# Calculate concentrations in a system made up of molecules in order to calculate ionic strength

I understand how to calculate ionic strength where there are concentration data in molality or molarity units available and where the system is made up of only two components. However, I have been struggling for a few days with what is probably a very simple problem: how to calculate ionic strength where there are more than two components and where the concentrations first have to be calculated because the system is made up of molecules. I'm not a chemist, but need ionic strength for another calculation.

In one kg of water there are 55 moles (1000 g / 18 g mol-1) of water molecules. To simplify calculations, my system has 55 water molecules. As an example, let's say it also has one Na atom, three Cl atoms, and one Ga atom.

I think that this (below) is the expression to use, where z is charge and m is the molal concentration of species i. I want to use a molality-based expression because I want to do calculations at a range of temperatures and expect that it would not be right to use molarities, where temperature can affect volume.

$$\large I_\mathrm{m}=\frac 12 \sum_i z_i^2 m_i$$

So it looks like I only need to know the charges and m for all of my species. But here's where I'm stuck on something simple. How do I calculate m? Is it true that in my system, there would be one mole of Na and Ga and 3 moles of Cl? So m for Na (for example) would be 3 moles/1 kg water?

Another thing I'd like to check that I haven't been able to find out online is whether the "1/2" in the expression is only valid for two-component systems. Is this expression valid for my system? Thank you for any guidance.

Update: I have realized that I have been confused. I want to find out which reactions happen in my system at different temperatures. As part of that, I want to find out the activity coefficients of the various species involved and the standard state free energies. This requires knowing the ionic strength.

I was confused by something I read online and basically thought that my ionic strength calculation for each temperature would only involve the species participating in the reactions happening at each temperature. But I now realize (hope) that actually, the ionic strength is the same at ALL of the temperatures I'm dealing with because calculating ionic strength means accounting for all species present in the system.

So if I'm now on the right track, my ionic strength would be 6.5 mol kg-1 based on using the equation above with concentrations of Ga, Na, and Cl of 1, 1, and 3 mol kg-1, respectively, and charges of 3, 1, and -1.

• (1) Yes you are right. $m_i$ is the molarity (or molality) of ion species $i$. (2) The 1/2 is because there are cations and anions irregardless of the relative numbers. (3) You do realize that this only partially works. The gist is that the moles of solute should be small compared to the moles of solvent. (4) The $I_m$ means that you are calculating the ionic strength of compound $m$. You can have solutions with more than one solute too. – MaxW Jul 4 '18 at 19:49
• Thank you, Max W; that is helpful. Please forgive my ignorance of your point (3). I don't understand what you mean. Are you saying that in a system with more Na, Ga, and Cl, there would be interactions like repulsion (for example) that would complicate things? Do you have any thoughts on whether the expression would lead to valid ionic strength results for my system? – Ant Jul 4 '18 at 19:56
• The equation defines ionic strength. The problem comes in trying to use that to predict some behavior like the solubility product of a salt. The "reactivity" of a ion doesn't depend on its molarity (or molality) but rather on its chemical activity. So the overall ionic strength of the solution gives a means to correct for the activity of an ion from a particular salt that is dissolved or in a mixture of salts. See Debye–Hückel theory. But is doesn't work for concentrated solutions. So AgCl has a $K_{sp}$ but NaCl doesn't. – MaxW Jul 4 '18 at 20:09
• Thank you for that. I'm going to have to take time to read what's at your link in order to understand your final sentence. I do need to use the ionic strength to calculate activity coefficients, so it sounds like I need to consider whether my solution is too concentrated to do that. If anyone has any further advice/simplification, I'm all ears. Thanks a lot, Max. – Ant Jul 4 '18 at 20:27
• The point that I was trying to make is that you could use ionic strength to make corrections for the $\ce{a_{Ag+}}$ and $\ce{a_{Cl-}}$ so that the $K_{sp}$ was still useful in a KCl solution, or a CaCl2 solution. However trying to theoretically figure out how much NaCl will dissolve in a KCl solution, or a CaCl2 solution is hopeless. – MaxW Jul 4 '18 at 20:40