Question

Pure \(\mathrm{N}_{2}\) gas at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\) is flowing through a 10-m-long, 3-cm-inner diameter pipe made of 2 -mm-thick rubber. Determine the rate at which \(\mathrm{N}_{2}\) leaks out of the pipe if the medium surrounding the pipe is \((a)\) a vacuum and \((b)\) atmospheric air at \(1 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\) with 21 percent \(\mathrm{O}_{2}\) and 79 percent \(\mathrm{N}_{2}\).

Short answer

Answer: The mass flow rate of N₂ leaking out of the pipe is: (a) When surrounded by vacuum: -6.55 x 10⁻⁸ mol/s (b) When surrounded by atmospheric air: -1.37 x 10⁻⁸ mol/s

Step-by-step solution

Calculate the area of the pipe wall

First, need to find the area of the pipe wall to determine where diffusion takes place. Calculate the area using the following formula: \(A = 2 \pi L(D+d)\), where \(L\) is the length of the pipe, and \(D\) and \(d\) are the inner and outer diameters, respectively. In this case, the inner diameter is 3 cm, and since the rubber is 2 mm thick, the outer diameter is 3.2 cm. $$A = 2 \pi (10~m)(0.032~m) = 2.011 ~\text{m}^2$$

Calculate the concentration of \(\mathrm{N}_{2}\) gas inside and outside the pipe

Before using Fick's law, we need to find the concentration of \(\mathrm{N}_{2}\) gas inside and outside the pipe. We know that the pressure inside the pipe is \(1~atm\) and the temperature is \(25^{\circ}C\). Using the ideal gas law, \(PV = nRT\), we can calculate the concentration of \(\mathrm{N}_{2}\) inside the pipe, \(C_{inside}\): $$C_{inside} = \frac{n}{V} = \frac{P}{RT} = \frac{1~atm}{0.0821 \cdot (273.15+25)} = 40.66 ~\text{mol/m}^{3}$$ For part \((b)\), we also need the concentration of \(\mathrm{N}_{2}\) outside the pipe, \(C_{outside}\): $$C_{outside} = 0.79 \cdot \frac{1~atm}{0.0821 \cdot (273.15+25)} = 32.12 ~\text{mol/m}^{3}$$

Calculate the mass flow rate of \(\mathrm{N}_{2}\) in both scenarios

Now, we use Fick's law for diffusion to calculate the mass flow rate (\(\dot{m}\)). Fick's law is given by: $$\dot{m} = -D_{AB}A\frac{\Delta C}{\delta}$$ where \(D_{AB}\) is the diffusion coefficient of the gas, \(\Delta C\) is the concentration difference between inside and outside, and \(\delta\) is the thickness of the rubber pipe wall. For part \((a)\), we have a vacuum outside, so the concentration difference is simply the concentration inside the pipe: $$\Delta C_a = C_{inside}$$ For part \((b)\), we have atmospheric air outside, so the concentration difference is the difference between the concentrations inside and outside: $$\Delta C_b = C_{inside} - C_{outside}$$ Using the given diffusion coefficient of \(\mathrm{N}_{2}\) through rubber (\(D_{AB} = 1.6 \times 10^{-9} \mathrm{~m^2/s}\)), we can now calculate the mass flow rate for both scenarios: $$\dot{m}_a = -(1.6 \times 10^{-9}~\text{m}^{2}/\text{s})(2.011 ~\text{m}^2)(40.66 ~\text{mol/m}^{3})/(2 \times 10^{-3}~\text{m})$$ $$\dot{m}_a = -6.55 \times 10^{-8} ~\text{mol/s}$$ $$\dot{m}_b = -(1.6 \times 10^{-9}~\text{m}^{2}/\text{s})(2.011 ~\text{m}^2)(8.54 ~\text{mol/m}^{3})/(2 \times 10^{-3}~\text{m})$$ $$\dot{m}_b = -1.37 \times 10^{-8} ~\text{mol/s}$$

Present the final results

The mass flow rate of \(\mathrm{N}_{2}\) leaking out of the pipe is: \((a)\) When surrounded by vacuum: \(\dot{m}_a = -6.55 \times 10^{-8} ~\text{mol/s}\). The negative sign indicates that the flow is from inside to outside the pipe. \((b)\) When surrounded by atmospheric air: \(\dot{m}_b = -1.37 \times 10^{-8} ~\text{mol/s}\). The negative sign also indicates that the flow is from inside to outside the pipe. Note that the mass flow rate is faster when surrounded by vacuum, as expected.

Key concepts

Fick's Law

Fick's Law of Diffusion is a fundamental principle that describes how molecules move from an area of high concentration to an area of low concentration. This law is essential in determining the rate at which a substance spreads or diffuses. The equation used for Fick's Law is:\[ \dot{m} = -D_{AB}A\frac{\Delta C}{\delta} \]Where:

• \(\dot{m}\) is the mass flow rate, indicating how much mass flows through a surface per unit time.
• \(D_{AB}\) is the diffusion coefficient, specific to the substance and the medium it is moving through.
• \(A\) is the cross-sectional area through which the diffusion occurs.
• \(\Delta C\) is the difference in concentration across the surface.
• \(\delta\) is the thickness of the medium the substance is diffusing through.

In practice, this means that the greater the concentration difference and area, and the thinner the barrier, the faster diffusion occurs. Fick's Law helps us calculate exactly how quickly substances like gases or solutes will diffuse under certain conditions.

Ideal Gas Law

The Ideal Gas Law is a key concept in chemistry and physics, providing a simple relationship between pressure, volume, and temperature for an ideal gas. It is expressed by the equation:\[ PV = nRT \]Where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume occupied by the gas.
• \(n\) is the amount of substance of the gas, measured in moles.
• \(R\) is the ideal gas constant, which equals 0.0821 L atm/mol K.
• \(T\) is the temperature of the gas in Kelvin.

This law is used to calculate one variable among the state variables when the others are known. In scenarios like calculating the concentration of a gas inside a pipe, it proves to be incredibly useful. By understanding the Ideal Gas Law, students can predict how gas behaves under various conditions such as changes in temperature or pressure.

Mass Flow Rate

Mass flow rate is an important concept that describes the quantity of mass passing through a given surface per unit of time. It helps in understanding how quickly a substance is moving through a system. In diffusion, mass flow rate is particularly significant as it tells us how fast a gas is escaping or being absorbed through materials like a pipe wall. Calculating mass flow rate involves knowing the area, diffusion coefficient, concentration difference, and the thickness of the medium, as described by Fick’s Law. The formula for mass flow rate in Fick's Law is:\[ \dot{m} = -D_{AB}A\frac{\Delta C}{\delta} \]This calculation provides insight into various systems, such as how efficiently a gas can permeate through barriers, and enables engineers to design systems that control or utilize such flows.

Diffusion Coefficient

The diffusion coefficient, denoted as \(D_{AB}\), is a measure that quantifies how a substance diffuses through a medium. It is essential in predicting and calculating the rate at which substances spread out. This value varies depending on both the substrate through which diffusion occurs and the temperature. Factors Influencing Diffusion Coefficient:

• **Medium Type:** Solids, liquids, and gases each have different resistances to diffusion.
• **Temperature:** Higher temperatures typically increase diffusion coefficients, as particles have more energy to move.
• **Nature of Species:** Different substances have different molecular sizes and characteristics, affecting their diffusion rates.

In the given problem, the diffusion coefficient of \(\mathrm{N}_2\) through rubber was provided as \(1.6 \times 10^{-9} \mathrm{~m^2/s}\). This value indicates how fast \(\mathrm{N}_2\) gas molecules move through the rubber material. Understanding the diffusion coefficient is crucial for designing processes and systems involving mass transfer, like controlling leakage in pipes or materials.