# 24.7: Activities of Nonideal Solutions

As seen before activities are a way to account for deviation from ideal behavior while still keeping the formulism for the ideal case intact. For example in a ideal solution we have:

\[μ^{sln} = μ^* + RT \ln x_i\]

is replaced by

\[μ^{sln} = μ6* + RT \ln a_i\]

The relationship between \(a_i\) and \(x_i\) is often written using an activity coefficient \(γ\):

\[a_i= γ_ix_i\]

### Raoult versus Henry

Implicitly we have made use of Raoult's law here because we originally used

\[x_i = \dfrac{P_i}{P^*_i}\]

In the case of a solvent this makes sense because Raoult's law is still valid in the limiting case, but for the solute it would make more sense to use Henry's law as a basis for the definition of activity:

\[a_{solute,H} ≡ \dfrac{P_{solute}}{K_{x,H}}\]

This does mean that the \(μ^*\) now becomes a \(μ^{*Henry}\) because the extrapolation of the Henry law all the way to the other side of the diagram where \(x_{solute}=1\) points to a point that is not the equilibrium vapor pressure of this component. In fact it represents a virtual state of the system that cannot be realized. This however does not affect the usefulness of the convention.

### Various concentration units

The subscript X was added to the K value because we are still using mole fractions. However Henry's law is often used with other concentration measures. The most important are:

- molarity
- molality
- mole fraction

Both the numerical values and the dimensions of K will differ depending on which concentration measure is used. In addition the pressure units can differ. For example for oxygen in water we have:

- K
_{x,H}= 4.259 10^{4}atm - K
_{cp,H}= 1.3 10^{-3}mol/lit.atm - K
_{pc,H}= 769.23 lit.atm/mol

As you can see K_{cp,H} is simply 1/K_{pc,H}, both conventions are used..

Note that in this case a choice based on Raoult is really not feasible. At room temperature we are far above the critical point of oxygen which make the equilibrium vapor pressure a non-existent entity. Returning to activities we could use each of the versions of K as a basis for the activity definition. This means that when using activities it must be specified what scale we are using. Activities and Henry coefficients of dissolved gases in water (both fresh and salt) are quite important in geochemistry, environmental chemistry etc.

### Non-volatile solutes

A special case arises if the vapor pressure of a solute is negligible. For example if we dissolve sucrose in water. In that case we can still use the Henry based definition

\[a_{solute,H} ≡ \dfrac{P_{solute}}{K_{x,H}}\]

Even though both \(K\) and \(P\) will be exceedingly small their ratio is still finite. However how do we determine either?

The answer lies in the solvent. Even if the vapor pressure of sucrose is immeasurably small, the water vapor pressure above the solution can be measured. The Gibbs-Duhem equation can then be used to translate one into the other. We can use Raoult Law to define the activity of the solvent:

\[a_1 = \dfrac{P_1}{P^*_1}\]

We can measure the pressures as a function of the solute concentrations. At low concentrations

\[\ln a_1 \ln x-1 ≈ -x-2\]

At higher concentrations we will get deviations, we can write:

\[\ln \dfrac{P_1}{P^*_1}=\ln a_1 ≈ -x_2φ\]

The 'fudge factor' \(φ\) is known as the *osmotic coefficient* and can thus be determined as a function of the solute concentration from the pressure data. What we are really interested in is \(a_2\), not \(a_1\):

\[a_2= γ_2x_2\]

Using Gibbs-Duhem we can convert \(φ\) into \(γ_2\). Usually this is done in terms of molalities rather than mole fractions and it leads to this integral:

\[\ln γ_{2,m} = φ – 1 + \int_{m'=0}^m \dfrac{φ – 1}{m'} dm'\]