# 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/dm$^3$ 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 6.02214076×10$^{23}$ mol$^{−1}$. 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.

### Formality or analytical concentration

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 (Na$_2$CO$_3$) has a formal concentration of c(Na$_2$CO$_3$) = 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.

## Units

In the International System of Units (SI), the coherent unit for molar concentration is mol/m$^3$. However, most chemical literature traditionally uses mol/dm$^3$, 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/m$^3$ = 10$^{−3}$ mol/dm$^3$ = 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:
| Name | Abbreviation | Concentration | Concentration |
| Name | Abbreviation | (mol/L) | (mol/m$^3$) |
| millimolar | mM | 10$^{−3}$ | 10$^0$=1 |
| micromolar | μM | 10$^{−6}$ | 10$^{−3}$ |
| nanomolar | nM | 10$^{−9}$ | 10$^{−6}$ |
| picomolar | pM | 10$^{−12}$ | 10$^{−9}$ |
| femtomolar | fM | 10$^{−15}$ | 10$^{−12}$ |
| attomolar | aM | 10$^{−18}$ | 10$^{−15}$ |
| zeptomolar | zM | 10$^{−21}$ | 10$^{−18}$ |
| yoctomolar | yM | 10$^{−24}$ (6 particles per 10 L) | 10$^{−21}$ |
| rontomolar | rM | 10$^{−27}$ | 10$^{−24}$ |
| quectomolar | qM | 10$^{−30}$ | 10$^{−27}$ |

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

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

### 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.

## Examples

* 11.6 g of NaCl is dissolved in 100 g of water. The final mass concentration ρ(NaCl) is 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 Here, 58 g/mol is the molar mass of NaCl.
    ρ(NaCl) = ⁠11.6 g/11.6 g + 100 g⁠ = 0.104 g/g = 10.4 %.
    c(NaCl) = ⁠11.6 g/58 g/mol⁠ / 104.3 mL = 0.00192 mol/mL = 1.92 mol/L.
* 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 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.
    m(NaCl) = 2 mol/L × 0.1 L × 58 g/mol = 11.6 g.
* 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 Likewise, the concentration of solid hydrogen (molar mass = 2.02 g/mol) is The concentration of pure osmium tetroxide (molar mass = 254.23 g/mol) is
    c(H$_2$O) = ⁠1000 g/L/18.02 g/mol⁠ ≈ 55.5 mol/L.
    c(H$_2$) = ⁠88 g/L/2.02 g/mol⁠ = 43.7 mol/L.
    c(OsO$_4$) = ⁠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 The molar concentration is
    C = 60 / (10$^{−15}$ L) = 6×10$^{16}$ L$^{−1}$.
    c = ⁠C/N$_A$⁠ = ⁠6×10$^{16}$ L$^{−1}$/6×10$^{23}$ mol$^{−1}$⁠ = 10$^{−7}$ mol/L = 100 nmol/L.
* Reference ranges for blood tests, sorted by molar concentration: