## Activity & Ionic Strength

*Activity vs. Concentration (Non-ideal Solutions)*

Ions in solution interact with each other and with H_{2}O 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 *activity*:

(1) | activity a_{i} = effective concentration ≤ real concentration c_{i} |

In (infinitely) dilute solutions, i.e. at low concentrations c_{i} and at low background salt concentrations, the ionic interactions can be ignored, and we have

(2) | ideal solution: a_{i} = c_{i} |

Usually, solutions are *non-ideal*. Hence, all hydrochemical calculations with *aqion* are based on activities (rather than concentrations).

*Activity Coefficient*

Once we know the concentration of the free ion c_{i} we convert it to the activity a_{i} by the free-ion activity coefficient γ_{i}:

(3) | a_{i} = γ_{i} c_{i} |
( activity = γ_{i} × concentration ) |

In the very limit of infinitely dilute systems the activity coefficient becomes 1:

(4) | ideal solution: γ_{i} = 1 |

γ_{i} corrects for electrostatic shielding by other ions; hence, γ_{i} depends on the *ionic strength* (i.e. the concentration of electrical charge). There are several approaches to calculate the activity coefficients.

*Ionic Strength*

The ionic strength of a solution is a function of the concentration of all ions present in a solution:

(5) | \(\large I = \frac{1}{2} \, \sum\limits_{i}z_{i}^{2} \, c_{i}\) |

Here, c_{i} and z_{i} are the molar concentration and the charge of ion i. The sum is taken over all ions in the solution. Due to the square of z_{i}, multivalent ions contribute strongly to the ionic strength. [Note: In literature the ionic strength, *I*, is also abbreviated by the Greek symbol μ.]

*aqion* displays the ionic strength of each aqueous solution in the output tables. For comparison: Typical ionic strentgths of natural waters are

surface water | I = 0.001 – 0.005 M |

potable water / groundwater | I = 0.001 – 0.02 M |

seawater | I = 0.7 M |

The ionic strength is related to both EC and TDS, respectively.

*Example: Ionic Strength of CaCl _{2}*

The ionic strength of a CaCl_{2} solution is calculated as follows:

\(\begin{align*} I \ &= \ \frac{1}{2} \left\{ z_{Ca}^2 \,[Ca^{+2}] + z_{Cl}^2\, [Cl^{-}] \,\right \} \\ &= \ \frac{1}{2} \left\{ z_{Ca}^2\, [CaCl_2] + z_{Cl}^2 \,2\, [CaCl_2] \,\right \} \\ &= \ \frac{1}{2} \left\{ 2^2 \,[CaCl_2] + (-1)^2 \,2\, [CaCl_2] \,\right \} \\ &= \ \ 3 \ [CaCl_2] \\ \end{align*}\) |

Here, rectangular brackets [..] symbolize molar concentrations. Please note the stoichiometric factor 2 of Cl^{-} in the second line.

Based on this equation, a 0.5 molar CaCl_{2} solution has an ionic strength of I = *3 × 0.5 M = 1.5 M*.

[You can check this result with aqion: Click on *New* and activate checkbox *Mol*, then enter Ca = *500 mM* and Cl = 1000 mM, then click on *Start*. In the output table, row “ionic strength” you will find: *1.5 mol/L*.]