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as the mole was historically defined such that the avogadro number the number of atomic scale entities comprising one mole was exactly equal to the number of daltons in a gram g da this means that for all practical purposes 1 mol g da ent the relationship between the molar mass of carbon 12 m 12 c 12 g mol and its atomic mass m a 12 c 12 da can be expressed as m 12 c m a 12 c n a rearranging and substituting the given values into the equation yields the following expression for the avogadro constant n a 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 one entity atom molecule formula unit in da i e m a x da a r x so that m x a r x g mol the equivalence was exact before the redefinition 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 of water is about 18 0153 da 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 calculated using the relative atomic mass of the element usually given by the standard atomic weight indicated in the periodic table thus for example the molar mass of iron is about 55 845 g mol calculation edit molar masses of elements edit main articles relative atomic mass and standard atomic weight the molar mass m x of atoms of an element x is given by the relative atomic mass a r x of the element multiplied by the molar mass constant m u which for all practical purposes is equal to 1 g mol m x a r x m u 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 m he 4 002602 2 m u 4 002602 2 g mol m ne 20 1797 6 m u 20 1797 6 g mol m fe 55 845 2 m u 55 845 2 g mol m cu 63 546 3 m u 63 546 3 g mol m ag 107 8682 2 m u 107 8682 2 g mol displaystyle begin array lll m ce he 4 002602 2 times m_ mathrm u 4 002602 2 text g mol m ce ne 20 1797 6 times m_ mathrm u 20 1797 6 text g mol m ce fe 55 845 2 times m_ mathrm u 55 845 2 text g mol m ce cu 63 546 3 times m_ mathrm u 63 546 3 text g mol m ce ag 107 8682 2 times m_ mathrm u 107 8682 2 text g mol end array multiplying by the molar mass constant ensures that the calculation is dimensionally correct relative atomic masses and standard atomic weights are dimensionless quantities i e pure numbers whereas molar masses have units in this case grams per mole some elements are usually encountered as molecules e g hydrogen h 2 nitrogen n 2 oxygen o 2 sulfur s 8 chlorine cl 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 2 2 1 00794 7 m u 2 01588 14 g mol m n 2 2 14 0067 2 m u 28 0134 4 g mol m o 2 2 15 9994 3 m u 31 9988 6 g mol m s 8 8 32 065 5 m u 256 52 4 g mol m cl 2 2 35 453 2 m u 70 906 4 g mol displaystyle begin array lll m ce h2 2 times 1 00794 7 times m_ mathrm u 2 01588 14 text g mol m ce n2 2 times 14 0067 2 times m_ mathrm u 28 0134 4 text g mol m ce o2 2 times 15 9994 3 times m_ mathrm u 31 9988 6 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 molar masses of compounds edit main article molecular mass calculation the molar mass m x of a compound is given by the sum of the relative atomic masses a r x i of the elements each multiplied by the number of atoms n i per element which form the compound multiplied by the molar mass constant m u 1 g mol m x m r x m u m u i n i a r x i displaystyle m text x m_ text r text x cdot m_ text u m_ text u sum _ i n_ i a_ text r text x _ i here m r x is the relative molar mass also called molecular weight or 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 m nacl 22 98976928 2 35 453 2 m u 58 443 2 g mol m c 12 h 22 o 11 12 12 0107 8 22 1 00794 7 11 15 9994 3 m u 342 297 14 g mol displaystyle begin array lll m ce nacl 22 98976928 2 35 453 2 times m_ text u 58 443 2 text g mol m ce c12h22o11 12 times 12 0107 8 22 times 1 00794 7 11 times 15 9994 3 times m_ text u 342 297 14 text g mol end array average molar mass of mixtures edit an average molar mass may be defined for mixtures of substances 1 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 3 4 the average molar mass of mixtures m displaystyle overline m can be calculated from the mole fractions x i of the components and their molar masses m i m i x i m i displaystyle overline m sum _ i x_ i m_ i it can also be calculated from the mass fractions w i of the components 1 m i w i m i displaystyle frac 1 overline m sum _ i frac w_ i m_ i as an example the average molar mass of dry air is 28 9647 g mol 5 related quantities edit molar mass is closely related to the molecular weight m w for molecular compounds and formula weight f w for non molecular compounds older terms for what is now more correctly called the relative molar mass m r 6 a dimensionless quantity i e a pure number without units equal to the molar mass divided by the molar mass constant calculated from the 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 symbol da is also sometimes used as a unit of molecular weight and formula weight now called relative molar mass especially in biochemistry despite the fact that the quantities are dimensionless as relative masses 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 7 molecular mass edit main article molecular mass the molecular mass m is the mass of a given molecule it is usually measured in daltons da or u 7 different molecules of the same compound may have different molecular masses because they contain different isotopes 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 masses of each