Editing Molar mass

{{Short description|Mass per amount of substance}}
{{Distinguish|Molecular mass|Mass number}}
{{Infobox physical quantity
| bgcolour = {default}
| name = Molar mass
| image = Mass versus moles of iron vs gold.svg
| caption = A diagram comparing [[Mole (unit)|moles]] and molar masses of [[iron]] and [[gold]] samples that have equal [[Mass|masses]]
| unit = [[Kilogram|kg]]/[[Mole (unit)|mol]]
| otherunits = [[Gram|g]]/[[Mole (unit)|mol]]
| symbols = {{mvar|M}}
| dimension = '''M''' '''N'''<sup>−1</sup>
}}

In [[chemistry]], the '''molar mass''' ({{mvar|M}}) (sometimes called '''molecular weight''' or '''formula weight''', but see [[Molar mass#Related quantities|related quantities]] for usage) of a [[chemical compound]] is defined as the ratio between the [[mass]] and the [[amount of substance]] (measured in [[mole (unit)|moles]]) of any sample of the compound.<ref name="GreenBook">{{GreenBook2nd|page=41}}</ref> The molar mass is a bulk, not molecular, [[physical property|property]] of a substance. The molar mass is an ''[[average]]'' of many instances of the compound, which often vary in mass due to the presence of [[isotope]]s. Most commonly, the molar mass is computed from the [[standard atomic weight]]s and is thus a terrestrial average and a function of the relative abundance of the [[isotope]]s of the constituent atoms on Earth. 

For a sample of a substance X, the molar mass, ''M''(X), is appropriate for converting between the mass of the substance, ''m''(X), and the amount of the substance, ''n''(X), for bulk quantities: ''M''(X) = ''m''(X)/''n''(X). If ''N''(X) is the number of entities in the sample, ''m''(X) = ''N''(X)''m''{{sub|a}}(X) and ''n''(X) = ''N''(X)/''N''{{sub|A}} = ''N''(X) ent, where ent is an atomic-scale unit of amount equal to one entity. So ''M''(X) = ''m''{{sub|a}}(X)/ent, the atomic-scale entity mass per entity, which is self evident. Since ''m''{{sub|a}}(X) = ''A''{{sub|r}}(X) Da, molar mass can be written in units of dalton per entity as ''M''(X) = ''A''{{sub|r}}(X) Da/ent. One mole is an aggregate of an Avogadro number of entities, and (for all practical purposes) the Avogadro number is g/Da. So (for all practical purposes) Da/ent = g/mol. And the molar mass can be calculated from ''M''(X) = ''A''{{sub|r}}(X) Da/ent = ''A''{{sub|r}}(X) g/mol = ''A''{{sub|r}}(X) kg/kmol.  
 
The [[molecular mass]] (for molecular compounds) and formula mass (for non-molecular compounds, such as [[ionic salt]]s) are commonly used as synonyms of molar mass, differing only in units ([[Dalton (unit)|dalton]] vs Da/ent or g/mol); however, the most authoritative sources define it differently. The difference is that molecular mass is the mass of one specific particle or molecule, while the molar mass is an average over many particles or molecules.
 
The molar mass is an [[intensive property]] of the substance, that does not depend on the size of the sample. In the [[International System of Units]] (SI), the [[coherent unit]] of molar mass is kg/mol. However, for historical reasons, molar masses are almost always expressed in g/mol.
 
The mole was defined in such a way that the numerical value of the molar mass of a compound in g/mol, i.e. ''M''(X)/(g/mol), was equal to the numerical value of the average atomic-scale mass of one entity (atom, molecule, formula unit, . . .) in Da, i.e. ''m''{{sub|a}}(X)/Da = ''A''{{sub|r}}(X). Specifically: ''M''(X) = ''A''{{sub|r}}(X) g/mol. It was exactly equal before the [[2019 revision of the SI#Mole|redefinition of the mole in 2019]], and is now only approximately equal, but the difference is negligible for all practical purposes. Thus, for example, the average mass of a molecule of [[properties of water|water]] is about 18.0153&nbsp;Da, and the molar mass of water is about 18.0153&nbsp;g/mol.
 
For chemical elements without isolated molecules, such as [[carbon]] and metals, the molar mass is still computed using ''M''(X) = ''A''{{sub|r}}(X) g/mol. Thus, for example, the molar mass of [[iron]] is about 55.845&nbsp;g/mol.
 
