What is the true motivation behind diffusion of components in solution?

Question

I am very well aware that chemical potential or concentration is typically said to be the reason for molecular diffusion but consider a case, We have a cube filled with gas containing minor amount of nitrogen, say nitrogen is a solute and other molecules are solvents, opposite faces of cube are maintained at different temperatures, concentration of gas should be less near the hotter face and should be more near the cooler face. Since we have concentration gradient of nitrogen available, by ficks law it's obvious that diffusion(i mean net diffusion) should be observed but this isn't the case*.

Hence the question what is the true(it's just an adjective and doesn't ask for opinion) motivation behind diffusion of components in solution. I am interested in macroscopic treatment of subject.

*Here we neglect the coupling effects of concentration and temperature gradients on diffusion.

Answer 1

We tend to more likely observe situations that are more likely to occur. Agreed? That's arguably a tautology.

All particles in the matter around us are continually in motion by virtue of having a positive temperature; thus, they tend to explore different arrangements.

The number of available arrangements is quantified by the thermodynamic entropy.

The very first statement above can be stated more precisely by the Second Law of Thermodynamics: We tend to see the entropy $S$ of an isolated system increase. (With increasing system size and time scale, as for the gas experiments you describe, the tendency becomes absolute: $dS>0$, and $d^2S<0$ at equilibrium, referring to a maximum point.)

This gives us Fick's laws for simple, isolated systems of ideal gases, for example: For a given conceptual plane inserted into the gas, the flux corresponds to the molecules passing through the plane due to their random thermal motion. Concentrations even out because this maximizes entropy (there are more arrangements for particles that can be in any region of a container relative to only a subset of the container). Steeper gradients even out more quickly because there are more particles on one side gradient tending to move randomly to the other side.

But no system is truly isolated; systems exchange energy (and mass) with the universe around them. This introduces many ways in which Fick's laws can be violated.

• The gas may condense into a liquid, producing a sharp concentration gradient at the phase interface. The latent heat that's released tends to heat the surroundings, increasing their entropy. Entropy maximization is satisfied even though Fick's laws aren't.

• The gas may selectively bond with (in this case, nitridize) different regions of the container walls. This again releases a heat of reaction that compensates for the configurational entropy reduction. The chemical potential is often a more appropriate parameter than the concentration in all but the simplest cases—i.e., it's the chemical potential of matter that tends to even out rather than the concentration. Fick's law's can't accommodate this aspect and indeed often fail when chemical bonding is possible.

• The gas may settle in a gravity field, with an emergent barometic pressure gradient. The field has done work on the gas and increased its temperature and thus its entropy. Entropy maximization is still satisfied even though Fick's laws aren't.

• Alternatively, we could push ionized molecules to one size of the container using an electric field—another example of applied work.

• As in your example, we may drive heat transfer through the gas, rarefying the hotter area relative to the cooler area. Entropy is generated during this heat transfer process, as it is for any spontaneous process. The entropy generation compensates for the appearance of concentration gradients.

Thus, you may consider using the broader framework of entropy maximization to analyze cases in which Fick's laws are violated. I would call this a reasonable "true motivation."

Answer 2

Firstly, the concept of diffusion in chemistry is primarily driven by the tendency of particles to move from regions of higher concentration to regions of lower concentration, seeking to achieve a state of equilibrium. This movement is indeed often described by Fick's laws of diffusion.

Now, consider your scenario: a cube filled with a gas, predominantly a solvent with a minor component of nitrogen acting as a solute. You have ingeniously set up a situation where there is a temperature gradient across the cube — one face is hotter than the other. This setup leads to an interesting observation. While we might intuitively expect a concentration gradient for nitrogen due to this temperature difference, it appears that this is not the case. This observation leads us to question the "true motivation" behind the diffusion of components in a solution, or in this case, a gaseous mixture.

The key here lies in understanding how temperature gradients affect gas behavior. In a gaseous system, temperature is a crucial factor influencing particle movement. Hotter temperatures impart more kinetic energy to the gas molecules, causing them to move more vigorously. This increased movement leads to a decrease in the gas's density near the heated side of the cube. As a result, there is a lower concentration of all gas components, including nitrogen, near the hotter face of the cube.

However, the twist in your scenario is the observation that despite a concentration gradient, diffusion of nitrogen does not occur as expected. The reason for this lies in the dynamic equilibrium established in the system. While nitrogen molecules do indeed move from a region of higher concentration (cooler side) to a region of lower concentration (hotter side), the overall movement of all gas molecules due to the temperature gradient must also be considered. The increased kinetic energy on the hotter side leads to a more uniform distribution of all gas molecules, counteracting the concentration gradient effect specifically for nitrogen.

In essence, the "true motivation" behind diffusion in your scenario is not solely dictated by concentration gradients, as Fick's laws would suggest, but is significantly influenced by the temperature gradient. This temperature gradient affects the kinetic energy and thus the distribution of all gas molecules in the system, not just the nitrogen. The interplay between concentration gradients and temperature-induced kinetic energy distribution is a delicate and fascinating aspect of thermodynamics and statistical mechanics, highlighting the complexity of real-world physical systems.

Therefore, in systems where temperature gradients are significant, the simplistic view of diffusion-driven only by concentration gradients must be expanded to include the effects of thermal energy distribution and the resulting dynamic equilibrium of all components in the system.

Answer 3

If the gas starts in an equilibrium state then the initial condition is a uniform concentration. No concentration gradients.

To eliminate heat transfer by mass transport (convection), I'll add the restriction that this experiment occurs in a gravity-free environment so convection doesn't occur.

Finally, I will allow the cube to expand with expanding gas. Constant pressure.

Now a temperature gradient is created by heating one surface and keeping the other face at equilibrium temperature. This gradient is not created instantaneously but develops over time. Gases near the hotter surface acquire additional kinetic energy which increases the average distance between gas molecules but does not induce flow ( its a more vigorous vibration of molecules). This lowers the density of the gas near the surface and creates a concentration gradient whose boundary conditions are the hot surface and the equilibrium boundary (where the gas is still at equilibrium temperature).

The analogy is a growing depletion layer where the layer is thermally depleted of analyte.

As soon as the depletion layer begins to grow, Diffusion begins and analytes move into the depletion region at a rate governed by Fick's equations. The rate of diffusion is dependent on temperature so the rate of diffusion will be the highest at the hot surface. If analyte diffuses in as fast as the depletion layer grows, then any small concentration gradients are quickly neutralized by on going diffusion. Note that the boundary conditions for diffusion move towards to colder face as concentration gradients are neutralized due to the temperature difference.

When the temperature gradient is finally established and stable, the gas concentration is uniform despite the continuing thermal conditions. The density gradient remains and is constant but the concentration reaches uniformity by the time this steady state is achieved.

Diffusion distributes the number of analyte molecules so as the volume expands due to heating, the number of analyte molecules increases keeping the concentration constant.

The motivation behind diffusion is the ability to achieve equilibrium without the input of additional energy. Without diffusion equilibrium would not be a natural state. It would always require additional energy to drive a system into equilibrium. Without diffusion the 2nd Law doesn't work since heat tranfer alone does not induce mass transport.

Answer 4

When there is a zero net flux of particle movement/diffusion, then there is a balance between the amount of particles that move from two opposite directions. These two amounts are not only determined by the particle density, but also by the particle speed.

On the warmer side the particles are moving faster than on the colder side. This faster movement will increases the amount of particles per second that will pass through some area, and (albeit the lower density) the same amount of particles move from hot side to cold side, as particles are moving from cold to hot.