Mass per amount of substance

This is not to be confused with Molecular mass or Mass number. Honestly, the distinction is crucial. One deals with the abstract concept of a substance's quantity, the other with the concrete mass of a single entity or its nucleus. It’s like confusing the idea of a crowd with the weight of a single person in it.

Molar masses

Imagine comparing two samples, one of iron and one of gold. If they happen to have the exact same mass, the diagram might show you how many moles of each you're dealing with. Iron, being less dense, would take up more space for the same weight, and you’d have a different number of moles. Gold, denser, would pack more atoms into that same mass, resulting in fewer moles. It’s a visual representation of how different elements behave at the atomic level, even when presenting the same macroscopic weight.

Common symbols

The SI unit for molar mass is the kilogram per mole, symbolized as kg/mol. It’s the standard, the official designation. But in the trenches of a lab, you’ll almost always see grams per mole (g/mol). It’s more convenient, less cumbersome. The dimension of molar mass is M (mass) divided by N (amount of substance), which is kg/mol in SI.

In chemistry

In the realm of chemistry, the molar mass, denoted by M, is a fundamental concept. It’s not about a single atom or molecule, but about a sample of a substance. Think of it as the average weight of a vast collection of those particles. It’s defined as the ratio between the total mass (m) of that sample and the amount of substance it contains, measured in moles. So, M = m / n. This isn't some fleeting molecular whim; it's a stable, macroscopic property.

This molar mass is a weighted average, a concession to the fact that elements exist as isotopes. Earth’s average isotopic composition dictates the standard atomic weights, making the molar mass a terrestrial average. It’s a snapshot of what we typically find here, not necessarily what you’d find on a distant moon.

Often, people throw around terms like molecular mass or formula weight as if they’re interchangeable. For practical purposes, their numerical values often align perfectly. The difference, however, is subtle yet profound. Molecular mass refers to the mass of one specific particle—a microscopic quantity. Molar mass, on the other hand, is the average mass of many particles—a macroscopic quantity. It's the difference between weighing a single grain of sand and weighing a beach.

Molar mass is an intensive property. This means it doesn't change regardless of how much of the substance you have. A gram of water has the same molar mass as a ton of water. The coherent SI unit is kg/mol, but the practical, everyday unit is g/mol. It’s a matter of convenience, like using inches instead of meters for a quick measurement.

The Mole and its Definition

Since 1971, the SI has treated the "amount of substance" as a distinct dimension of measurement. Before the 2019 redefinition, the mole was anchored to the mass of carbon-12. Specifically, it was the amount of substance containing as many constituent particles as there are atoms in exactly 12 grams of carbon-12. The dalton, the unit of atomic mass, was defined as precisely one-twelfth the mass of a carbon-12 atom. This created a neat numerical equivalence: the molar mass of a substance in g/mol was, for all practical intents and purposes, the same as the average mass of one of its entities (atom, molecule, etc.) in daltons.

However, the 2019 SI redefinition shifted the mole's definition. It's now defined by a fixed number of entities – exactly 6.02214076×10^23. This fixes the numerical value of the Avogadro constant N_A. The dalton, though, remains tied to the experimentally determined mass of a carbon-12 atom. This subtle shift means the numerical equivalence between molar mass in g/mol and average entity mass in daltons is now approximate, though the agreement is remarkably close – within about one part per billion. It’s a slight divergence, like a perfectly aligned set of gears that develop a minuscule wobble over time.

Technical Background

For a pure substance, say 'X', if you know its molar mass, M(X), you can easily calculate the amount of substance, n(X), from the sample's mass, m(X), using the formula: n(X) = m(X) / M(X). This is straightforward enough.

Now, consider the total number of entities in the sample, N(X), and the mass of a single entity, m_a(X) (which could be an atomic mass, molecular mass, or formula mass). The total mass of the sample is simply m(X) = N(X) * m_a(X). The amount of substance, n(X), can also be expressed as n(X) = N(X) / N_A, where N_A is the Avogadro constant. Alternatively, we can think of an "elementary amount," n_a, which is simply one entity. So, n(X) = N(X) * n_a.

By substituting these into the molar mass equation, M(X) = m(X) / n(X), we get M(X) = [N(X) * m_a(X)] / [N(X) / N_A] = m_a(X) * N_A. This means the molar mass is the mass of a single entity multiplied by the Avogadro constant. Or, viewed differently, M(X) = m_a(X) / n_a, the mass per elementary amount.

