Molar concentration

Units

While there is clear consensus on the equivalence of units:

1 mol/m³ = 10⁻³ mol/dm³ = 10⁻³ mol/L = 10⁻³ M = 1 mM = 1 mmol/L,

guidance on unit names and abbreviations varies:

[A]mount concentration ... [a]lso called amount-of-substance concentration, substance concentration (in clinical chemistry) and in older literature molarity. ...The common unit is mole per cubic decimetre (mol dm⁻³) or mole per litre (mol L⁻¹) sometimes denoted by M.

— IUPAC, "Gold Book" 5ed

In the older literature this quantity was often called molarity, a usage that should be avoided due to the risk of confusion with the quantity molality. Units commonly used for amount concentration are mol L⁻¹ (or mol dm⁻³), mmol L⁻¹, mmol L⁻¹ etc., often denoted M, mM, μM etc. (pronounced molar, millimolar, micromolar).

— IUPAC, "Green Book" 3ed

The term molarity and the symbol M should no longer be used because they, too, are obsolete. One should use instead amount-of-substance concentration of B and such units as mol/dm³, kmol/m³, or mol/L. (A solution of, for example, 0.1 mol/dm³ was often called a 0.1 molar solution, denoted 0.1 M solution. The molarity of the solution was said to be 0.1 M.)

— NIST, Guide to the SI, Chapter 8

The SI prefix "mega" (symbol M) has the same symbol. However, the prefix is never used alone, so "M" unambiguously denotes molar. Sub-multiples, such as "millimolar" (mM) and "nanomolar" (nM), consist of the unit preceded by an SI prefix:

Related quantities

Number concentration

The conversion to number concentration ${\displaystyle C_i}$ is given by

${\displaystyle C_i=c_i{N}_{\text{A}},}$

where ${\displaystyle N_{\text{A}}}$ is the Avogadro constant.

Mass concentration

The conversion to mass concentration ${\displaystyle {\rho }_i}$ is given by

${\displaystyle {\rho }_i=c_i{M}_i,}$

where ${\displaystyle M_i}$ is the molar mass of constituent ${\displaystyle i}$.

Mole fraction

The conversion to mole fraction ${\displaystyle x_i}$ is given by

${\displaystyle x_i=c_i\frac{\overline{M}}{\rho },}$

where ${\displaystyle \overline{M}}$ is the average molar mass of the solution, ${\displaystyle \rho }$ is the density of the solution.

A simpler relation can be obtained by considering the total molar concentration, namely, the sum of molar concentrations of all the components of the mixture:

${\displaystyle x_i=\frac{c_i}{c}=\frac{c_i}{\sum _j{c}_j}.}$

Mass fraction

The conversion to mass fraction ${\displaystyle w_i}$ is given by

${\displaystyle w_i=c_i\frac{M_i}{\rho }.}$

Molality

For binary mixtures, the conversion to molality ${\displaystyle b_2}$ is

${\displaystyle b_2=\frac{c_2}{\rho -c_1{M}_1},}$

where the solvent is substance 1, and the solute is substance 2.

For solutions with more than one solute, the conversion is

${\displaystyle b_i=\frac{c_i}{\rho -\sum _{j\ne i}{c}_j{M}_j}.}$

Properties

Sum of molar concentrations – normalizing relations

The sum of molar concentrations gives the total molar concentration, namely the density of the mixture divided by the molar mass of the mixture or by another name the reciprocal of the molar volume of the mixture. In an ionic solution, ionic strength is proportional to the sum of the molar concentration of salts.

Sum of products of molar concentrations and partial molar volumes

The sum of products between these quantities equals one:

${\displaystyle \sum _i{c}_i\overline{V_i}=1.}$

Dependence on volume

The molar concentration depends on the variation of the volume of the solution due mainly to thermal expansion. On small intervals of temperature, the dependence is

${\displaystyle c_i=\frac{c_{i,T_0}}{1+\alpha \mathrm{\Delta }T},}$

where ${\displaystyle c_{i,T_0}}$ is the molar concentration at a reference temperature, ${\displaystyle \alpha }$ is the thermal expansion coefficient of the mixture.

Examples

• 11.6 g of NaCl is dissolved in 100 g of water. The final mass concentration ρ(NaCl) is

  ρ(NaCl) = ⁠11.6 g/11.6 g + 100 g⁠ = 0.104 g/g = 10.4 %.

  The volume of such a solution is 104.3mL (volume is directly observable); its density is calculated to be 1.07 (111.6g/104.3mL)

  The molar concentration of NaCl in the solution is therefore

  c(NaCl) = ⁠11.6 g/58 g/mol⁠ / 104.3 mL = 0.00192 mol/mL = 1.92 mol/L.

  Here, 58 g/mol is the molar mass of NaCl.
• A typical task in chemistry is the preparation of 100 mL (= 0.1 L) of a 2 mol/L solution of NaCl in water. The mass of salt needed is

  m(NaCl) = 2 mol/L × 0.1 L × 58 g/mol = 11.6 g.

  To create the solution, 11.6 g NaCl is placed in a volumetric flask, dissolved in some water, then followed by the addition of more water until the total volume reaches 100 mL.
• The density of water is approximately 1000 g/L and its molar mass is 18.02 g/mol (or 1/18.02 = 0.055 mol/g). Therefore, the molar concentration of water is

  c(H₂O) = ⁠1000 g/L/18.02 g/mol⁠ ≈ 55.5 mol/L.

  Likewise, the concentration of solid hydrogen (molar mass = 2.02 g/mol) is

  c(H₂) = ⁠88 g/L/2.02 g/mol⁠ = 43.7 mol/L.

  The concentration of pure osmium tetroxide (molar mass = 254.23 g/mol) is

  c(OsO₄) = ⁠5.1 kg/L/254.23 g/mol⁠ = 20.1 mol/L.

• A typical protein in bacteria, such as E. coli, may have about 60 copies, and the volume of a bacterium is about 10⁻¹⁵ L. Thus, the number concentration C is

  C = 60 / (10⁻¹⁵ L) = 6×10¹⁶ L⁻¹.

  The molar concentration is

  c = ⁠C/N_A⁠ = ⁠6×10¹⁶ L⁻¹/6×10²³ mol⁻¹⁠ = 10⁻⁷ mol/L = 100 nmol/L.

• Reference ranges for blood tests, sorted by molar concentration: