particle type. The formula gives the number of the particles to which the amount of substance refers. One mole of water contains $6.022\times{}10^{23}$ water molecules, but each molecule consists of three atoms, so it holds $3\cdot{}6.022\times{}10^{23} \approx{} 1.8\times{}10^{24}$ atoms. One mole of oxygen gas O₂ contains N_A molecules and $2\cdot{}N_{A}$ oxygen atoms. For salts you count formula units or ions directly: one mole of CaCl₂ delivers one mole of $Ca^{2+}$ and two moles of Cl⁻ ions. Exam questions often target precisely this difference; so read carefully whether molecules, atoms or ions are asked for.

Avogadro's hypothesis of 1811 states: equal volumes of any gases contain equal numbers of particles at the same pressure and temperature. That is remarkable because the particles themselves can differ hugely in mass. From this follows the molar volume: at standard temperature and pressure (0 °C, 1013 hPa) one mole of any ideal gas occupies about 22.4 L, at 25 °C and 1 bar about 24.8 L (from V = nRT/p = 8.314·298.15/100,000 m³). This lets you translate gas volumes directly into amounts: 11.2 L of hydrogen at STP are 0.5 mol. The law appears as V ∝ n in the ideal gas law $pV = nRT$ and explains why reaction equations for gases also describe volume ratios.