# Molar concentration

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Molar concentration

In chemistry, the molar concentration, ci is defined as the amount of a constituent ni divided by the volume of the mixture V : $c_i = \frac {n_i}{V}$

It is also called molarity, amount-of-substance concentration, amount concentration, substance concentration, or simply concentration. The volume V in the definition ci = ni / 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.

## 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 below:

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 10−24 mol/dm3
(1 molecule per 1.6 L)
10−21 mol/m3

## Related Quantities

### Number concentration

The conversion to number concentration Ci is given by: $C_i = c_i \cdot N_{\rm A}$

where NA is the Avogadro constant, approximately 6.022×1023 mol−1.

### Mass concentration

The conversion to mass concentration ρi is given by: $\rho_i = c_i \cdot M_i$

where Mi is the molar mass of constituent i.

### Mole fraction

The conversion to mole fraction xi is given by: $x_i = c_i \cdot M / \rho$

where M is the average molar mass of the solution and ρ is the density of the solution.

### Mass fraction

The conversion to mass fraction wi is given by: $w_i = c_i \cdot M_i / \rho$

### Molality

The conversion to molality (for binary mixtures) is: $m_2 = \frac{{c_2}}{{\rho - c_2 \cdot M_2}} \,$

where the solute is assigned the subscript 2.

For solutions with more than one solute, the conversion is: $m_i = \frac{{c_i}}{{\rho - \sum c_i \cdot M_i}} \,$

## Properties

### Dependence on volume

Molar concentration depends on the variation of the volume of the solution due mainly to thermal expansion.

## 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 * 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. 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: Proteins in bacteria, such as E. coli, usually occur at 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

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(CO32-) = 1 mol/L because the salt dissociates into these ions.

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