If you have the relative atomic mass, A_r(X) – essentially the atomic weight – of an entity, its mass in daltons is m_a(X) = A_r(X) * Da. The unit of amount is the "entity" (symbol ent). So, molar mass can be expressed as M(X) = A_r(X) * Da/ent. The molar mass constant, M_u, is defined as 1 Da/ent. For practical purposes, M_u is equal to 1 g/mol. This equivalence arises because historically, the mole was defined such that the Avogadro number (the number of entities in a mole) was numerically equal to the number of daltons in a gram. Thus, 1 mol is equivalent to (g/Da) ent.

Take carbon-12 as an example. Its molar mass, M(¹²C), is 12 g/mol, and its atomic mass, m_a(¹²C), is 12 Da. The relationship M(¹²C) = m_a(¹²C) * N_A holds true. Rearranging this gives us N_A = (g/Da) mol⁻¹, which means the Avogadro number is precisely the number of daltons in a gram, and, by definition, the number of atoms in 12 grams of carbon-12.

The mole was historically defined so that the numerical value of the molar mass in g/mol matched the numerical value of the average mass of an entity in daltons. So, if a water molecule has an average mass of about 18.0153 Da, its molar mass is about 18.0153 g/mol. For elements like carbon or metals, which don't typically exist as discrete molecules, we use their relative atomic mass (or standard atomic weight from the periodic table) to find the molar mass. For instance, iron has a molar mass of approximately 55.845 g/mol.

Calculation

Molar masses of elements

The molar mass M(X) of an element X, when it exists as individual atoms, is simply its relative atomic mass, A_r(X), multiplied by the molar mass constant, M_u, which is practically 1 g/mol. For typical terrestrial samples, the standard atomic weight serves as a very good approximation for the relative atomic mass.

For example:

• Helium (He): M(He) = 4.002602(2) × M_u = 4.002602(2) g/mol
• Neon (Ne): M(Ne) = 20.1797(6) × M_u = 20.1797(6) g/mol
• Iron (Fe): M(Fe) = 55.845(2) × M_u = 55.845(2) g/mol
• Copper (Cu): M(Cu) = 63.546(3) × M_u = 63.546(3) g/mol
• Silver (Ag): M(Ag) = 107.8682(2) × M_u = 107.8682(2) g/mol

Multiplying by M_u ensures dimensional correctness. Relative atomic masses are dimensionless, but molar masses carry units (g/mol).

However, some elements commonly exist as molecules. Think of hydrogen (H₂), nitrogen (N₂), oxygen (O₂), sulfur (S₈), or chlorine (Cl₂). For these, the molar mass of the molecule is the molar mass of the atom multiplied by the number of atoms in the molecule.

• M(H₂) = 2 × 1.00794(7) × M_u = 2.01588(14) g/mol
• M(N₂) = 2 × 14.0067(2) × M_u = 28.0134(4) g/mol
• M(O₂) = 2 × 15.9994(3) × M_u = 31.9988(6) g/mol
• M(S₈) = 8 × 32.065(5) × M_u = 256.52(4) g/mol
• M(Cl₂) = 2 × 35.453(2) × M_u = 70.906(4) g/mol

Molar masses of compounds

For a compound X, its molar mass M(X) is the sum of the relative atomic masses of all the constituent atoms, multiplied by the molar mass constant M_u (approximately 1 g/mol). If nᵢ is the number of atoms of element Xᵢ in the compound, then:

M(X) = M_r(X) ⋅ M_u = M_u Σᵢ nᵢ A_r(Xᵢ)

Here, M_r(X) is the relative molar mass, often called molecular weight or formula weight. For typical terrestrial samples, we use the standard atomic weights.

Let's look at some examples:

• Sodium chloride (NaCl): M(NaCl) = [22.98976928(2) + 35.453(2)] × M_u = 58.443(2) g/mol
• Sucrose (C₁₂H₂₂O₁₁): M(C₁₂H₂₂O₁₁) = [12 × 12.0107(8) + 22 × 1.00794(7) + 11 × 15.9994(3)] × M_u = 342.297(14) g/mol

Average molar mass of mixtures

For mixtures, we can define an average molar mass, denoted as $\overline{M}$. This is particularly relevant in polymer science, where polymers often have a distribution of molecular sizes.

If we know the mole fractions, xᵢ, and molar masses, Mᵢ, of the components:

$\overline{M} = \sum_{i} x_{i} M_{i}$

Alternatively, using mass fractions, wᵢ:

$1 / \overline{M} = \sum_{i} w_{i} / M_{i}$

For instance, the average molar mass of dry air is approximately 28.965 g/mol.

Related quantities

Molar mass is intimately linked to terms like molecular weight (M.W.) and formula weight (F.W.), though these are older terms for what is now more precisely called relative molar mass (M_r). M_r is dimensionless, being the molar mass divided by the molar mass constant. It’s derived from standard atomic weights.

However, it’s vital to distinguish this from the molecular mass. Molecular mass refers to the mass of a single molecule, often measured in daltons. While their numerical values are often the same, the scale is different: microscopic versus macroscopic. The dalton (Da) is sometimes used as a unit for relative molar mass, especially in biochemistry, even though it's technically dimensionless.

Obsolete terms like "gram atomic mass" (for a mole of atoms) and "gram molecular mass" (for a mole of molecules) are relics of the past. Similarly, "gram-atom" and "gram-molecule" are historical terms for the mole.

Molecular mass

The molecular mass (m) is the mass of a single molecule, typically expressed in daltons (Da or u). Since elements exist as isotopes, molecules of the same compound can have slightly different molecular masses. This is distinct from molar mass, which is the average mass of all molecules in a macroscopic sample.

Molecular masses are calculated using the atomic masses of specific nuclides, whereas molar masses use the standard atomic weights of elements, which account for their natural isotopic distribution. For example, while water has a molar mass of about 18.0153 g/mol, individual water molecules can range in mass from 18.0105646863 Da (¹H₂¹⁶O) to 22.0277364 Da (²H₂¹⁸O).

This distinction is crucial. Mass spectrometry can measure relative molecular masses with incredible precision, often to parts per million. This accuracy is sufficient to directly determine a molecule's chemical formula.

DNA synthesis usage

In DNA synthesis, the term "formula weight" takes on a specific nuance. A nucleobase building block (a phosphoramidite) has protecting groups attached, and its molecular weight includes these. The "formula weight," in this context, refers to the molecular weight added to the DNA polymer by that nucleobase, after the protecting groups have been removed. It’s the weight that actually becomes part of the chain.

Precision and uncertainties

The precision of a molar mass calculation hinges on the precision of the atomic masses used and, to a lesser extent, the molar mass constant. Most atomic masses are known with high accuracy, often better than one part in ten thousand. This is generally sufficient for most chemical applications; it often surpasses the purity of reagents and the precision of analyses.

The real limitation on precision comes from the knowledge of an element's isotopic distribution. If you need a highly accurate molar mass for a specific sample, you must know its isotopic composition, which might deviate from the standard. Furthermore, isotopic distributions aren't always independent; processes like distillation can selectively enrich lighter isotopes of all elements present. This complexity makes calculating the standard uncertainty in molar mass a non-trivial task.

For everyday laboratory work, a practical convention is to use molar masses rounded to two decimal places. This provides ample accuracy and prevents accumulating rounding errors in calculations. For molar masses exceeding 1000 g/mol, one decimal place is usually sufficient. These conventions are widely adopted in tabulated data.

Measurement

Direct measurement of molar mass is rare. It’s almost always calculated from atomic weights and found in catalogues or safety data sheets (SDS). Molar masses typically fall within these ranges:

• 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, and similar large molecules.

While calculation is the norm, historical methods did exist for measurement, though they are far less precise than modern techniques like mass spectrometry. These methods relied on colligative properties, and any dissociation of the solute had to be accounted for.

Vapour density

Measuring molar mass via vapour density relies on Amedeo Avogadro's principle: equal volumes of gases at the same temperature and pressure contain equal numbers of particles. This is embedded in the ideal gas law: pV = nRT.

Vapour density (ρ) is defined as mass per unit volume: ρ = nM / V. Combining this with the ideal gas law allows us to express molar mass in terms of vapour density under known pressure (p) and temperature (T):

M = (RTρ) / p

Freezing-point depression

The freezing point of a solution is lower than that of the pure solvent. This freezing-point depression (ΔT) is proportional to the amount concentration in dilute solutions. When expressed in terms of molality, the proportionality constant is the cryoscopic constant, K_f. For a solute with mass fraction w and assuming no dissociation, the molar mass is:

M = (wK_f) / ΔT

Boiling-point elevation

Similarly, the boiling point of a solution containing a non-volatile solute is higher than that of the pure solvent. This boiling-point elevation (ΔT) is proportional to the amount concentration in dilute solutions. Using molality, the proportionality constant is the ebullioscopic constant, K_b. For a solute with mass fraction w and assuming no dissociation, the molar mass is:

M = (wK_b) / ΔT

See also

• Mole map (chemistry)