## Concentration Units

The word “concentration” is so often used in chemistry that we sometimes overlook its real meaning and the potential pitfalls. The preferred units for concentrations in chemistry are moles per liter solution, i.e. molarity M (as an abbreviation for mol/L). This statement captures the essence, but there is more to say.

Four types of concentrations will be discussed:

Molarity vs. Mass Concentration

Molar concentrations (molarity) are defined as the number of moles of a substance i per liter of the solution, while mass concentrations refer to the mass of that substance:

 (1) molarity: $c_i \, = \,\dfrac{moles\ of\ substance\ i}{\textrm{}volume \ of\ solution}\, =\, \dfrac{n_i}{V_S}$ in $\left[ \dfrac{\textrm{mol}}{\textrm{L}}\right]$ (2) mass concentration: $c_i^{m} \,=\, \dfrac{\textrm{}mass\ of\ substance\ i}{volume\ of\ solution}\, =\, \dfrac{m_i}{V_S}$ in $\left[\dfrac{\textrm{g}}{\textrm{L}}\right]$

Each concentration type has its own domain. While stoichiometric calculations are based on molar concentrations, analytical data (measured in the lab) are usually provided in mass concentrations. Both quantities are related by:

 (3) cim = Mi · ci with   Mi = molar mass of substance i

Mi is the mass of one mole of substance i; it is a well-known, tabulated quantity (unit: g/mol).1 So it’s quite easy to jump between molar and mass concentrations. [To recapitulate: One mole represents NA entities (atoms, molecules, ions, electrons), where NA = 6.02·1023 mol-1 is the Avogadro constant.2 The mole is a SI base unit.]

[Note. Instead of mol/L and g/L the program outputs concentrations in finer-scale units: mmol/L (= mM) and mg/L, which are more appropriate for common aqueous solutions (e.g. natural waters).]

Limitations. Irrespective of all advantages molar concentrations have the concept of molarity alone is insufficient to describe all phenomena in hydrochemistry:

• Charge balance calculations require equivalent concentrations.
• From physical point of view, molality is the more fundamental concept (due to its strict temperature independence).
• In non-ideal solutions molar concentrations have to be replaced by ion activities.

Normality vs. Molarity

Normality, or equivalent concentration differs from its cousin molarity in that it refers to equivalents (eq) instead of moles. Equivalents are something like reactive units, i.e. the number of valences taking place in a reaction. The equivalent of a substance is the number of moles multiplied by its valence z:

 moles: n number of entities (molecules, ions, electrons) equivalents: neq = z · n number of reactive units

In this way, the equivalent concentration (normality) provides information about the number of reactive units per liter solution (in eq/L). Mathematically, it is the molar concentration ci multiplied by zi (i.e. an integer ≥ 1):3

 (4) normality: $c_{i}^{eq} \, = \, z_i \cdot c_i$ in $\, \left[\dfrac{\textrm{eq}}{\textrm{L}}\right]$

Thus, to calculate the normality you should know two things: the exact molarity and the valence zi which is related to the stoichiometry of the reaction the substance undergoes. The same solution or substance can have different normalities for different types of reaction (acid-base reaction, redox reaction or precipitation).

Example: 1M sulfuric acid solution is 2N for acid-base reactions (z = 2), but it is 1N in the reaction of barium sulfate precipitation (z = 1). On the other hand, hydrochloric acid is 1N regardless of whether it is used for neutralization or chlorides precipitation.

Equivalent concentrations are used in charge-balance calculations, where the symbol zi refers to (the absolute value of) the electrical charge of ion i. The conversion table between meq/L (= 10-3 eq/L) and mM (= 10-3 mol/L) is simple:

 monovalent ions (zi = 1) 1 meq/L = 1.00 mM divalent ions (zi = 2) 1 meq/L = 0.50 mM trivalent ions (zi = 3) 1 meq/L = 0.33 mM

Molality vs. Molarity

The drawback of molarity (molar concentration defined in 1) is that it depends on temperature: When temperature rises volumes increase and, hence, molar concentration decreases. In contrast to the volume of solution that enters 1, masses are independent of temperature T. Thus, replacing the volume by the solvent mass yields a concentration quantity which doesn’t depend on temperature:

 molarity: moles per liter solution (depends on T) molality: moles per kilogram solvent (independent of T)

Mathematically, molality or molal concentration is defined as 4

 (5) molality: $b_i \, = \,\dfrac{moles\ of\ substance\ i}{mass \ of\ solvent}\, =\, \dfrac{n_i}{m_{solv}}$ in $\left[ \dfrac{\textrm{mol}}{\textrm{kg}}\right]$

In hydrochemistry the solvent is water. Therefore, molality is often expressed in units of [mol/kgw], where kgw = kg water. Principally, molality (given in mmol/kg) has the same dimension as the inverse molar mass, namely [g/mol]-1 = [kg/mmol]-1.

Given a solution with N solutes (i.e. substances or species that run from i=1 to N) there is a strict relation between molarity ci and molality bi:

 (6) $b_i \, = \,\dfrac{c_i}{\tilde{\rho}_S - c_i M_i}\ \ \Longleftrightarrow \ \ c_i \, = \, \dfrac{b_i\, \tilde{\rho}_S}{1 + b_i M_i}$ with $\tilde{\rho}_S = \rho_S - \sum\limits^N_{j\neq i} c_j M_j$
 where bi molality of species i ci molarity of species i Mi molar mass of species i ρS density of the solution

The derivation of 6 is presented in the Appendix. For the special case of a solution with only one single substance, 6 simplifies drastically:

 (7) $b \, = \,\dfrac{c}{{\rho}_S - c M}\ \ \ \Longleftrightarrow \ \ \ c \, = \, \dfrac{b\, {\rho}_S}{1 + b M}$

These are the standard equations you may usually encounter in textbooks.

