Doubts about how to apply Fick's law to gas

Question

According to Fick's law the flux of gas through a membrane depends on the concentration gradient and then on the pressure gradient:

$$\Delta P = P_a - P_b$$

where $P_a$ is the pressure of container $A$ and $P_b$ is the pressure of container $B$, $A$ and $B$ are separated by a membrane of conductance $C_{ab}$.

My question: is Fick's law still applicable if $P_b$ is altered by increasing the temperature (without modifying the density of the gas in $B$)?

Furthermore, assuming three containers in series $A$, $B$ and $C$ separated by membranes of conductances $C_{ab}$ and $C_{bc}$, at the equilibrium (gas flux is the same in the 3-container assembly) $P_b$ is modified, increasing the temperature of $B$. This seems to lead to a paradox as the flux between $A$ and $B$ should decrease as the gradient $\Delta P_{ab}$ decreases, while the flux between $B$ and $C$ increases as $\Delta P_{bc}$ increases..? Why is this reasoning wrong?

Answer 1

Broadly, matter shifts to equalize the chemical potential (or molar Gibbs free energy) $\mu_i\equiv\left(\frac{\partial G}{\partial N}\right)_{T,P}$.

It is convenient to define an activity $a$ such that

$$\mu\equiv\mu_0+RT\ln a,$$

where $\mu_0$ is the chemical potential at a reference temperature and pressure (e.g., STP, for which $\mu_0$ values are often tabulated).

The activity of a gas is called the fugacity. It so happens that for an ideal gas, the fugacity is $$a=\frac{P}{P_0},$$ where $P$ is the partial pressure (the total pressure for a pure gas) and $P_0$ is some reference pressure to make the logarithm argument dimensionless (this reference pressure can be arbitrarily set as long as consistency is maintained).

This gives a broader framework for describing equilibrium and kinetics in cases where the temperature varies (or the materials aren't pure, for example). Instead of $P_\text A=P_\text B$ at equilibrium and the flux $q$ scaling to first order with $q\sim P_\text{A}-P_\text B$, we have $\mu_\text A=\mu_\text B$ at equilibrium and the flux scaling to first order with $q\sim\mu_\text A-\mu_\text B=RT\ln\left(\frac{P_\text A}{P_\text B}\right)$. The coefficient mediating this relationship depends on the material properties and geometry of the membrane, the surrounding conditions, and the gas concentrations (e.g., $q=M_\text{AB}c_\text{AB}\nabla\mu_\text{AB}$, where $M$ is a mobility, $c$ is a gas concentration, and $\nabla$ is the spatial gradient, all at the membrane location).

For constant temperature and small differences in pressure $\delta_\text{AB}=\frac{P_\text{A}-P_\text B}{P_\text B}\ll 1$, we have $\ln\left(1+\delta_\text{AB}\right)\approx \delta_\text{AB}$, recovering Fick's law $q\sim P_\text{A}-P_\text B$.

In your example, changing the pressure or temperature in one of the containers while at steady state could cause a new steady-state flow to be established, which can be estimated using the above framework. I'm not seeing a contradiction or paradox in your prediction. If you increase the pressure of an intermediate container, the flow entering from a higher-pressure container will decrease, and the flow exiting to a lower-pressure container will increase.