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

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


  • $W$ 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.

In any case you should again look up the definition of activity (or relative humidity) in D'Arcy and Watt's paper, because the activity defined for the above relation could be larger than 1.

  • $\begingroup$ 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? $\endgroup$ – Lisa_Clare Jan 18 '17 at 4:49
  • $\begingroup$ @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. $\endgroup$ – Martin - マーチン Jan 18 '17 at 5:12

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