# When can I use concentrations instead of activities in equilibrium calculations?

To calculate an equilibrium constant, it is said that you should actually use activities $$a_i$$ instead of concentrations $$c_i$$. But it is also said that within a certain range, you can use concentrations instead. This is also what is usually done because it's easier.

So what is this range? When can I safely use concentrations and when should I definitely switch to using activities?

To calculate any equilibrum constant we need to be sure of the fugacity of each species. Note that fugacities are defined for every common state of matter (gas, liquid and solid), although they are most commonly used for gases. Look at the end of this answer for more details.

The activity of the coumpound $$i$$ written $$a_i$$ is defined as:

$$a_i=\exp\left(\frac{\mu_i -\mu_i^{\circ}}{RT}\right)$$

where $$\mu_i$$ is the chemical potential, which depends on the fugacity.

However this expression can be hard to use if you have not a strong idea of what you are doing because the are a lot of different ways to calculate things, depending on the formulas and laws or theories used.

What you must remember all the time

Activity can also be defined as $$a_i=\gamma_i \cdot \frac{X_i}{X_i^\circ}$$

where $$X_i$$ can be a concentration, a partial pressure, a molar fraction, a molality, and so on, and where $$\gamma_i$$ is the activity coefficient. Be careful; this coefficient depends on what $$X_i$$ is.

The role of $$\gamma_i$$ is to correct for the interactions between each constituent of the mixture.

If you are in the ideal regime, which means that the amount of your solute is almost negligible with respect to the entire mixture, then $$\gamma_i \approx 1$$. Only then can you use $$X_i$$ in place of $$a_i$$.

In general, the determination of this coefficient is not quite simple, especially for ions: see the example here for hydrogen ion.

Ionic solutions

Electrolyte solutions (e.g. a solution of $$\ce{NaCl}$$) can be modelled by Debye–Hückel theory. There are different versions of this model depending on the precision you want. One of the most used is given in the paper about hydrogen ion:

$$\log(\gamma_i)=-\frac{A_m z_i^2\sqrt{I_m}}{1+\sqrt{I_m}}$$

where $$I_m$$ is the molar ionic strength, given by:

$$I_m=\frac{1}{2}\sum_i z_i^2m_i$$

where $$m_i$$ is the molality of $$i$$.

To answer the other part of your question, whether your system is behaving "ideally" or not ultimately depends on the precision you want.

For example if you have a reaction constant expressed as:

$$K=\frac{a_\ce{A}^2 a_\ce{B}^3}{a_\ce{C}}=\frac{\gamma_\ce{A}^2 \gamma_\ce{B}^3}{\gamma_\ce{C}} \cdot \frac{(c_\ce{A}/c^\circ)^2 (c_\ce{B}/c^\circ)^3}{c_\ce{C}/c^\circ}=\frac{\gamma_\ce{A}^2 \gamma_\ce{B}^3}{\gamma_\ce{C}} K^\circ$$

which corresponds to the reaction $$\ce{C -> 2A + 3B}$$, you will have a large error if you assume you are under ideal conditions when you aren't.

For example if you assume ideality in calculating the $$\mathrm{pH}$$ of $$\pu{5 mol/L}$$ of $$\ce{HCl}$$ solution, like I did here, the answer will not be the same as if you were to consider the activity of the ion, like this.

So at least for strong acid, if the concentration is much than $$\pu{1 mol/L}$$, you must use activity coefficients. Depending on your problem you'll need to think by yourself and do calculations with and without assuming ideality. If you are unsure as to whether your solution can be considered as being ideal, it's good to do a check (and feel free to use a computer in hard cases).

I would add, as an example, if one of your compounds is 10% of the solution in mass, the solution must no longer be considered as ideal. But, as said earlier, the limit of what is ideal is fixed by you, depending on the precision you want.