Since 1971, [[SI]] defined the "amount of substance" as a separate [[dimensional analysis|dimension 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&nbsp;grams of [[carbon-12]]. That meant that, during that period, the molar mass of carbon-12 was thus ''exactly'' 12&nbsp;g/mol, by definition: ''M''({{sup|12}}C) = 12 g/mol (exactly). Inverting this gives an expression for the (original) definition of the mole in terms of the international prototype of the kilogram (IPK) and the molar mass of carbon-12: 1 mol = (0.012 IPK)/''M''({{sup|12}}C). Because the dalton was (and still is) defined as 1 Da = ''m''{{sub|a}}({{sup|12}}C)/12 and ''M''({{sup|12}}C) = ''m''{{sub|a}}({{sup|12}}C)''N''{{sub|A}}, the original mole definition can be written as 1 mol = (g/Da)(1/''N''{{sub|A}}), where (g/Da) is the (1971 definition of the) Avogadro number—the number of carbon-12 atoms in 12 grams of carbon-12—and (1/''N''{{sub|A}}) is an amount of one entity. Since 2019, a mole of any substance has been [[2019 revision of the SI|redefined in the SI]] as the amount of that substance containing an exactly defined number of entities: 1 mol = {{physconst|NA|unit=no|ref=no}}(1/''N''{{sub|A}}). This is still in the same form as the previous definition, one mole = (Avogadro number)(amount of one entity), but because the dalton is still defined in terms of the (now inexactly known) mass of the carbon-12 atom, the Avogadro number is no longer ''exactly'' equal to (g/Da). The numerical value of the molar mass of a substance expressed in g/mol thus is (for all practical purposes) still equal to the numerical value of the mass of this number of entities (i.e. an amount of one mole) of the substance expressed in grams—(the relative discrepancy is only of order 10{{sup|–9}}).

== Molar masses of elements ==
{{main|Relative atomic mass|Standard atomic weight}}

The molar mass of [[atom]]s of an [[Chemical element|element]] is given by the relative atomic mass of the element multiplied by the [[molar mass constant]], {{physconst|Mu|symbol=yes|round=6|unit=kg/mol|ref=no}} ≈ 1&nbsp;g/mol. For normal samples from Earth with typical isotope composition, the atomic weight can be approximated by the standard atomic weight<ref name="AtWt">{{AtWt 2005}}</ref> or the conventional atomic weight.<!-- generates a named reference that can be reused as <ref name="CODATA2010" /> -->
: <math chem>\begin{array}{lll}
M(\ce{H})  &= 1.00797(7) \times M_\mathrm{u} &= 1.00797(7) \text{ g/mol} \\
M(\ce{S})  &= 32.065(5) \times M_\mathrm{u}  &= 32.065(5) \text{ g/mol} \\
M(\ce{Cl}) &= 35.453(2) \times M_\mathrm{u}  &= 35.453(2) \text{ g/mol} \\
M(\ce{Fe}) &= 55.845(2) \times M_\mathrm{u}  &= 55.845(2) \text{ g/mol}
\end{array}</math>

Multiplying by the molar mass constant ensures that the calculation is [[dimension]]ally correct: standard relative atomic masses are dimensionless quantities (i.e., pure numbers) whereas molar masses have units (in this case, grams per mole).

Some elements are usually encountered as [[molecule]]s, e.g. [[hydrogen]] ({{chem2|H2}}), [[sulfur]] ({{chem2|S8}}), [[chlorine]] ({{chem2|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:
: <math chem>\begin{array}{lll}
M(\ce{H2})  &= 2\times 1.00797(7) \times M_\mathrm{u} &= 2.01595(4) \text{ g/mol} \\
M(\ce{S8})  &= 8\times 32.065(5) \times M_\mathrm{u}  &= 256.52(4) \text{ g/mol} \\
M(\ce{Cl2}) &= 2\times 35.453(2) \times M_\mathrm{u}  &= 70.906(4) \text{ g/mol}
\end{array}</math>

== Molar masses of compounds ==
The molar mass of a [[Chemical compound|compound]] is given by the sum of the [[relative atomic mass]] {{math|''A''{{sub|r}}}} of the [[atom]]s which form the compound multiplied by the [[molar mass constant]] <math>M_u \approx 1 \text{ g/mol}</math>:
: <math>M = M_{\rm u} M_{\rm r} = M_{\rm u} \sum_i {A_{\rm r}}_i.</math>

