thermodynamics - How to measure quantities at standard state when this state is a hypothetical one?

Consider the reaction

$$\ce{A(g) + H2O(l) <=> B(aq) + C(g)}$$

If all gases and solutions were ideal, the standard state would be pure water, the solute B at a concentration of 1 M, the gases A and C each at a pressure of 1 bar, and the system at a pressure of 1 bar.

For non-ideal gases and solutions, it would be the hypothetical state where all the activities are 1 (i.e. equal to the activity of an ideal gas at 1 bar, activity of a solvent with infinitely diluted solutes, and equal to the activity of a solute behaving ideally at a concentration of 1 M).

This hypothetical state can be quite different from a real state. How can the pressure be 1 bar while two gases are at a partial pressure of 1 bar each? What if the solute B is barely soluble, how do you get an activity of 1? How can you have pure water when it is acting as a solvent?

So how to you measure quantities like $\Delta_\mathrm{r} G^\circ$ when the standard state is a hypothetical one?

Answer

The standard states are limiting states. In these limits you can write the chemical potential of the component as

$\mu = \mu^\circ+RT\log(x)$

where $x$ is a measure of the concentration of solute (for Henry's law, where the standard state is a hypothetical state related to the behavior as $x \rightarrow 0$) or solvent (for Raoult's law, where the standard state is approached as $x \rightarrow 1$). In other words, in the limiting states the activity goes to 1. The meaning of $\mu^\circ$ depends on the limiting law (hypothetical with properties of an infinitely dilute solution or pure solvent).

A standard state can be hypothetical (as may happen when Henry's law is invoked), but it can also be real (as in Raoult's law, for a pure solvent). It is a state in which the sample behaves ideally as defined by the above equation, in the limit of infinite dilution or of a pure solvent. This limiting behavior provides a convenient and predictable reference, like setting the potential energy to zero when particles are separated by an infinite distance. In the case of Raoult's law, the standard state is the (not particularly mysterious) pure substance. In the case of Henry's law, the reference state is found by extrapolating behaviour at infinite dilution to the standard concentration (1 molal or 1 M, conventionally). This may be a hypothetical state.

Using reference standard states (at 1 bar of course) can simplify combining or comparing data for different substances or conditions. Of course when considering real solutions, activities may diverge from concentrations and activity coefficients deviate from unity.

For a mixture of substances, for each component you can describe individual thermodynamic properties relative to the ideal standard states. By ideal it is also meant that individual solutes do not interact.