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Molar mass
In chemistry, the molar mass (M) (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance (element or compound) is defined as the ratio between the mass (m) and the amount of substance (n, measured in moles) of any sample of the substance: M = m/n. The molar mass is a bulk, not molecular, property of a substance. The molar mass is a weighted average of many instances of the element or compound, which often vary in mass due to the presence of isotopes. Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth.
The molecular mass (for molecular compounds) and formula mass (for non-molecular compounds, such as ionic 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 (a microscopic quantity), while the molar mass is an average over many particles or molecules (a macroscopic quantity).
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 with the unit g/mol (or equivalently in kg/kmol).
Since 1971, SI defined the "amount of substance" as a separate 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 grams of carbon-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 was exactly equal to the numerical value of the average mass of an entity (atom, molecule, formula unit) of the substance expressed in daltons.
Since 2019, the mole has been redefined in the SI as the amount of any substance containing exactly 6.02214076×1023 entities, fixing the numerical value of the Avogadro constant NA 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 a part per billion).
For a pure sample of a substance X, 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). If N(X) is the number of entities of the substance in the sample, and ma(X) is the mass of each entity of the substance (atomic mass, molecular mass, or formula mass), then the mass of the sample is m(X) = N(X) ⋅ ma(X), and the amount of substance is n(X) = N(X)/NA = N(X) ⋅ na, where na is the elementary amount, an amount consisting of exactly one atomic-scale entity of any kind (atom, molecule, formula unit), analogous to the elementary charge e. Since the elementary amount is the reciprocal of the Avogadro constant, using the relationship M(X) = m(X)/n(X), the molar mass is then given by M(X) = ma(X) ⋅ NA = ma(X)/na (dimension M/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, or formula weight) Ar(X) of an entity of a substance X, its mass expressed in daltons is ma(X) = Ar(X) Da, where the atomic-scale unit of mass is defined as 1 Da = mu = ma(12C)/12 (dimension M). The corresponding atomic-scale unit of amount of substance is the entity (symbol ent), defined as 1 ent = na (dimension N). So, with Ar(X) known, the molar mass can be expressed in daltons per entity as M(X) = Ar(X) Da/ent. Thus, the molar mass of a substance X can be calculated as M(X) = Ar(X) ⋅ Mu, with the molar mass constant Mu equal to exactly 1 Da/ent, which (for all practical purposes) is equal to 1 g/mol, 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(12C) = 12 g/mol, and its atomic mass, ma(12C) = 12 Da, can be expressed as M(12C) = ma(12C) · NA. Rearranging and substituting the given values into the equation yields the following expression for the Avogadro 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).
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Molar mass
In chemistry, the molar mass (M) (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance (element or compound) is defined as the ratio between the mass (m) and the amount of substance (n, measured in moles) of any sample of the substance: M = m/n. The molar mass is a bulk, not molecular, property of a substance. The molar mass is a weighted average of many instances of the element or compound, which often vary in mass due to the presence of isotopes. Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth.
The molecular mass (for molecular compounds) and formula mass (for non-molecular compounds, such as ionic 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 (a microscopic quantity), while the molar mass is an average over many particles or molecules (a macroscopic quantity).
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 with the unit g/mol (or equivalently in kg/kmol).
Since 1971, SI defined the "amount of substance" as a separate 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 grams of carbon-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 was exactly equal to the numerical value of the average mass of an entity (atom, molecule, formula unit) of the substance expressed in daltons.
Since 2019, the mole has been redefined in the SI as the amount of any substance containing exactly 6.02214076×1023 entities, fixing the numerical value of the Avogadro constant NA 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 a part per billion).
For a pure sample of a substance X, 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). If N(X) is the number of entities of the substance in the sample, and ma(X) is the mass of each entity of the substance (atomic mass, molecular mass, or formula mass), then the mass of the sample is m(X) = N(X) ⋅ ma(X), and the amount of substance is n(X) = N(X)/NA = N(X) ⋅ na, where na is the elementary amount, an amount consisting of exactly one atomic-scale entity of any kind (atom, molecule, formula unit), analogous to the elementary charge e. Since the elementary amount is the reciprocal of the Avogadro constant, using the relationship M(X) = m(X)/n(X), the molar mass is then given by M(X) = ma(X) ⋅ NA = ma(X)/na (dimension M/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, or formula weight) Ar(X) of an entity of a substance X, its mass expressed in daltons is ma(X) = Ar(X) Da, where the atomic-scale unit of mass is defined as 1 Da = mu = ma(12C)/12 (dimension M). The corresponding atomic-scale unit of amount of substance is the entity (symbol ent), defined as 1 ent = na (dimension N). So, with Ar(X) known, the molar mass can be expressed in daltons per entity as M(X) = Ar(X) Da/ent. Thus, the molar mass of a substance X can be calculated as M(X) = Ar(X) ⋅ Mu, with the molar mass constant Mu equal to exactly 1 Da/ent, which (for all practical purposes) is equal to 1 g/mol, 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(12C) = 12 g/mol, and its atomic mass, ma(12C) = 12 Da, can be expressed as M(12C) = ma(12C) · NA. Rearranging and substituting the given values into the equation yields the following expression for the Avogadro 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).