Here, {{math|''M''{{sub|r}}}} is the relative molar mass, also called formula weight. For normal samples from earth with typical isotope composition, the [[standard atomic weight]] or the conventional atomic weight can be used as an approximation of the relative atomic mass of the sample. Examples are: <math chem display=block>\begin{array}{ll}
M(\ce{NaCl}) &= \bigl[22.98976928(2) + 35.453(2)\bigr] \times 1 \text{ g/mol} \\
             &= 58.443(2) \text{ g/mol} \\[4pt]
M(\ce{C12H22O11})  &= \bigl[12 \times 12.0107(8) + 22 \times 1.00794(7) + 11 \times 15.9994(3)\bigr] \times 1 \text{ g/mol} \\ 
                   &= 342.297(14) \text{ g/mol}
\end{array}</math>

An average molar mass may be defined for mixtures of compounds.<ref name="GreenBook" /> This is particularly important in [[polymer science]], where there is usually a [[molar mass distribution]] of non-uniform polymers so that different polymer molecules contain different numbers of [[monomer]] units.<ref>{{cite journal | title = International union of pure and applied chemistry, commission on macromolecular nomenclature, note on the terminology for molar masses in polymer science| year = 1984 | journal = Journal of Polymer Science: Polymer Letters Edition| volume = 22 | pages = 57 | issue=1 | doi=10.1002/pol.1984.130220116 |bibcode = 1984JPoSL..22...57.}}</ref><ref>{{cite book | last = Metanomski | first =  W. V. | title = Compendium of Macromolecular Nomenclature | year = 1991 | publisher = [[Blackwell Science]] | location = Oxford | pages = 47–73 | isbn = 0-632-02847-5}}</ref>

== Average molar mass of mixtures ==

The average molar mass of mixtures <math>\overline{M}</math> can be calculated from the [[mole fraction]]s {{mvar|x{{sub|i}}}} of the components and their molar masses {{mvar|M{{sub|i}}}}:
: <math>\overline{M} = \sum_i x_i M_i.</math>

It can also be calculated from the [[mass fraction (chemistry)|mass fractions]] {{mvar|w{{sub|i}}}} of the components:
: <math>\frac{1}{\overline{M}} = \sum_i\frac{w_i}{M_i}.</math>

As an example, the average molar mass of dry air is 28.96&nbsp;g/mol.<ref>The Engineering ToolBox  [http://www.engineeringtoolbox.com/molecular-mass-air-d_679.html Molecular Mass of Air]</ref>

== Related quantities ==

Molar mass is closely related to the '''relative molar mass''' ({{math|''M''{{sub|r}}}}) of a compound and  to the [[Standard atomic weight|standard atomic weights]] of its constituent elements. However, it should be distinguished from the [[molecular mass]] (which is confusingly ''also'' sometimes known as molecular weight), which is the mass of ''one'' molecule (of any ''single'' isotopic composition), and to the [[atomic mass]], which is the mass of ''one'' atom (of any ''single'' isotope). The [[dalton (unit)|dalton]], symbol Da, is also sometimes used as a unit of molar mass, especially in [[biochemistry]], with the definition 1&nbsp;Da&nbsp;= 1&nbsp;g/mol, despite the fact that it is strictly a unit of mass (1&nbsp;Da&nbsp;= 1&nbsp;u&nbsp;= {{val|1.66053906892e−27|(52)|u=kg}}, as of 2022 CODATA recommended values).<ref>{{Cite web |title=CODATA Value: atomic mass constant |url=https://physics.nist.gov/cgi-bin/cuu/Value?u |access-date=2024-06-21 |website=physics.nist.gov}}</ref>

Obsolete terms for molar mass include '''gram atomic mass''' for the mass, in grams, of one mole of atoms of an element, and '''gram molecular mass''' for the mass, in grams, of one mole of molecules of a compound. The '''gram-atom''' is a former term for a mole of atoms, and '''gram-molecule''' for a mole of molecules.<ref name="SI" />