An alternative formulation of the general case in 6 can be obtained from A4 in the Appendix:

 (8) $b_i \, = \,\dfrac{c_i}{\tilde{\rho}}\ \ \ \Longleftrightarrow \ \ \ c_i \, = \, \tilde{\rho}\cdot b_i$ with $\tilde{\rho} = \rho_S - \sum\limits_{j} c_j M_j$

While the molar/molal conversion formula looks so simple, the density $\tilde{\rho}$ that enters this equation is neither the solvent density nor the solution density. However, when the amount of dissolved species ni is very much smaller than the amount of the solvent H2O then $\tilde{\rho}$ approaches the density of pure water:

 (9) $\tilde{\rho}\ \approx \ \rho_{w}$ when $c_i \ll c_{H2O}$ for all i

This assumption is sufficiently fulfilled for natural waters where cH2O = 55.5 M and the molar concentrations of dissolved species are in the order of 10-3 M, which is 4 to 5 orders of magnitude smaller. In other words, for common aqueous solutions, the molality and molarity of dissolved species have the same numerical value:

 (10) dissolved species: ci $\left[ \dfrac{\textrm{mmol}}{\textrm{L}}\right]$  =  ρw · bi  =  1.0 $\dfrac{\textrm{kg}}{\textrm{L}}$ · bi $\left[ \dfrac{\textrm{mmol}}{\textrm{kg}}\right]$

Important. In aqion, all numerical calculations are based on molalities.5 Even the output concentrations are in mmol/kgw, but in the display we always write mmol/L – using the approximation in 10, i.e. keeping the numerical value unchanged. So, please read “mmol/L” as “mmol/kgw” in the output tables.

Why this caprice? Because most people are more familiar with mmol/L than mmol/kgw.

Ion Activity vs. Molarity

Molar concentrations (molarities) are measurable quantities. But molarities (or molalities) are insufficient to perform accurate hydrochemistry calculations that are based on the Law of Mass Action algorithm.

This is because ions in solution interact with each other and with H2O molecules. In this way, ions behave chemically like they are less concentrated than they really are (or measured). This effective concentration, which is available for reactions, is called ion activity. The conversion to ion activity looks simple:

 (11) ion activity: ai   =   γi · ci

but it requires quite sophisticated models to calculate the activity coefficient γi. Only in the special case of ideal solutions (i.e. waters with low ionic strength), γi = 1 and we have ai = ci.

Notation. To keep the notation straight (i.e. to get rid of too much indices) molar concentrations and ion activities are abbreviated by square and curly braces:

 molar concentration: ci  =  [i] ion activity: ai  =  {i}

Appendix

The aim is to derive the molal/molar conversion formula in Eq. (6). This will be done for the general case of a N-solute system (think of an aqueous solution with N dissolved species).

The mass of the solution is the sum of the mass of the solvent (e.g. pure water) plus the masses of all solutes (e.g. dissolved species):

 (A1) $m_S \, = \, m_{solv} + \sum\limits^N_{i=1} m_i \ = \ m_{solv} + \sum\limits^N_{i=1} n_i M_i$

On the right-hand side, the masses are replaced by their molar amounts via mi = ni Mi, where Mi is the molar mass. Now, we divide 1 by 5, and enter A1 into the numerator:

 (A2) $\dfrac{c_i}{b_i} \, = \, \dfrac{m_{solv}}{V_S} \, = \, \dfrac{m_S - \sum_i n_i M_i}{V_S} \ = \ \dfrac{m_S}{V_S} - \sum\limits_i c_i M_i$

In the last equation we used the definition of molarity, ci = ni/VS. Furthermore, the ratio of solution mass to solution volume represents the density of the solution, ρS = mS/VS. This yields

 (A3) $\dfrac{c_i}{b_i} \, = \, \rho_S - \sum\limits^N_{i=1} c_i M_i \ = \ \tilde{\rho}_S - c_i M_i$ with $\tilde{\rho}_S = \rho_S - \sum\limits^{N}_{j\neq i} c_j M_j$

Here we extracted the term ciMi from the sum and inserted the remaining part of the sum into $\tilde{\rho}_S$. Finally, re-arrangement of A3 for bi or for ci yields the result shown in 6 above.

An alternative result is obtained when we do not extract the term ciMi from the sum:

 (A4) $\dfrac{c_i}{b_i} \, = \, \tilde{\rho}$ with $\tilde{\rho} = \rho_S - \sum\limits^{N}_{j = 1} c_j M_j$

But now we have to recognize that the density itself depends on ci, i.e. $\tilde{\rho} = \tilde{\rho}(c_i)$.

Remarks & Footnotes

1. Once the elemental composition of substance i is known (stoichiometric formula), the molar mass M(i) can be directly obtained from the atomic mass in the periodic table.

2. NA is the number of atoms in 12 g of 12C.

3. The reciprocal of z is also known as ‘equivalence factor’ feq = 1/z.

4. Traditionally, molarity is abbreviated by M and molality by m. To avoid confusion with other symbols (M as mol/L, M as molar mass, m as mass) we prefer the small-letter symbols c (for molarity) and b (for molality).

5. More precisely: Molalities are the input from which ion activities are calculated. The latter enter the law-of-mass-action numerical solver.