Argon diffuses through a high-density polyethylene (HDPE) sheet \(40 \mathrm{~mm}\) thick at a rate of \(4.0 \times 10^{-7}\left(\mathrm{~cm}^{3} \mathrm{STP}\right) / \mathrm{cm}^{2} \cdot \mathrm{s}\) at \(325 \mathrm{~K}\). The pressures of argon at the two faces are \(5000 \mathrm{kPa}\) and \(1500 \mathrm{kPa}\), which are maintained constant. Assuming conditions of steady state, what is the permeability coefficient at \(325 \mathrm{~K}\) ?

Fick's First Law is a principle that describes the relationship between the diffusion flux and the concentration gradient within a medium. It's an essential concept for students delving into the field of material science and physics. In simplified terms, it states that the rate of transfer of a substance through a medium is directly proportional to the concentration gradient present in that medium.

The law can be mathematically expressed as:
\[ J = -D \frac{dC}{dx} \]
where:

• \(J\) is the diffusion flux (amount of substance per unit area per unit time),
• \(D\) is the diffusion coefficient (a property of the system that quantifies the diffusion mobility),
• \( \frac{dC}{dx} \) represents the concentration gradient across the distance \(x\).

In the case of argon diffusing through HDPE, our diffusion rate \(J\) is given, as well as the pressures at two points which help us determine the concentration gradient, a crucial aspect we need to plug into the equation to determine the system's permeability coefficient.

Diffusion in polymers is notably different from diffusion in gases or liquids due to the complexity and variability of polymeric structures. As students of chemistry and materials science will learn, polymers can be semi-crystalline, amorphous, or a combination of both, impacting the way molecules pass through them.

Polymers have tortuous pathways and varying degrees of free volume which can significantly influence the diffusion coefficient \(D\). In the example involving argon and HDPE, while gas diffusion is straightforward, determining how argon atoms move through the solid HDPE sheet involves understanding the complex interactions within the polymer matrix. Factors such as temperature, molecular size of the diffusing species, and the nature of the polymer itself all play a critical role in how diffusion occurs. The steady-state diffusion in such systems indicates that the diffusion rate has reached an equilibrium and is constant over time.

Steady-state diffusion is a term that denotes the diffusion process after it has reached equilibrium. Meaning, the flux of material passing through the medium—be it a membrane, a solid, or any other barrier—does not change with time.

In our problem, argon diffuses through an HDPE sheet at a constant rate, and pressures on each side of the sheet are maintained. This implies the system has achieved a steady state, a key assumption for using Fick's First Law to compute the permeability coefficient. Understanding the conditions necessary for steady-state diffusion is fundamental for students since it simplifies the calculations of diffusion problems and is often a condition seen in industrial applications.

The concentration gradient is a vital concept in understanding diffusion processes. It refers to the variation in the concentration of a substance across a region. In the context of diffusion, it is the driving force that determines the direction and rate at which particles spread out.

To put it simply, molecules tend to move from areas of higher concentration to areas of lower concentration, trying to reach a state of equilibrium. The steeper the gradient, the faster the diffusion rate. In our HDPE sheet problem, by calculating the pressure difference and subsequently the concentration gradient across the sheet, we determine the magnitude of the driving force for argon atoms to pass from one side of the HDPE to the other. Grasping the significance of concentration gradients allows students to predict how substances will behave when encountering different environmental conditions.