Molar concentration

In chemistry, the molar concentration, $c_{i}$ is defined as the amount of a constituent $n_{i}$ (usually measured in moles – hence the name) divided by the volume of the mixture $V$ :^[1]

c_{i}={\frac {n_{i}}{V}}

It is also called molarity, amount-of-substance concentration, amount concentration, substance concentration, or simply concentration. The volume $V$ in the definition $c_{i}=n_{i}/V$ refers to the volume of the solution, not the volume of the solvent. One litre of a solution usually contains either slightly more or slightly less than 1 litre of solvent because the process of dissolution causes volume of liquid to increase or decrease.

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

Notation

In addition to the notation $c_{i}$ there is a notation using brackets and the formula of a compound like [A]. This notation is encountered especially in equilibrium constants and reaction quotients.

Units

The SI unit is mol/m³. However, more commonly the unit mol/L is used. A solution of concentration 1 mol/L is also denoted as "1 molar" (1 M).

1 mol/L = 1 mol/dm³ = 1 mol dm⁻³ = 1 M = 1000 mol/m³.

An SI prefix is often used to denote concentrations. Commonly used units are listed in the table hereafter:

Name	Abbreviation	Concentration	Concentration (SI unit)
millimolar	mM	10⁻³ mol/dm³	10⁰ mol/m³
micromolar	μM	10⁻⁶ mol/dm³	10⁻³ mol/m³
nanomolar	nM	10⁻⁹ mol/dm³	10⁻⁶ mol/m³
picomolar	pM	10⁻¹² mol/dm³	10⁻⁹ mol/m³
femtomolar	fM	10⁻¹⁵ mol/dm³	10⁻¹² mol/m³
attomolar	aM	10⁻¹⁸ mol/dm³	10⁻¹⁵ mol/m³
zeptomolar	zM	10⁻²¹ mol/dm³	10⁻¹⁸ mol/m³
yoctomolar	yM^[2]	10⁻²⁴ mol/dm³ (1 particle per 1.6 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_{\rm {A}}

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

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 _{i}x_{i}M_{i}}{\rho }}

x_{i}=c_{i}\cdot {\frac {\sum 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 relation

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.

\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

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

\rho

(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 * 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.

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 $10^{-15}$ L. Thus, the number concentration $C$ is:

C

= 60 / (10⁻¹⁵ L)= 6×10¹⁶ L⁻¹

The molar concentration is:

c=C/N_{A}

= 6×10¹⁶ L⁻¹ / (6×10²³ mol⁻¹) = 10⁻⁷ mol/L = 100 nmol/L

Reference ranges for blood tests, sorted by molar 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 $c$ (CO₃^2-) = 1 mol/L because the salt dissociates into these ions.

References

^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "amount concentration, c". doi:10.1351/goldbook.A00295
^ David Bradley. "How low can you go? The Y to Y".

External links

[GoldBook-1] IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "amount concentration, c". doi:10.1351/goldbook.A00295

[2] David Bradley. "How low can you go? The Y to Y".

[1]

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v t e Chemical solutions
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Concentration and related quantities	Molar concentration Mass concentration Number concentration Volume concentration Normality Molality Mole fraction Mass fraction Isotopic abundance Mixing ratio Ternary plot
Solubility	Solubility equilibrium Total dissolved solids Solvation Solvation shell Enthalpy of solution Lattice energy Raoult's law Henry's law Solubility table (data) Solubility chart Miscibility
Solvent	(Category) Acid dissociation constant Protic solvent Polar aprotic solvent Inorganic nonaqueous solvent Solvation List of boiling and freezing information of solvents Partition coefficient Polarity Hydrophobe Hydrophile Lipophilic Amphiphile Lyonium ion Lyate ion