Molar Mass

Molar mass is the mass of a substance divided by its amount of substance, conventionally denoted by $M$. It connects the macroscopic mass of a sample with the quantity of its specified chemical entities. In chemistry, these entities may be atoms, molecules, ions, or formula units. Molar mass is expressed in mass-per-mole units, most commonly grams per mole, and must not be confused with the mass of an individual particle or a dimensionless relative mass. (media.iupac.org)

Definition and units

The defining relation is

where $m$ is sample mass and $n$ is amount of substance. The coherent unit in the International System of Units is kilogram per mole, written $\mathrm{kg\,mol^{-1}}$. Laboratory calculations generally use $\mathrm{g\,mol^{-1}}$, with $1\ \mathrm{g\,mol^{-1}}=10^{-3}\ \mathrm{kg\,mol^{-1}}$. Rearrangement gives $m=nM$ and $n=m/M$. (media.iupac.org)

The entity being counted must be identified. One mole of oxygen atoms and one mole of oxygen molecules are different quantities of material: the latter contains twice as many oxygen atoms. Similarly, an ionic solid can be described using formula units without implying that it consists of discrete molecules. Specifying the entity prevents ambiguity when converting between mass and amount. (bipm.org)

Relationship to particle mass

The Avogadro constant, $N_{\mathrm A}$, links the number of entities $N$ to amount of substance:

One mole contains exactly $6.02214076\times10^{23}$ specified entities. Consequently, if an entity has mass $m_{\mathrm e}$, or if $m_{\mathrm e}$ represents the mean entity mass in a sample,

This relation connects individual atoms or molecules with bulk measurements. (bipm.org)

Individual particle masses are often expressed in daltons, symbol Da. The dalton is one twelfth of the mass of a free carbon-12 atom in its ground state and at rest. By contrast, relative atomic mass and relative molecular mass are dimensionless ratios. Although their numerical values closely match corresponding molar masses expressed in grams per mole, the quantities and units are distinct. (bipm.org)

Calculation from chemical composition

For routine chemical calculations, a compound’s molar mass is obtained by summing the contributions of its constituent elements. If its chemical formula contains $\nu_i$ atoms of element $i$, then

where $A_{\mathrm r,i}$ is the relative atomic mass appropriate to the material. Values commonly displayed in the periodic table are standard atomic weights, recommended for normal materials rather than every possible isotopic composition. (bipm.org)

For example, using rounded values of 1.008 for hydrogen, 15.999 for oxygen, and 12.011 for carbon, the calculated molar mass of water, $\mathrm{H_2O}$, is approximately $18.015\ \mathrm{g\,mol^{-1}}$. The corresponding calculation for carbon dioxide, $\mathrm{CO_2}$, gives $44.009\ \mathrm{g\,mol^{-1}}$. These are practical approximate values derived from the tabulated atomic weights, not exact constants. (ciaaw.org)

Isotopic composition and precision

Different isotopes of a chemical element have different atomic masses. The molar mass of an element therefore depends on its isotopic abundances. For isotope amount fractions $x_j$,

An isotopically enriched sample can consequently have a different molar mass from ordinary material of the same element. Natural isotopic variation also means that some standard atomic weights are reported as intervals rather than single values with symmetric uncertainties. (ciaaw.org)

The SI definition of the mole changed on May 20, 2019, fixing the Avogadro constant rather than defining the mole through a mass of carbon-12. The molar mass constant, $M_{\mathrm u}=N_{\mathrm A}m_{\mathrm u}$, is therefore no longer exactly $1\ \mathrm{g\,mol^{-1}}$. The exact relationship is $M=M_{\mathrm r}M_{\mathrm u}$. The resulting distinction is negligible for ordinary chemical calculations but matters in high-precision metrology. (bipm.org)

Applications in chemical calculations

Molar mass enables mass measurements to be used in stoichiometry. Coefficients in a balanced chemical equation specify ratios of amounts of substance, not generally ratios of masses. Converting measured reactant masses into moles allows those ratios to be applied; product amounts can then be converted back into masses. This procedure depends on using the correct chemical identity and composition. (media.iupac.org)

For a solution of volume $V$, molar concentration is $c=n/V=m/(MV)$. This follows directly from the definition of molar mass. In precise measurements, the mass attributable to the intended substance must be distinguished from total sample mass: impurities and uncertain isotopic composition can limit the accuracy of the conversion. The BIPM identifies substance purity as a frequent limitation in realizing amounts of substance through weighing. (old.iupac.org)

Mixtures and polymer averages

A mixture has a mean molar mass determined by its composition. For component mole fractions $x_i$, it is $\overline M=\sum_i x_iM_i$. Polymers require additional care because a sample may contain chains of many different lengths. Their molar-mass averages depend on how those chains are weighted. (media.iupac.org)

For $N_i$ chains of molar mass $M_i$, the number-average and mass-average, also called weight-average, values are

The second assigns greater influence to heavier chains. Reporting the averaging convention is therefore essential: these values describe different features of a molar-mass distribution and need not coincide. (media.iupac.org)