We have derived an expression for the general form of the chemical potential of any solvent:
Recall that pA* is the vapour pressure of the pure liquid, and pA is the vapour pressure of the substance when it is a component of a solution. For an ideal solution, the solvent obeys Raoult’s Law at all concentrations, so it satisfies this equation:
Note that the standard state of the solvent is the pure liquid.
This is obtained when xA is 1, at which point the logarithm is equal to zero and the chemical potential of the solvent is equal to the chemical potential of the standard state.
When the solution is not ideal (does not obey Raoult’s law at all compositions), we may preserve the form of the last equation by writing:
where aA is the activity of A, an effective mole fraction. It may be viewed as the mole fraction adjusted to take into account the interactions of particles, which alter the extent to which molecules are free to take part in reactions.
By comparison with the first equation, we can see that
which provides a simple way of measuring the activity of a solvent.
We have stated that all solvents obey Raoult’s Law increasingly closely as the mole fraction of the solute approaches zero. This implies that the activity of the solution must approach the mole fraction xA as the mole fraction approaches 1. This convergence is taken into account by the introduction of the activity coefficient, γA, with the following definition:
This is true at all temperatures and pressures. The chemical potential of the solvent is then given by:
Note that the standard state of the solvent, when it is pure, comes when xA = 1 , at which point γA also equals one (from the above definition). The two logarithmic terms in the above expression then vanish, giving us the correct result.
Note all deviations from ideality are contained in the final term, RT ln γA. For an ideal solution, the activity is equal to the mole fraction, which implies that the activity coefficient is one, and hence that this term is zero for an ideal solution. This gives back the equation obtained above for an ideal solution.
For solutes, ideal-dilute behaviour is approached as their mole fraction approaches zero (unlike solvents, which approach ideal behaviour as their mole fraction approaches one).
This necessitates a modification of the calculations. A solute, B, that obeys Henry’s Law has a vapour pressure that is given by the expression pB = KBxB . The chemical potential of B is then given by:
Now, KB and pB* are both characteristic of the solute under consideration, so the second term on the right hand side must be characteristic of the solute, and may be combined with the chemical potential of the pure substance, μB*, to give a new standard chemical potential, denoted μB#:
from which it immediately follows that:
Thus far, we have not considered the possibility of deviation from ideal-dilute (Henry’s Law) behaviour. If we now replace the mole fraction of B with its activity in the equation above, we obtain:
All deviations from ideality are contained within the activity, aB. The activity at any given concentration may be be calculated from the following expression:
which may be obtained from inspection of the preceding equations. Again, in a similar fashion to the solvent, it is convenient to introduce an activity coefficient, γB:
Now all the deviations from ideal behaviour are contained within the activity coefficient.
Since solutes approach Henry’s Law behaviour at low concentrations, it follows that
that is, as the concentration of solute is lowered, the activity coefficient gets closer and closer to one, and the value of the activity gets closer and closer to the value of the mole fraction.
This implies deviations from solute ideality decrease as the concentration is lowered.