# Doubts about how to apply Fick's law to gas

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?

## 1 Answer

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.

• Many thanks for your answer. About the 3 containers, starting from a stable flux, if on one side the flux decreases and on the other side increases, there is no stable flux. Is that possible?
– odis
Commented Jul 9, 2022 at 19:15
• Agreed, unless you're changing the pressure of the middle container by adding or removing the balance through a separate orifice, of course. Commented Jul 9, 2022 at 19:17
• I think we'd want to specify what the boundary conditions are—i.e., what's keeping the pressure in A higher and the pressure in C lower. Commented Jul 9, 2022 at 19:29
• I was actually thinking of changing the pressure of the middle container acting only on its temperature. In principle the new fluxes will tend to deplete the container B and then to eastablish a new flux with fewer particles in the B container. Is this interpretation correct?
– odis
Commented Jul 9, 2022 at 19:37
• All else being equal (e.g., the temperature dependence of the membrane mobilities to the gas), that seems intuitive, yes. Commented Jul 9, 2022 at 21:21