'''Molecular weight''' (M.W.) (for molecular compounds) and '''formula weight''' (F.W.) (for non-molecular compounds), are older terms for what is now more correctly called the '''relative molar mass''' ({{math|''M''{{sub|r}}}}).<ref>{{GoldBookRef|title=relative molar mass|file=R05270}}</ref> This is a [[dimension]]less quantity (i.e., a pure number, without units) equal to the molar mass divided by the [[molar mass constant]].<ref group="notes">The technical definition is that the relative molar mass is the molar mass measured on a scale where the molar mass of unbound [[carbon 12]] atoms, at rest and in their electronic ground state, is 12. The simpler definition given here is equivalent to the full definition because of the way the [[molar mass constant]] is itself defined.</ref>

=== Molecular mass ===
{{main|Molecular mass}}
The molecular mass ({{mvar|m}}) is the mass of a given molecule: it is usually measured in [[Dalton (unit)|dalton]]s (Da or u).<ref name="SI">{{SIbrochure8th|page=126}}</ref> Different molecules of the same compound may have different molecular masses because they contain different [[isotope]]s 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 the [[atomic mass]]es of each [[nuclide]], while molar masses are calculated from the [[standard atomic weight]]s<ref>{{cite web | title = Atomic Weights and Isotopic Compositions for All Elements | url = http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=html&isotype=some | publisher = [[NIST]] | access-date = 2007-10-14}}</ref> of each [[Chemical element|element]]. The standard atomic weight takes into account the [[Isotope|isotopic distribution]] of the element in a given sample (usually assumed to be "normal"). For example, [[water (molecule)|water]] has a molar mass of {{val|18.0153|(3)|u=g/mol}}, but individual water molecules have molecular masses which range between {{val|18.0105646863|(15)|u=Da}} ({{chem2|^{1}H2^{16}O}}) and {{val|22.0277364|(9)|u=Da}} ({{chem2|^{2}H2^{18}O}}).

The distinction between molar mass and molecular mass is important because relative molecular masses can be measured directly by [[mass spectrometry]], often to a precision of a few [[Part per million|parts per million]]. This is accurate enough to directly determine the [[chemical formula]] of a molecule.<ref>{{cite web | title = Author Guidelines – Article Layout | url = http://www.rsc.org/Publishing/ReSourCe/AuthorGuidelines/ArticleLayout/sect3.asp | publisher = [[Royal Society of Chemistry|RSC Publishing]] | access-date = 2007-10-14}}</ref>

=== DNA synthesis usage ===
The term '''formula weight''' has a specific meaning when used in the context of DNA synthesis: whereas an individual [[phosphoramidite]] nucleobase to be added to a DNA polymer has protecting groups and has its ''molecular 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's ''formula weight'' (i.e., the molecular weight of this nucleobase within the DNA polymer, minus protecting groups).{{citation needed|date=August 2022}}

== Precision and uncertainties ==
The precision to which a molar mass is known depends on the precision of the [[atomic mass]]es from which it was calculated (and very slightly on the value of the [[molar mass constant]], which depends on the measured value of the [[Dalton (unit)|dalton]]). Most atomic masses are known to a precision of at least one part in ten-thousand, often much better<ref name="AtWt"/> (the atomic mass of [[lithium]] is a notable, and serious,<ref>{{Greenwood&Earnshaw|page=21}}</ref> exception). This is adequate for almost all normal uses in chemistry: it is more precise than most [[chemical analysis|chemical 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 the [[Isotope|isotopic 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 been [[Distillation|distilled]] will be [[Isotopic enrichment|enriched]] in the lighter [[isotope]]s of all the elements present. This complicates the calculation of the [[standard uncertainty]] in the molar mass.

A useful convention for normal laboratory work is to quote molar masses to two [[decimal place]]s for all calculations. This is more accurate than is usually required, but avoids [[rounding error]]s during calculations. When the molar mass is greater than 1000&nbsp;g/mol, it is rarely appropriate to use more than one decimal place. These conventions are followed in most tabulated values of molar masses.<ref>See, e.g., {{RubberBible53rd}}</ref><ref>
{{cite journal
 |title=Interpreting and propagating the uncertainty of the standard atomic weights (IUPAC Technical Report)
 |journal=Pure and Applied Chemistry
 |volume=90|issue=2|pages=395–424
 |date=2018-01-04
 |doi=10.1515/pac-2016-0402
 |first1=Antonio |last1=Possolo
 |first2=Adriaan M. H. |last2=van der Veen
 |first3=Juris |last3=Meija
 |first4=D. Brynn |last4=Hibbert
 |s2cid=145931362
 |url=https://nrc-publications.canada.ca/eng/view/fulltext/?id=f1f2ca7f-9b5f-4da0-bbba-d196cc190911
 |doi-access=free
}}</ref>

