Inchemistry, themolar mass (M) (sometimes calledmolecular weight orformula weight, but seerelated quantities for usage) of achemical substance (element orcompound) is defined as the ratio between themass (m) and theamount of substance (n, measured inmoles) of any sample of the substance:M =m/n.[1] The molar mass is a bulk, not molecular,property of a substance. The molar mass is aweightedaverage of many instances of the element or compound, which often vary in mass due to the presence ofisotopes. Most commonly, the molar mass is computed from thestandard atomic weights and is thus a terrestrialaverage and a function of the relativeabundance of theisotopes of the constituent atoms on Earth.
Themolecular mass (for molecular compounds) and formula mass (for non-molecular compounds, such asionic salts) are commonly used as synonyms of molar mass, as the numerical values are identical (for all practical purposes), differing only in units (dalton vs. g/mol or kg/kmol). However, the most authoritative sources define it differently. The difference is that molecular mass is the mass of one specific particle or molecule (amicroscopic quantity), while the molar mass is an average over many particles or molecules (amacroscopic quantity).
The molar mass is anintensive property of the substance, that does not depend on the size of the sample. In theInternational System of Units (SI), thecoherent unit of molar mass iskg/mol. However, for historical reasons, molar masses are almost always expressed with the unitg/mol (or equivalently in kg/kmol).
Since 1971,SI defined the "amount of substance" as a separatedimension of measurement. Until 2019, the mole was defined as the amount of substance that has as many constituent particles as there are atoms in 12 grams ofcarbon-12, with the dalton defined as+1/12 of the mass of a carbon-12 atom. Thus, during that period, the numerical value of the molar mass of a substance expressed in g/mol wasexactly equal to the numerical value of the average mass of anentity (atom,molecule,formula unit) of the substance expressed in daltons.
Since 2019, the mole has beenredefined in the SI as the amount of any substance containing exactly6.02214076×1023 entities, fixing the numerical value of theAvogadro constantNA when expressed in the unit mol−1, but because the dalton is still defined in terms of the experimentally determined mass of a carbon-12 atom, the numerical equivalence between the molar mass of a substance and the average mass of an entity of the substance is now only approximate, but equality may still be assumed with high accuracy—(the relative discrepancy is only of order 10–9, i.e. within apart per billion).
For a pure sample of a substanceX, the known molar mass,M(X), is used for calculating the amount of the substance in the sample,n(X), given the mass of the sample,m(X), through the equation:n(X) =m(X)/M(X). IfN(X) is thenumber of entities of the substance in the sample, andma(X) is the mass of each entity of the substance (atomic mass,molecular mass, orformula mass), then the mass of the sample ism(X) =N(X) ⋅ma(X), and the amount of substance isn(X) = N(X)/NA =N(X)⋅na, wherena is the elementary amount, an amount consisting of exactly one atomic-scale entity of any kind (atom, molecule, formula unit), analogous to the elementary chargee. Since the elementary amount is the reciprocal of the Avogadro constant, using the relationshipM(X) =m(X)/n(X), the molar mass is then given byM(X) =ma(X) ⋅NA =ma(X)/na (dimensionM/N), i.e. the atomic-scale mass of one entity of the substance per elementary amount.
Given the relative atomic-scale mass (atomic weight,molecular weight, orformula weight)Ar(X) of an entity of a substanceX, its mass expressed in daltons isma(X) =Ar(X) Da, where the atomic-scale unit of mass is defined as 1 Da =mu =ma(12C)/12 (dimensionM). The corresponding atomic-scale unit of amount of substance is the entity (symbol ent), defined as 1 ent =na (dimensionN). So, withAr(X) known, the molar mass can be expressed in daltons per entity asM(X) =Ar(X) Da/ent. Thus, the molar mass of a substanceX can be calculated asM(X) =Ar(X) ⋅Mu, with themolar mass constantMu equal to exactly 1 Da/ent, which (for all practical purposes) is equal to 1 g/mol, as themole was historically defined such that the Avogadro number (the number of atomic-scale entities comprising one mole) was exactly equal to the number ofdaltons in agram (g/Da). This means that (for all practical purposes): 1 mol = (g/Da) ent.
