It is important not to lose track of the basic principles of activity amongst all the detailed derivations of the equations. Thus it is appropriate to provide here a brief summary of the salient points of the previous two pages:

Activities are an adjusted form of the mole fraction (or molality or concentration) of a species in a solution, be it the solvent or solute. The adjustment is required to take into account the interactions between molecules in solution, which can alter the extent to which they are free to participate in reaction.

Activities are commonly written as an activity coefficient, γ, multiplying the mole fraction (or molality) that the activity is to replace. These coefficients change with the composition of the solution.

Solvents

For a solvent which is the component in excess in a two species mixture, we define the standard state as the pure liquid. The chemical potential of the pure liquid is given the symbol μ*. The chemical potential of the solvent at any composition is given by:

In this instance, the activity is defined in two ways. It is used in calculations as:

and it must be noted that as the mole fraction tends to one (as the purity of the liquid increases) the activity coefficient also tends to one (the deviations from ideality in the solvent's behaviour decrease, and its behaviour becomes more in line with Raoult's Law).
The activity of a solvent may be measured experimentally using the other definition:

Solutes

For a solute there are two different approaches.
The first method defines the standard state of the solute as a hypothetical state of the pure solute. This does not mean it is defined as the pure state of the solute, but as a state with different properties to the pure state (eg a different chemical potential) that behaves as a pure liquid in certain situations. The chemical potential of this standard state is given the symbol μ^#. The chemical potential of the solute at any composition is now given by:

The activity is again defined in two ways. The first definition is the same as for the solvent case:

but for the solute, the activity coefficient tends to one as the mole fraction tends to zero. i.e. the more dilute the solution, the more in accord with Henry's Law is the behaviour of the solute.
The activity of the solute may be measured from its second definition:

where K is the Henry's Law constant for the solute.

The second method for dealing with the solute defines the standard state as a hypothetical state of the solute when its molality is bº (1 mol kg^-1). The chemical potential of the solute in this standard state is given the symbol μº. The chemical potential of the solute at any molality is now given by:

The activity is defined in terms of the molality, b, of the solution:

where bº is the molality of the standard state (1 mol kg^-1). Note that, in this definition, the activity coefficient tends to one as the molality of the solution tends to zero (i.e. as the solution gets more dilute, its behaviour conforms more with Henry's Law).