Molar Mass

Add up the molar masses of all atoms according to the formula. Read the values from the periodic table: $M(H) = 1.008$, $M(C) = 12.01$, $M(O) =\frac{ 16.00 g}{mol}$. For carbon dioxide CO₂ this gives $M = 12.01 + 2\cdot{}16.00 =\frac{ 44.01 g}{mol}$, for water $2\cdot{}1.008 + 16.00 =\frac{ 18.02 g}{mol}$. With brackets, multiply the entire bracket content by the index: Ca(OH)₂ yields $40.08 + 2\cdot{}(16.00 + 1.008) =\frac{ 74.10 g}{mol}$. Water of crystallization counts too: CuSO₄·5H₂O contains five additional water units. Writing down each atom type step by step avoids the typical careless index errors.

The molecular mass describes a single particle and is given in the atomic mass unit u; the molar mass describes a whole mole of particles and carries the unit $\frac{g}{mol}$. Conveniently the numerical value is identical: one water molecule weighs 18.02 u, one mole of water weighs 18.02 g. This is no coincidence but the design principle of the mole: the Avogadro number was historically chosen so that exactly this match arises. The two quantities are linked via $M = m_{T}\cdot{}N_{A}$. For salts such as NaCl one does not speak of molecular mass, because no molecules exist, but of the mass of the formula unit; the molar mass works there just the same.

Measure the mass and amount of a sample and form the quotient $M =\frac{ m}{n}$. For gases you obtain the amount conveniently via the ideal gas law: $n =\frac{ pV}{(RT)}$. Example: 3.55 g of an unknown gas correspond to 0.050 mol, so $M =\frac{ 3.55}{0.050} =\frac{ 71.0 g}{mol}$. If the gas is known to be diatomic, look for an element with $M \approx{}\frac{ 35.5 g}{mol}$ and find chlorine: Cl₂ at $\frac{70.9 g}{mol}$. Mass spectrometry uses the same logic with high precision: it separates particles by mass-to-charge and reads off the molecular mass directly. In exams the gas variant is the standard route, often combined with density data.

Because the tabulated values are averages over the natural isotope distribution. Chlorine consists of about 76 % Cl-35 and 24 % Cl-37; weighted, this gives the mean atomic mass of 35.45 u. Nuclear binding energy also shifts the masses slightly relative to the pure nucleon count. For stoichiometry this is the correct number, since every real sample contains the natural isotope mixture. Only when working with enriched isotopes, for example in nuclear chemistry or with labelled compounds, must you use the mass of the specific isotope. For exams: read the values from the periodic table and round sensibly, usually to two decimals.

For ideal gases, $pV = nRT$ with $n =\frac{ m}{M}$ yields the relation $\rho{} =\frac{ m}{V} =\frac{ p\cdot{}M}{(R\cdot{}T)}$. The density of a gas is therefore proportional to its molar mass: at the same pressure and temperature, carbon dioxide $(\frac{44 g}{mol})$ is considerably denser than air (about $\frac{29 g}{mol}$ on average), which is why CO₂ collects near the ground while helium $(\frac{4 g}{mol})$ rises. Conversely you can determine the molar mass from a measured gas density: $M =\frac{ \rho{}\cdot{}R\cdot{}T}{p}$. Example: a gas with $\rho{} =\frac{ 1.25 g}{L}$ at 273 K and 101,325 Pa gives $M =\frac{ 1.25\cdot{}8.314\cdot{}273}{101.325} \approx{}\frac{ 28 g}{mol}$, which matches nitrogen N₂.