Mass concentration (chemistry)

In chemistry, the mass concentration $\rho_i$ (or $\gamma_i$) is defined as the mass of a constituent $m_i$ divided by the volume of the mixture s$V$:^[1]

$\rho_i = \frac {m_i}{V}.$

For a pure chemical the mass concentration equals its density (mass divided by volume); thus the mass concentration of a component in a mixture can be called the density of a component in a mixture. This explains the usage of ρ (the lower case Greek letter rho), the symbol most often used for density.

Definition and properties

The volume $V$ in the definition 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. Sometimes the mass concentration is called titer.

Notation

The notation common with mass density underlines the connection between the two quantities (the mass concentration being the mass density of a component in the solution), but it can be a source of confusion especially when they appear in the same formula undifferentiated by an additional symbol (like a star superscript, a bolded symbol or varrho).

Dependence on volume

Mass concentration depends on the variation of the volume of the solution due mainly to thermal expansion. On small intervals of temperature the dependence is :

$\rho_i = \frac {{\rho_{i,T_0}}}{{(1 + \alpha \cdot \Delta T)}}$

where $\rho_{i,T_0}$ is the mass concentration at a reference temperature, $\alpha$ is the thermal expansion coefficient of the mixture.

Sum of mass concentrations - normalizing relation

The sum of the mass concentrations of all components (including the solvent) gives the density $\rho$ of the solution:

$\rho = \sum_i \rho_i \,$

Thus, for pure component the mass concentration equals the density of the pure component.

Sum of products mass concentrations - partial specific volumes

The sum of products between these quantities equals one.

$\sum_i \rho_i \cdot \bar{v_i} = 1$

Units

The SI-unit for mass concentration is kg/m³ (kilogram/cubic metre). This is the same as mg/ml and g/l. Another commonly used unit is g/100mL, which is identical to g/dL (gram/decilitre).

Usage in biology

In biology, the unit "%" is sometimes incorrectly used to denote mass concentration, also called "mass/volume percentage." A solution with 1 g of solute dissolved in a final volume of 100 mL of solution would be labeled as "1 %" or "1 % m/v" (mass/volume). The notation is mathematically flawed because the unit "%" can only be used for dimensionless quantities. "Percent solution" or "percentage solution" are thus terms best reserved for "mass percent solutions" (m/m = m% = mass solute/mass total solution after mixing), or "volume percent solutions" (v/v = v% = volume solute per volume of total solution after mixing). The very ambiguous terms "percent solution" and "percentage solutions" with no other qualifiers, continue to occasionally be encountered.

This common usage of % to mean m/v in biology is because of many biological solutions being dilute and water-based or an aqueous solution. Liquid water has a density of approximately 1 g/cm³ (1 g/ml) (water density). Thus 100 ml of water is equal to approximately 100 g. Therefore, a solution with 1 g of solute dissolved in final volume of 100 ml aqueous solution may also be considered 1% m/m (1 g solute in 99 g water). This approximation breaks down as the solute concentration is increased. For an example, refer to the densities of water-NaCl mixtures (density of water with dissolved NaCl). High solute concentrations are often not physiologically relevant, but are occasionally encountered in pharmacology, where the mass per volume notation is still sometimes encountered. An extreme example is saturated solution of potassium iodide (SSKI) which attains 100 "%" m/v potassium iodide mass concentration (1 gram KI per mL solution) only because the solubility of the dense salt KI is extremely high in water, and the resulting solution is very dense (1.72 times as dense as water).

Although there are examples to the contrary, it should be stressed that the commonly used "units" of % w/v are grams/milliliters (g/ml). 1% m/v solutions are sometimes thought of as being gram/100 ml but this detracts from the fact that % m/v is g/ml; 1 g of water has a volume of approximately 1 ml (at standard temperature and pressure) and the mass concentration is said to be 100%. To make 10 ml of an aqueous 1% cholate solution, 0.1 grams of cholate are dissolved in 10 ml of water. Volumetric flasks are the most appropriate piece of glassware for this procedure as deviations from ideal solution behavior can occur with high solute concentrations.

In solutions, mass concentration is commonly encountered as the ratio of mass/[volume solution], or m/volume. In water solutions containing relatively small quantities of dissolved solute (as in biology), such figures may be "percentivized" by multiplying by 100 a ratio of grams solute per mL solution. The result is given as "mass/volume percentage". Such a convention expresses mass concentration of 1 gram of solute in 100 mL of solution, as "1 m/v %."

Related quantities

Density of pure component

The relation between mass concentration and density of a pure component (mass concentration of single component mixtures) is:

$\rho_i = \rho_i^* \frac{V_i}{V}\,$

where $\rho_i^*$ is the density of the pure component, $V_i$ the volume of the pure component before mixing.

Specific volume (or mass-specific volume)

Specific volume is the inverse of mass concentration only in the case of pure substances, for which mass concentration is the same as the density of the pure-substance:

$\ \nu = \frac{V}{m}\ = {\rho}^{-1}$

Molar concentration

The conversion to molar concentration $c_i$ is given by:

$c_i = \frac{\rho_i}{M_i}$

where $M_i$ is the molar mass of constituent $i$.

Mass fraction

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

$w_i = \frac{\rho_i}{\rho}$

Mole fraction

The conversion to mole fraction $x_i$ is given by:

$x_i = \frac{\rho_i}{\rho} \cdot \frac{M}{M_i}$

where $M$ is the average molar mass of the mixture.

Molality

For binary mixtures, the conversion to molality $b_i$ is given by:

$b_i = \frac{\rho_i}{M_i (\rho - \rho_i)}$

Spatial variation and gradient

The values of (mass and molar) concentration different in space triggers the phenomenon of diffusion.

References

1. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "mass concentration".