Why is concentration used as a proxy for activity instead of mole fraction?

Question

I am trying to understand how the Gibbs energy change of a chemical reaction is computed and I do not understand why concentrations (mol/L) are used as a proxy for activity rather than mole fractions (mol/total mol).

From the Wikipedia article on the chemical potential, in an ideal solution, the chemical potential of the chemical species $i$ is expressed as

$$\mu_i \approx \mu_i^\circ + RT \ln x_i,$$

where $\mu_i^\circ$ is the chemical potential of species $i$ in standard state, $R$ is the gas constant, $T$ is the temperature and $x_i$ is the mole fraction of $i$ in the system. Because real solutions are not ideal, the activity of $i$ ($a_i = \gamma_i x_i$) is used rather than its mole fraction $x_i$ in the calculation of $\mu_i$.

When $T$ and $P$ are constant, the Gibbs energy change of a reaction is formulated as

$$\Delta_r G = \sum_i \mu_i dn_i,$$

where $dn_i$ is the change in the number of particles of $i$ caused by the reaction (its stoichiometric coefficient). Consequently;

$$\Delta_r G = \sum_i \mu_i^\circ \cdot dn_i + \sum_i dn_i \cdot RT \ln \gamma_i x_i.$$

$\sum_i \mu_i^\circ \cdot dn_i$ is equivalent to $\Delta_r G^\circ$, while $\sum_i dn_i \cdot RT \ln \gamma_i x_i$ is equivalent to $RT \ln Q$ in the more usual formulation $\Delta_r G = \Delta_rG^\circ + RT \ln Q$.

Molar concentration (mol of species per volume of solution) are often considered as a proxy for the activity of solute species, while partial pressures are used for gas species (which gives the "mass action ratio" $\Gamma$ instead of the reaction quotient $Q$). While partial pressures are a concept similar to mol fractions (they are fractions of a total), concentrations and mol fractions are not the same quantity and I do not think they scale the same way. So why are $\Delta_r G$ quantities often computed using concentrations instead of mole ratios?

EDIT: Thinking about it, I think I found a simple justification for the use of concentration as a proxy for activity.

Let's consider a fictive reaction $A \leftrightharpoons B$. The chemical potential change associated with this reaction is

$$\Delta_r \mu = \Delta \mu_B - \Delta \mu_A$$

$$\Delta_r \mu = \Delta \mu^\circ_B - \Delta \mu^\circ_A + RT \ln x_B - RT \ln x_A$$

$$\Delta_r \mu = \Delta \mu^\circ + RT \ln \frac{x_B}{x_A}$$

$$\Delta_r \mu = \Delta \mu^\circ + RT \ln Q$$

mole fractions $x_A$ and $x_B$ are respectively $\frac{n_A}{n_{total}}$ and $\frac{n_A}{n_{total}}$ with $n_{total} = n_A + n_B$, so $Q = \frac{n_B}{n_A}$ ($n_{total}$ cancels). Obviously $Q$ has the same value if concentrations ($\frac{n}{V}$) are used since the volume $V$ is the same for both species and then cancels itself. This justification is obvious for a 1:1 stoichiometry. However, with other stoichiometries, either $n_{total}$ or $V$ remains as a factor in $Q$ (to the power $\sum_i \nu_i$, where $\nu_i$ is the signed stoichiometric coefficient of species $i$). I am pretty sure that $n_{total}^{\sum_i \nu_i}$ is not always equal to $V^{\sum_i \nu_i}$; this is where the difference may happen.

Answer 1

The posted notation seems unorthodox if not incorrect.

For the Gibbs free energy of reaction one usually writes

$$\Delta_r G = \left(\frac{d \Delta G}{d \xi}\right) = \sum_i \nu_i\mu_i^\circ + RT\sum_i \nu_i \ln \gamma_i x_i$$ $$ = \Delta _r G^\circ + RT \ln Q$$

where $ \xi$ is the reaction progress, $Q=\sum_i \nu_i \ln \gamma_i x_i$ is the reaction quotient, and one can equate $\Delta _r \mu= \sum_i \nu_i\mu_i $ with $\Delta_r G$. For the reaction $A \leftrightharpoons B$ one would write $\Delta \mu = \mu_B - \mu_A$.

Changing concentration units is ok but a set of units usually implies a choice of thermodynamic reference state. The reference state generally defines the limit in which $\gamma \rightarrow 1$ (the limit of ideal behavior). If you decide to express the reaction quotient Q in terms of molar concentration rather than mole fraction but want to retain the same general form of the above equations, then it becomes necessary to modify both Q and $\Delta _r G^\circ$. Some reference states are better choices than others under certain circumstances, and it is not accurate to say that mole fraction is not used, it is, just not always. For solids or solvents a mole fraction unit is often convenient. When dealing with solutions, and particularly with solutes as a minor component, it is generally more convenient to deal in molal and molar units (the prior being favored), in combination with Henry's law. In addition, at small concentrations of solute the mole fraction, molarity and molality are linearly proportional, differing by a multiplicative constant. For instance, at low concentration we can write the mole fraction of solute as

$$\chi _2 = \frac{n_2}{n_1 + \sum_i n_i} \approx \frac{n_2}{n_1} = \frac{c_2}{c_1} = \frac{c_2}{1000\delta_1/M_1} \approx \frac{M_1}{1000 \delta_1 ^\circ}c_2$$

where 1 and 2 label solvent and solute and $\delta_1 ^\circ$ is the mass density of the pure solvent, $M_1$ its molar mass.