nuclide while molar masses are calculated from the standard atomic weights 8 of each element the standard atomic weight takes into account the isotopic distribution of the element in a given sample usually assumed to be normal for example water has a molar mass of 18 0153 3 g mol but individual water molecules have molecular masses which range between 18 010 564 6863 15 da 1 h 2 16 o and 22 027 7364 9 da 2 h 2 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 parts per million this is accurate enough to directly determine the chemical formula of a molecule 9 dna synthesis usage edit 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 precision and uncertainties edit the precision to which a molar mass is known depends on the precision of the atomic masses 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 most atomic masses are known to a precision of at least one part in ten thousand often much better 2 the atomic mass of lithium is a notable and serious 10 exception this is adequate for almost all normal uses in chemistry it is more precise than most 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 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 distilled will be enriched in the lighter isotopes 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 places for all calculations this is more accurate than is usually required but avoids rounding 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 measurement edit 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 sheets sds molar masses typically vary between 1 238 g mol for atoms of naturally occurring elements 10 1000 g mol for simple chemical compounds 1000 5 000 000 g mol for polymers proteins 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 spectrometric measurements of atomic weights and molecular masses and are of mostly historical interest all of the procedures rely on colligative properties and any dissociation of the compound must be taken into account vapour density edit main article 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 p v n r t displaystyle pv nrt where n is the amount of substance the vapour density ρ is given by ρ n m v displaystyle rho nm over v combining these two equations gives an expression for the molar mass in terms of the vapour density for conditions of known pressure and temperature m r t ρ p displaystyle m rt rho over p freezing point depression edit main article 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 f and is characteristic for each solvent if w represents the mass fraction of the solute in solution and assuming no dissociation of the solute the molar mass is given by m w k f δ t displaystyle m wk_ text f over delta t boiling point elevation edit main article boiling point elevation the boiling point of a solution of an involatile 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 b and is characteristic for each solvent if w represents the mass fraction of the solute in solution and assuming no dissociation of the solute the molar mass is given by m w k b δ t displaystyle m wk_ text b over delta t see also edit mole map chemistry references edit 1 2 international union of pure and applied chemistry 1993 quantities units and symbols in physical chemistry 2nd edition oxford blackwell science isbn 0 632 03583 8 p 41 electronic version 1 2 wieser m e 2006 atomic weights of the elements 2005 pdf pure and applied chemistry 78 11 2051 2066 doi 10 1351 pac200678112051 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 metanomski w v 1991 compendium of macromolecular nomenclature oxford blackwell science pp 47 73 isbn 0 632 02847 5 the engineering toolbox molecular mass of air iupac compendium of chemical terminology 5th ed the gold book 2025 online version 2006 relative molar mass doi 10 1351 goldbook r05270 1 2 international bureau of weights and measures 2006 the international system of units si pdf 8th ed p 126 isbn 92 822 2213 6 archived pdf from the original on 2021 06 04 retrieved 2021 12 16 atomic weights and isotopic compositions for all elements nist retrieved 2007 10 14 author guidelines article layout rsc publishing retrieved 2007 10 14 greenwood norman n earnshaw alan 1997 chemistry of the elements 2nd ed butterworth heinemann p 21 doi 10 1016 c2009 0 30414 6 isbn 978 0 08 037941 8 see e g weast r c ed 1972 handbook of chemistry and physics 53rd ed cleveland ohio chemical rubber co possolo antonio van der veen adriaan m h meija juris hibbert d brynn 2018 01 04 interpreting and propagating the uncertainty of the standard atomic weights iupac technical report pure and applied chemistry 90 2 395 424 doi 10 1515 pac 2016 0402 s2cid 145931362 notes edit external links edit molecular weight calculator calculate molecular weight molar mass from chemical formulas online molar mass calculator with the uncertainty of m and all the calculations shown molar mass calculator online molar mass and elemental composition calculator stoichiometry add in for microsoft excel archived 2011 05 11 at the wayback machine for calculation of molecular weights reaction coefficients and stoichiometry it includes both average atomic weights and isotopic weights molar mass chemistry second level course v t e mole concepts constants avogadro constant boltzmann ...
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