Molar concentration

In chemistry, the molar concentration, $$c_i$$ is defined as the amount of a constituent $$n_i$$ 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 $$c_i = n_i/V$$ refers to the volume of the solution, not the volume of the solvent. One liter of a solution usually contains either slightly more or slightly less than 1 liter 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:

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 mol&minus;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:


 * $$x_i = c_i \cdot M / \rho$$

where $$M$$ is the average molar mass of the solution and $$\rho$$ is the density of the solution.

Mass fraction
The conversion to mass fraction $$w_i$$ is given by:


 * $$w_i = c_i \cdot M_i / \rho$$

Molality
The conversion to molality (for binary mixtures) is:


 * $$ m_2 = \frac \,$$

where the solute is assigned the subscript 2.

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


 * $$ m_i = \frac \,$$

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 $$\rho$$(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. 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 L−1

The molar concentration is:


 * $$c = C / N_A$$ = 6 L−1 / (6 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.