Molar concentration

Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular, of a solute in a solution, in terms of amount of substance per unit volume of solution. In chemistry, the most commonly used unit for molarity is the number of moles per liter, having the unit symbol mol/L or mol/dm3 in SI units. A solution with a concentration of 1 mol/L is said to be 1 molar, commonly designated as 1 M or 1 M. Molarity is often depicted with square brackets around the substance of interest; for example, the molarity of the hydrogen ion is depicted as [H+].

Definition
Molar concentration or molarity is most commonly expressed in units of moles of solute per litre of solution. 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 $$c$$:


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

Here, $$n$$ is the amount of the solute in moles, $$N$$ is the number of constituent particles present in volume $$V$$ (in litres) of the solution, and $$N_\text{A}$$ is the Avogadro constant, since 2019 defined as exactly. The ratio $$\frac{N}{V}$$ is the number density $$C$$.

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.

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

If a molecular entity dissociates in solution, the concentration refers to the original chemical formula in solution, the molar concentration is sometimes called formal concentration or formality (FA) or analytical concentration (cA). 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(CO3(2−)) = 1 mol/L because the salt dissociates into these ions.
 * Formality or analytical concentration

Units
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:

Number concentration
The conversion to number concentration $$C_i$$ is given by


 * $$C_i = c_i N_\text{A},$$

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

Mass concentration
The conversion to mass concentration $$\rho_i$$ is given by


 * $$\rho_i = c_i M_i,$$

where $$M_i$$ is the molar mass of constituent $$i$$.

Mole fraction
The conversion to mole fraction $$x_i$$ is given by


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

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


 * $$x_i = \frac{c_i}{c} = \frac{c_i}{\sum_j c_j}.$$

Mass fraction
The conversion to mass fraction $$w_i$$ is given by


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

Molality
For binary mixtures, the conversion to molality $$b_2$$ is


 * $$b_2 = \frac{c_2}{\rho - c_1 M_1},$$

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

For solutions with more than one solute, the conversion is


 * $$b_i = \frac{c_i}{\rho - \sum_{j\neq i} c_j M_j}.$$

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:
 * $$\sum_i c_i \overline{V_i} = 1.$$

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


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

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