Molar concentration

Molar concentration or molarity is most commonly expressed in units of moles of solute per litre of solution.^[1] For use in broader applications, it is defined as amount of substance of solute per unit volume of solution, or per unit volume available to the species, represented by lowercase ${\displaystyle c}$ :^[2]

${\displaystyle c={\frac {n}{V}}={\frac {N}{N_{\text{A}}\,V}}={\frac {C}{N_{\text{A}}}}.}$ 

Here, ${\displaystyle n}$  is the amount of the solute in moles,^[3] ${\displaystyle N}$  is the number of constituent particles present in volume ${\displaystyle V}$  (in litres) of the solution, and ${\displaystyle N_{\text{A}}}$  is the Avogadro constant, since 2019 defined as exactly 6.02214076×10²³ mol⁻¹. The ratio ${\displaystyle {\frac {N}{V}}}$  is the number density ${\displaystyle C}$ .

In thermodynamics, the use of molar concentration is often not convenient because the volume of most solutions slightly depends on temperature due to thermal expansion. This problem is usually resolved by introducing temperature correction factors, or by using a temperature-independent measure of concentration such as molality.^[3]

The reciprocal quantity represents the dilution (volume) which can appear in Ostwald's law of dilution.

If a molecule or salt dissociates in solution, the concentration refers to the original chemical formula in solution, the molar concentration is sometimes called formal concentration or formality (F_A) or analytical concentration (c_A). For example, if a sodium carbonate solution (Na₂CO₃) has a formal concentration of c(Na₂CO₃) = 1 mol/L, the molar concentrations are c(Na⁺) = 2 mol/L and c(CO2−3) = 1 mol/L because the salt dissociates into these ions.^[4]

In the International System of Units (SI), the coherent unit for molar concentration is mol/m³. However, most chemical literature traditionally uses mol/dm³, which is the same as mol/L. This traditional unit is often called a molar and denoted by the letter M, for example:

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

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:

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.

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 ${\displaystyle i}$ .

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:

${\displaystyle x_{i}={\frac {c_{i}}{c}}={\frac {c_{i}}{\sum _{j}c_{j}}}.}$ 

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

${\displaystyle w_{i}=c_{i}{\frac {M_{i}}{\rho }}.}$ 

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

${\displaystyle b_{i}={\frac {c_{i}}{\rho -\sum _{j\neq i}c_{j}M_{j}}}.}$ 

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.

The sum of products between these quantities equals one:

${\displaystyle \sum _{i}c_{i}{\overline {V_{i}}}=1.}$ 

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

${\displaystyle c_{i}={\frac {c_{i,T_{0}}}{1+\alpha \Delta T}},}$ 

where ${\displaystyle c_{i,T_{0}}}$  is the molar concentration at a reference temperature, ${\displaystyle \alpha }$  is the thermal expansion coefficient of the mixture.

• Molality
• Orders of magnitude (molar concentration)

• Molar Solution Concentration Calculator
• Experiment to determine the molar concentration of vinegar by titration