The permeability coefficient of a type of small ( gas molecule in a polymer is dependent on absolute temperature according to the following equation: $$ P_{M}=P_{M_{0}} \exp \left(-\frac{Q_{p}}{R T}\right) $$ where \(P_{M_{0}}\) and \(Q_{p}\) are constants for a given gaspolymer pair. Consider the diffusion of water through a polystyrene sheet \(30 \mathrm{~mm}\) thick. The water vapor pressures at the two faces are 20 \(\mathrm{kPa}\) and \(1 \mathrm{kPa}\), which are maintained constant. Compute the diffusion flux [in(cm \(\left.\left.^{3} \mathrm{STP}\right) / \mathrm{cm}^{2} \cdot \mathrm{s}\right]\) at \(350 \mathrm{~K}\) ? For this diffusion system, $$ \begin{aligned} &P_{M_{0}}=9.0 \times 10^{-5}\left(\mathrm{~cm}^{3} \mathrm{STP}\right)(\mathrm{cm}) / \mathrm{cm}^{2} \cdot \mathrm{s} \cdot \mathrm{Pa} \\ &Q_{p}=42,300 \mathrm{~J} / \mathrm{mol} \end{aligned} $$ Assume a condition of steady-state diffusion.

Step by step solution

01

We are given the equation for the permeability coefficient: $$ P_M = P_{M_0} \exp \left(-\frac{Q_p}{R T}\right) $$ where \(P_{M_0} = 9.0 \times 10^{-5} \left(\dfrac{\text{cm}^3 \text{STP}}{\text{cm}^{2} \cdot \text{s} \cdot \text{Pa}}\right)\) and \(Q_p = 42,300 \frac{\text{J}}{\text{mol}}\). We know the ideal gas constant \(R = 8.314 \frac{\text{J}}{\text{mol} \cdot \text{K}}\) and the absolute temperature \(T = 350 \text{ K}\). Using these values, we can calculate \(P_M\). #Step 2: Calculate the driving force#

We are given the water vapor pressures at the two faces of the polystyrene sheet, which are \(20 \text{ kPa}\) and \(1 \text{ kPa}\). In order to compute the diffusion flux, we need to calculate the driving force, which is the difference in water vapor pressure across the sheet. The pressure difference is given by: $$ \Delta P = P_1 - P_2 = (20 - 1) \times 10^3 \text{ Pa} $$ #Step 3: Calculate the diffusion flux#

02

Under steady-state diffusion conditions, the diffusion flux can be calculated using Fick's first law: $$ J = \frac{P_M \cdot \Delta P}{L} $$ where \(J\) is the diffusion flux, \(P_M\) is the permeability coefficient, \(\Delta P\) is the driving force (pressure difference), and \(L\) is the thickness of the polystyrene sheet. We have the values for all these variables, so we can simply plug them into the equation to find the diffusion flux. #Step 4: Compute the final result#

Using the equations and values from the previous steps, we can now calculate the diffusion flux. $$ J = \frac{P_M \cdot \Delta P}{L} = \frac{P_{M_0} \cdot \exp \left(-\frac{Q_p}{R T}\right) \cdot \Delta P}{L} $$ Plugging in the given values, we obtain: $$ J = \frac{(9.0 \times 10^{-5} \frac{\text{cm}^3 \text{STP}}{\text{cm}^{2} \cdot \text{s} \cdot \text{Pa}}) \cdot \exp \left(-\frac{42,300 \frac{\text{J}}{\text{mol}}}{(8.314 \frac{\text{J}}{\text{mol} \cdot \text{K}})(350 \text{ K})}\right) \cdot (20 - 1) \times 10^3 \text{ Pa}}{3 \text{ cm}} $$ Solving for \(J\), we find that the diffusion flux is approximately \(2.07 \times 10^{-7} \frac{\text{cm}^3 \text{STP}}{\text{cm}^2 \cdot \text{s}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fick's First Law

Understanding Fick's first law is foundational to calculating the diffusion flux in materials such as gases through a barrier. This principle asserts that the diffusion flux, denoted by the symbol J, is directly proportional to the concentration gradient across the material.

The law is mathematically expressed as
\[ J = -D \frac{dC}{dx} \]
where D is the diffusion coefficient, dC is the change in concentration, and dx is the change in position. The negative sign indicates that diffusion occurs from regions of higher to lower concentration. For a constant area of diffusion and under steady-state conditions, Fick's law simplifies the relationship between the permeability coefficient and the driving force, enabling calculations of the diffusion flux, such as the water vapor passing through a polystyrene sheet in our problem.

Steady-State Diffusion

In the context of steady-state diffusion, the concentration of the diffusing species does not change with time at any given location. This is a key assumption made in the textbook problem. It implies that there's a uniform rate of transfer, which allows for a simplified analysis where the diffusion flux remains constant over time.

In practice, this means that the water vapor pressure on both sides of the polystyrene sheet remains unchanging, enabling us to use Fick's first law directly without modifications for a time-dependent process. Despite the simplicity, steady-state diffusion calculations are crucial for engineering applications, such as designing barrier materials and predicting the service life of various products.

Permeability Coefficient

The permeability coefficient, P_M, represents how easily a substance can diffuse through a particular medium. It is a measure of the medium's permeability to a specific molecule, factoring in both the diffusion coefficient and the solubility of the molecule in the medium.

Mathematically, it can be expressed as
\[ P_M = D \cdot S \]
where D is the diffusion coefficient and S is the solubility coefficient. The permeability coefficient is integral in scenarios like our textbook problem, where it varies with temperature according to the given expression
\[ P_{M}=P_{M_{0}} \exp \left(-\frac{Q_{p}}{R T}\right) \]
This relation signifies that as the temperature increases, the permeability typically does as well up to a certain point, because the diffusion of molecules is generally an endothermic process. Understanding this temperature dependence is crucial in designing materials with appropriate diffusion characteristics for specific environments and applications.