View source for Molar mass

{{distinguish|Molecular mass|Mass number}}
{{Infobox physical quantity
| bgcolour = {default}
| name = Molar mass
| unit = kg/mol
| otherunits = g/mol
| symbols = ''M''
}}
In [[chemistry]], the '''molar mass''' of a [[chemical compound]] is defined as the [[mass]] of a sample of that compound divided by the [[amount of substance]] in that sample, measured in [[mole (unit)|moles]].<ref name="GreenBook">{{GreenBookRef|page=41}}</ref> The molar mass is a bulk, not molecular, 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 isotopes of the constituent atoms on earth. The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities.
 
The '''molecular weight''' is very commonly used as a synonym of molar mass, particularly for molecular compounds; however, the most authoritative sources define it differently (see [[molecular mass]]).
 
The '''formula weight''' is a synonym of molar mass that is frequently used for non-molecular compounds, such as [[ionic salt]]s.
 
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 [[base SI unit|base unit]] of molar mass is [[kilogram|kg]]/[[mole (unit)|mol]]. However, for historical reasons, molar masses are almost always expressed in [[gram|g]]/mol.
 
The mole was defined in such as way that the molar mass of a compound, in g/mol, is numerically equal (for all practical purposes) to the average mass of one molecule, in [[dalton (unit)|dalton]]s. Thus, for example, the average mass of a molecule of [[properties of water|water]] is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol.
 
For chemical elements without isolated molecules, such as [[carbon]] and metals, the molar mass is computed dividing by the number of moles of atoms instead. Thus, for example, the molar mass of [[iron]] is about 55.845 g/mol.
 
Between 1971 and 2019, SI defined the "amount of substance" as a separate [[dimensional analysis|dimension of measurement]], and the mole was defined as the amount of substance that has as many constituent particles as there are atoms in 12 grams of [[carbon-12]]. In that period, the molar mass of carbon-12 was thus ''exactly'' 12&nbsp;g/mol, by definition. Since 2019, a mole of any substance has been redefined in the SI as the amount of that substance containing an exactly defined number of particles, ''N'' = {{physconst|NA|unit=no|ref=no}}. Therefore, the molar mass of a compound now is simply the mass of this number of molecules of the compound.

== Molar masses of elements ==
{{main|Relative atomic mass|Standard atomic weight}}
Before reading this section, it must be understood that the [[2019 redefinition of the SI base units]] concluded that the [[molar mass constant]] is not exactly {{val|1|e=-3|u=kg/mol}}, but {{physconst|Mu|symbol=yes|after=.}}

The molar mass of [[atom]]s of an [[Chemical element|element]] is given by the [[standard atomic weight]] of the element<ref name="AtWt">{{AtWt 2005}}</ref> multiplied by the [[molar mass constant]], ''M''{{sub|u}}&nbsp;≈ 1&nbsp;×&nbsp;10<sup>−3</sup>&nbsp;kg/mol&nbsp;= 1&nbsp;g/mol:{{CODATA2010}}<!-- generates a named reference that can be reused as <ref name="CODATA2010" /> -->

The molar mass '''''M'''{{sub|m}}'' can be calculated from equation

    '''''M{{sub|m}} = m/n'''''

where '''''m''''' is the mass of substance and '''''n''''' is the number of moles of the substance. 

:''M''(H)&nbsp;= {{val|1.00797|(7)}}&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|1.00797|(7)|u=g/mol}}
:''M''(S)&nbsp;= {{val|32.065|(5)}}&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|32.065|(5)|u=g/mol}}
:''M''(Cl)&nbsp;= {{val|35.453|(2)}}&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|35.453|(2)|u=g/mol}}
:''M''(Fe)&nbsp;= {{val|55.845|(2)}}&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|55.845|(2)|u=g/mol}}.
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/mole).

Some elements are usually encountered as [[molecule]]s, e.g. [[hydrogen]] (H{{su|b=2}}), [[sulfur]] (S{{su|b=8}}), [[chlorine]] (Cl{{su|b=2}}). The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule:
:''M''(H{{su|b=2}})&nbsp;= 2&nbsp;× 1.007&nbsp;97(7)&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|2.01588|(14)|u=g/mol}}
:''M''(S{{su|b=8}})&nbsp;= 8&nbsp;× 32.065(5)&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|256.52|(4)|u=g/mol}}
:''M''(Cl{{su|b=2}})&nbsp;= 2&nbsp;× 35.453(2)&nbsp;× 1&nbsp;g/mol&nbsp;= {{val|70.906|(4)|u=g/mol}}.

