# How to convert the D'Arcy and Watt model from activity to concentration in mol/m^3

The D'Arcy and Watt 1970 model in "Analysis of sorption isotherms of non-homogeneous sorbents" for water adsorption is as follows:

$$W = \frac{KK'a}{1+Ka} + ca + \frac{kk'a}{1-ka},$$

where:

• $W$ is weight of vapour adsorbed per $1~\pu{g}$ of substrate per $1~\pu{g}$ sorbent ($\pu{g/g}$)
• $a$ is the water activity or relative humidity (no units, number between $0$ and $1$ or if relative humidity is used - is percentage)
• rest are constants with no units

I want to use this model to consider water concentration in $\pu{mol m^-3}$ instead of water activity.

How do I convert this?

The general relation between activity (effective concentration) and concentration is as follows (see Wikipedia for more detail): $$a_i = \gamma_i \frac{c_i}{c^\circ}$$ The standard concentration is defined as $c^\circ=1~\pu{mol dm^-3}$. For extremely dilute systems it is possible to just use the concentration instead of the activity (often used as a first order approximation, too): $$\lim_{c_i\to0}\gamma_i\approx1 \implies a_i\to c_i$$ Depending on the accuracy necessary for your model, you might have to look up the specific values.
• Ok thanks thats great! Also, you mentioned that $a$ could be larger than 1. They have defined $a$ as $p/p_0$. How much further can that go above 1? I know relative humidity in extreme cases can go higher than 100% to say 102%, is that what you mean? Commented Jan 18, 2017 at 4:49
• @Eloise It seems that your paper is not using the relative activity which is defined as $$a=\exp\left\{\frac{\mu-\mu^\circ}{RT}\right\}.$$ There is no (mathematical) reason, why there should be an upper limit. If they define it as $a=p/p_0$ and $p_0$ being the total pressure, it can indeed be 1 at maximum. Your question does not provide enough context to answer that. From the definition it is related to food chemistry, but I can't help you there. Commented Jan 18, 2017 at 5:12