The relationship between the molar mass ofcarbon-12,M(12C) = 12 g/mol, and its atomic mass,ma(12C) = 12 Da, can be expressed asM(12C) =ma(12C) ·NA. Rearranging and substituting the given values into the equation yields the following expression for theAvogadro constant:NA = (g/Da) mol−1, making the Avogadro number equal to the number of daltons in a gram, and equivalently the number of atoms in 12 grams of carbon-12 (as in the 1971 definition of the mole).
The mole was defined in such a way that the numerical value of the molar mass of a substance in g/mol, i.e.M(X)/(g/mol), was equal to the numerical value of the average mass of oneentity (atom,molecule,formula unit) in Da, i.e.ma(X)/Da =Ar(X), so thatM(X) =Ar(X) g/mol. The equivalence was exact before theredefinition of the mole in 2019, and is now only approximate, but equality may still be assumed with high accuracy. Thus, for example, the average mass of a molecule ofwater is about 18.0153 Da, and the molar mass of water is about 18.0153 g/mol. For chemical elements without isolated molecules, such ascarbon andmetals, the molar mass is calculated using therelative atomic mass of the element, usually given by thestandard atomic weight indicated in theperiodic table. Thus, for example, the molar mass ofiron is about 55.845 g/mol.
The molar massM(X) ofatoms of anelementX is given by the relative atomic massAr(X) of the element multiplied by themolar mass constant,Mu, which (for all practical purposes) is equal to 1 g/mol:M(X) =Ar(X) ⋅Mu. For normal samples from Earth with typical isotope composition, the atomic weight can be approximated by the standard atomic weight[2] or the conventional atomic weight.
Multiplying by the molar mass constant ensures that the calculation isdimensionally correct: relative atomic masses and standard atomic weights aredimensionless quantities (i.e., pure numbers), whereas molar masses have units (in this case,grams permole).
Some elements are usually encountered asmolecules, e.g.hydrogen (H2),nitrogen (N2),oxygen (O2),sulfur (S8),chlorine (Cl2). The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule:
Here,Mr(X) is the relative molar mass, also called molecular weight or formula weight. For normal samples from Earth with typical isotope composition, thestandard atomic weight or the conventional atomic weight can be used as an approximation of the relative atomic mass of the sample. Examples are:
An average molar mass may be defined for mixtures of substances.[1] This is particularly important inpolymer science, where there is usually amolar mass distribution of non-uniform polymers so that different polymer molecules contain different numbers ofmonomer units.[3][4] The average molar mass of mixtures can be calculated from themole fractionsxi of the components and their molar massesMi:
It can also be calculated from themass fractionswi of the components:
As an example, the average molar mass of dry air is 28.965 g/mol.[5]
Molar mass is closely related to themolecular weight (M.W.) (for molecular compounds) andformula weight (F.W.) (for non-molecular compounds), older terms for what is now more correctly called therelative molar mass (Mr),[6] adimensionless quantity (i.e., a pure number, without units) equal to the molar mass divided by themolar mass constant, calculated from thestandard atomic weights of its constituent elements. However, it should be distinguished from themolecular mass (which is confusinglyalso sometimes known as molecular weight), which is the mass ofone molecule (of anysingle isotopic composition), and to theatomic mass, which is the mass ofone atom (of anysingle isotope). Thedalton, symbol Da, is also sometimes used as a unit of molecular weight and formula weight (now called relative molar mass), especially inbiochemistry, despite the fact that the quantities are dimensionless as relative masses.
Obsolete terms for molar mass includegram atomic mass for the mass, in grams, of one mole of atoms of an element, andgram molecular mass for the mass, in grams, of one mole of molecules of a compound. Thegram-atom is a former term for a mole of atoms, andgram-molecule for a mole of molecules.[7]
The molecular mass (m) is the mass of a given molecule: it is usually measured indaltons (Da or u).[7] Different molecules of the same compound may have different molecular masses because they contain differentisotopes of an element. This is distinct but related to the molar mass, which is a measure of the average molecular mass of all the molecules in a sample and is usually the more appropriate measure when dealing with macroscopic (weigh-able) quantities of a substance.
Molecular masses are calculated from theatomic masses of eachnuclide, while molar masses are calculated from thestandard atomic weights[8] of eachelement. The standard atomic weight takes into account theisotopic distribution of the element in a given sample (usually assumed to be "normal"). For example,water has a molar mass of18.0153(3) g/mol, but individual water molecules have molecular masses which range between18.0105646863(15) Da (1H216O) and22.0277364(9) Da (2H218O).
The distinction between molar mass and molecular mass is important because relative molecular masses can be measured directly bymass spectrometry, often to a precision of a fewparts per million. This is accurate enough to directly determine thechemical formula of a molecule.[9]
The termformula weight has a specific meaning when used in the context of DNA synthesis: whereas an individualphosphoramidite nucleobase to be added to a DNA polymer has protecting groups and has itsmolecular weight quoted including these groups, the amount of molecular weight that is ultimately added by this nucleobase to a DNA polymer is referred to as the nucleobase'sformula weight (i.e., the molecular weight of this nucleobase within the DNA polymer, minus protecting groups).[citation needed]
The precision to which a molar mass is known depends on the precision of theatomic masses from which it was calculated (and very slightly on the value of themolar mass constant, which depends on the measured value of thedalton). Most atomic masses are known to a precision of at least one part in ten-thousand, often much better[2] (the atomic mass oflithium is a notable, and serious,[10] exception). This is adequate for almost all normal uses in chemistry: it is more precise than mostchemical analyses, and exceeds the purity of most laboratory reagents.
The precision of atomic masses, and hence of molar masses, is limited by the knowledge of theisotopic distribution of the element. If a more accurate value of the molar mass is required, it is necessary to determine the isotopic distribution of the sample in question, which may be different from the standard distribution used to calculate the standard atomic mass. The isotopic distributions of the different elements in a sample are not necessarily independent of one another: for example, a sample which has beendistilled will beenriched in the lighterisotopes of all the elements present. This complicates the calculation of thestandard uncertainty in the molar mass.
A useful convention for normal laboratory work is to quote molar masses to twodecimal places for all calculations. This is more accurate than is usually required, but avoidsrounding errors during calculations. When the molar mass is greater than 1000 g/mol, it is rarely appropriate to use more than one decimal place. These conventions are followed in most tabulated values of molar masses.[11][12]
Molar masses are almost never measured directly. They may be calculated from standard atomic masses, and are often listed in chemical catalogues and onsafety data sheets (SDS). Molar masses typically vary between:
1–238 g/mol for atoms of naturally occurring elements;
While molar masses are almost always, in practice, calculated from atomic weights, they can also be measured in certain cases. Such measurements are much less precise than modernmass spectrometric measurements of atomic weights and molecular masses, and are of mostly historical interest. All of the procedures rely oncolligative properties, and anydissociation of the compound must be taken into account.
The measurement of molar mass by vapour density relies on the principle, first enunciated byAmedeo Avogadro, that equal volumes of gases under identical conditions contain equal numbers of particles. This principle is included in theideal gas equation:
Thefreezing point of asolution is lower than that of the puresolvent, and the freezing-point depression (ΔT) is directly proportional to theamount concentration for dilute solutions. When the composition is expressed as amolality, the proportionality constant is known as thecryoscopic constant (Kf) and is characteristic for each solvent. Ifw represents themass fraction of thesolute in solution, and assuming no dissociation of the solute, the molar mass is given by
Theboiling point of asolution of an involatilesolute is higher than that of the puresolvent, and the boiling-point elevation (ΔT) is directly proportional to theamount concentration for dilute solutions. When the composition is expressed as amolality, the proportionality constant is known as theebullioscopic constant (Kb) and is characteristic for each solvent. Ifw represents themass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by
^"International union of pure and applied chemistry, commission on macromolecular nomenclature, note on the terminology for molar masses in polymer science".Journal of Polymer Science: Polymer Letters Edition.22 (1): 57. 1984.Bibcode:1984JPoSL..22...57..doi:10.1002/pol.1984.130220116.