== Molar masses of compounds ==
The molar mass of a [[Chemical compound|compound]] is given by the sum of the [[standard atomic weight]] (namely, the standard relative atomic mass) of the [[atom]]s which form the compound multiplied by the [[molar mass constant]], ''M''{{su|b=u}}:
:''M''(NaCl)&nbsp;= [22.989&nbsp;769&nbsp;28(2)&nbsp;+ 35.453(2)]&nbsp;× 1&nbsp;g/mol&nbsp;= 58.443(2)&nbsp;g/mol
:''M''(C{{su|b=12}}H{{su|b=22}}O{{su|b=11}})&nbsp;= ([12&nbsp;× 12.0107(8)]&nbsp;+ [22&nbsp;× 1.007&nbsp;94(7)]&nbsp;+ [11&nbsp;× 15.9994(3)])&nbsp;× 1&nbsp;g/mol&nbsp;= 342.297(14)&nbsp;g/mol.

An average molar mass may be defined for mixtures of compounds.<ref name="GreenBook"/> This is particularly important in [[polymer science]], where different polymer molecules may contain different numbers of [[monomer]] units (non-uniform polymers).<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>\bar{M}</math> can be calculated from the [[mole fraction]]s <math>x_i</math> of the components and their molar masses <math>M_i</math>:

:<math> \bar{M} = \sum_i  x_i M_i.</math>

It can also be calculated from the [[mass fraction (chemistry)|mass fractions]] <math>w_i</math> of the components:

:<math> 1/\bar{M} = \sum_i  \frac{{w_i}}{{M_i}}.</math>

As an example, the average molar mass of dry air is 28.97 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''' (''M''{{su|b=r}}) of a compound, to the older term '''formula weight''' (F.W.), and to the [[atomic mass|standard atomic masses]] of its constituent elements. However, it should be distinguished from the [[molecular mass]] (also known as molecular weight), which is the mass of ''one'' molecule (of any ''single'' isotopic composition) and is not directly related to the [[atomic mass]], 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;= 1.660&nbsp;539 066 60(50) ×10<sup>−27</sup>&nbsp;kg, as of 2018 CODATA recommended values).

'''Gram atomic mass''' is another term for the mass, in grams, of one mole of atoms of that element. "Gram atom" is a former term for a mole.

'''Molecular weight''' (M.W.) is an older term for what is now more correctly called the '''relative molar mass''' (''M''{{su|b=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>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 (''m'') is the mass of a given molecule: it is 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]] | accessdate = 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 18.0153(3)&nbsp;g/mol, but individual water molecules have molecular masses which range between 18.010&nbsp;564&nbsp;6863(15)&nbsp;u (<sup>1</sup>H{{su|b=2}}<sup>16</sup>O) and 22.027&nbsp;7364(9)&nbsp;u (<sup>2</sup>H{{su|b=2}}<sup>18</sup>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]] | accessdate = 2007-10-14}}</ref>

=== DNA synthesis usage ===
{{unreferenced section|date=December 2008}}
The term '''formula weight''' (F.W.) 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).

== 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 value of the molar mass constant. 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}}
</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 g/mol for atoms of naturally occurring elements;
:10–1000 g/mol for [[Small molecule|simple chemical compounds]];
:1000–5,000,000 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 ''n'' is the [[amount of substance]]. The vapour density (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]] is lower than that of the pure [[solvent]], and the freezing-point depression (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]] (''K''{{su|b=f}}) and is characteristic for each solvent. If ''w'' represents the [[mass fraction (chemistry)|mass fraction]] of the [[Solution|solute]] in solution, and assuming no dissociation of the solute, the molar mass is given by
:<math>M = {{wK_f}\over{\Delta T}}.\ </math>

=== Boiling-point elevation ===
{{main|Boiling-point elevation}}
The [[boiling point]] of a [[solution]] of an involatile [[Solution|solute]] is higher than that of the pure [[solvent]], and the boiling-point elevation (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]] (''K''{{su|b=b}}) and is characteristic for each solvent. If ''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_b}\over{\Delta T}}.\ </math>

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

==External links==
* [http://hox.io/molcalc HTML5 Molar Mass Calculator] 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] 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}}

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