# Molar concentration

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Molar concentration, also called molarity, amount concentration or substance concentration, is a measure of the concentration of a solute in a solution, or of any chemical species, in terms of amount of substance in a given volume. A commonly used unit for molar concentration in chemistry is the molar which is defined as the number of moles per litre (unit symbol: mol/L or m). A solution with a concentration of 1 mol/L is equivalent to 1 molar (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 solute per unit volume of solution, or per unit volume available to the species, represented by lowercase c:[1]

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

Here, n is the amount of the solute in moles,[2] N is the number of molecules present in the volume V (in litres), the ratio N/V is the number concentration C, and NA is the Avogadro constant, approximately 6.022×1023 mol−1.

Or more simply: 1 molar = 1 m = 1 mole/litre.

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.[2]

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, or mol dm−3, which is the same as mol/L. These traditional units are often denoted by a capital letter m (pronounced molar), sometimes preceded by an SI prefix 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 words "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[3] 10−24 mol/L
(1 particle per 1.6 L)
10−21 mol/m3

## Related quantities

### Number concentration

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

${\displaystyle C_{i}=c_{i}\cdot N_{\rm {A}}}$

where ${\displaystyle N_{\rm {A}}}$ is the Avogadro constant, approximately 6.022×1023 mol−1.

### Mass concentration

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

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

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:

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

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

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

### Mass fraction

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

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

### Molality

The conversion to molality (for binary mixtures) is:

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

where the solute is assigned the subscript 2.

For solutions with more than one solute, the conversion is:

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

## 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 molar concentrations-partial molar volumes

The sum of products between these quantities equals one.

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

### 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 :

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

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

## Spatial variation and diffusion

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

## Examples

Example 1: Consider 11.6 g of NaCl dissolved in 100 g of water. The final mass concentration ρ(NaCl) will be:

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 will be:

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.

Example 2: Another 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 x 0.1 L x 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.

Example 3: 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.

Example 4: A typical protein in bacteria, such as E. coli, may have about 60 copies, and the volume of a bacterium is about ${\displaystyle 10^{-15}}$ 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
Reference ranges for blood tests, sorted by molar concentration.

## 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 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.

## References

1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "amount concentration, c".
2. ^ a b Kaufman, Myron (2002). Principles of thermodynamics. CRC Press. p. 213. ISBN 0-8247-0692-7.
3. ^ David Bradley. "How low can you go? The Y to Y".