== Measurement ==
Molar masses are almost never measured directly. They may be calculated from standard atomic masses, and are often listed in chemical catalogues and on [[safety data sheet]]s (SDS). Molar masses typically vary between:
: 1–238&nbsp;g/mol for atoms of naturally occurring elements;
: {{val|10|–|1000|u=g/mol}} for [[Small molecule|simple chemical compounds]];
: {{val|1000|–|5000000|u=g/mol}} for [[polymer]]s, [[protein]]s, [[DNA]] fragments, etc.

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 modern [[Mass spectrometry|mass spectrometric]] measurements of atomic weights and molecular masses, and are of mostly historical interest. All of the procedures rely on [[Colligative property|colligative properties]], and any [[dissociation (chemistry)|dissociation]] of the compound must be taken into account.

=== Vapour density ===
{{main|Vapour density}}
The measurement of molar mass by vapour density relies on the principle, first enunciated by [[Amedeo Avogadro]], that equal volumes of gases under identical conditions contain equal numbers of particles. This principle is included in the [[ideal gas equation]]:
: <math>pV = nRT ,</math>
where {{mvar|n}} is the [[amount of substance]]. The vapour density ({{mvar|ρ}}) is given by
: <math>\rho = {{nM}\over{V}} .</math>
Combining these two equations gives an expression for the molar mass in terms of the vapour density for conditions of known [[pressure]] and [[temperature]]:
: <math>M = {{RT\rho}\over{p}} .</math>

=== Freezing-point depression ===
{{main|Freezing-point depression}}
The [[freezing point]] of a [[Solution (chemistry)|solution]] is lower than that of the pure [[solvent]], and the freezing-point depression ({{math|Δ''T''}}) is directly proportional to the [[amount concentration]] for dilute solutions. When the composition is expressed as a [[molality]], the proportionality constant is known as the [[cryoscopic constant]] ({{math|''K''{{sub|f}}}}) and is characteristic for each solvent. If {{mvar|w}} represents the [[mass fraction (chemistry)|mass fraction]] of the [[solute]] in solution, and assuming no dissociation of the solute, the molar mass is given by
: <math>M = {{wK_\text{f}}\over{\Delta T}}.\ </math>

=== Boiling-point elevation ===
{{main|Boiling-point elevation}}
The [[boiling point]] of a [[Solution (chemistry)|solution]] of an involatile [[solute]] is higher than that of the pure [[solvent]], and the boiling-point elevation ({{math|Δ''T''}}) is directly proportional to the [[amount concentration]] for dilute solutions. When the composition is expressed as a [[molality]], the proportionality constant is known as the [[ebullioscopic constant]] ({{math|''K''{{sub|b}}}}) and is characteristic for each solvent. If {{mvar|w}} represents the [[mass fraction (chemistry)|mass fraction]] of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by
: <math>M = {{wK_\text{b}}\over{\Delta T}}.\ </math>

== See also ==
* [[Mole map (chemistry)]]

== References ==
{{Reflist|30em}}

=== Notes ===
{{Reflist|group=notes}}

== External links ==
* [http://hox.io/molcalc HTML5 Molar Mass Calculator] {{Webarchive|url=https://web.archive.org/web/20170425114833/http://hox.io/molcalc/ |date=2017-04-25 }} web and mobile application.
* [http://www.chem4free.info/calculators/molarmass.htm Online Molar Mass Calculator] with the uncertainty of ''M'' and all the calculations shown
* [http://www.webqc.org/mmcalc.php Molar Mass Calculator] Online Molar Mass and Elemental Composition Calculator
* [http://chemistry-in-excel.jimdo.com Stoichiometry Add-In for Microsoft Excel] {{Webarchive|url=https://web.archive.org/web/20110511073820/http://chemistry-in-excel.jimdo.com/ |date=2011-05-11 }} for calculation of molecular weights, reaction coefficients and stoichiometry. It includes both average atomic weights and isotopic weights.
* [http://www.physics-chemistry-class.com/chemistry/molar-mass.html Molar mass: chemistry second-level course].

{{Mole concepts}}
{{Authority control}}

[[Category:Mass]]
[[Category:Molar quantities]]