# 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 litre, having the unit symbol mol/L. A solution with a concentration of 1 mol/L is said to be 1 molar, commonly designated as 1 M.

## 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:

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

Here, n is the amount of the solute in moles, N is the number of molecules present in vol (in litres), the ratio N/V is the number concentration C, and NA is the Avogadro constant, approximately 6.022×1023 mol−1.

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.

## Units

In the International System of Units (SI) the base unit for molar concentration is mol/m3. However, this is impractical for most laboratory purposes and most chemical literature traditionally uses mol/dm3, which is the same as mol/L. These traditional units are often denoted by the letter M, optionally preceded by an SI prefix as needed to denote sub-multiples, for example:

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

The adjectives "millimolar" and "micromolar" refer to mM and μM (10−3 mol/L and 10−6 mol/L), respectively.

Name Abbreviation Concentration Concentration (SI unit)
millimolar mM 10−3 mol/L 100 mol/m3
micromolar μM 10−6 mol/L 10−3 mol/m3
nanomolar nM 10−9 mol/L 10−6 mol/m3
picomolar pM 10−12 mol/L 10−9 mol/m3
femtomolar fM 10−15 mol/L 10−12 mol/m3
attomolar aM 10−18 mol/L 10−15 mol/m3
zeptomolar zM 10−21 mol/L 10−18 mol/m3
yoctomolar yM 10−24 mol/L
(6 particles per 10 L)
10−21 mol/m3

## Related quantities

### Number concentration

The conversion to number concentration $C_{i}$  is given by

$C_{i}=c_{i}\cdot N_{\text{A}},$

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

### Mass concentration

The conversion to mass concentration $\rho _{i}$  is given by

$\rho _{i}=c_{i}\cdot M_{i},$

where $M_{i}$  is the molar mass of constituent $i$ .

### Mole fraction

The conversion to mole fraction $x_{i}$  is given by

$x_{i}=c_{i}\cdot {\frac {M}{\rho }}=c_{i}\cdot {\frac {\sum _{j}x_{j}M_{j}}{\rho }},$
$x_{i}=c_{i}\cdot {\frac {\sum _{j}x_{j}M_{j}}{\rho -c_{i}M_{i}}},$

where $M$  is the average molar mass of the solution, $\rho$  is the density of the solution, and j is the index of other solutes.

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:

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

### Mass fraction

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

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

### Molality

The conversion to molality (for binary mixtures) is

$b_{2}={\frac {c_{2}}{\rho -c_{2}\cdot M_{2}}},$

where the solute is assigned the subscript 2.

For solutions with more than one solute, the conversion is

$b_{i}={\frac {c_{i}}{\rho -\sum c_{i}\cdot M_{i}}}.$

## 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 molar concentration of salts.

### Sum of products of molar concentrations and partial molar volumes

The sum of products between these quantities equals one:

$\sum _{i}c_{i}\cdot {\bar {V_{i}}}=1.$

### Dependence on volume

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

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

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

## Spatial variation and diffusion

Molar and mass concentration have different values in space where diffusion happens.

## 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 density of such a solution is 1.07 g/mL, thus its volume is:

V = 11.6 g + 100 g/1.07 g/mL = 104.3 mL

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 are 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/NA = 6×1016 L−1/6×1023 mol−1 = 10−7 mol/L = 100 nmol/L

## Formal concentration

If the concentration refers to original chemical formula in solution, the molar concentration is sometimes called formal concentration. 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.