Problem 39 - Physics

Question

Instead, they have a system of tiny tubes, called tracheae, through which oxygen diffuses into their bodies. The tracheae begin at the surface of the insect's body and penetrate into the interior. Suppose that a trachea is \(1.9 \mathrm{~mm}\) long with a cross-sectional area of \(2.1 \times 10^{-9} \mathrm{~m}^{2} .\) The concentration of oxygen in the air outside the insect is \(0.28 \mathrm{~kg} / \mathrm{m}^{3}\), and the diffusion constant is \(1.1 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). If the mass per second of oxygen diffusing through a trachea is \(1.7 \times 10^{-12} \mathrm{~kg} / \mathrm{s}\), find the oxygen concentration at the interior end of the tube.

Step-by-Step Solution

Answer

The oxygen concentration at the interior end of the tube is approximately \(0.42 \mathrm{~kg}/\mathrm{m}^3\).

Step 1: Identify Known Variables

First, let's list all given variables in the problem: \( L = 1.9 \mathrm{~mm} = 1.9 \times 10^{-3} \mathrm{~m} \) (length of the trachea), \( A = 2.1 \times 10^{-9} \mathrm{~m}^2 \) (cross-sectional area), \( C_{0} = 0.28 \mathrm{~kg}/\mathrm{m}^3 \) (outside concentration of oxygen), \( D = 1.1 \times 10^{-5} \mathrm{~m}^2/\mathrm{s} \) (diffusion constant), \( \frac{dm}{dt} = 1.7 \times 10^{-12} \mathrm{~kg}/\mathrm{s} \) (mass per second diffusing through the trachea).

Step 2: Use Fick's First Law of Diffusion

Fick's first law of diffusion is given by: \( \frac{dm}{dt} = -D \cdot A \cdot \frac{\Delta C}{\Delta x} \). Rearrange this to solve for the concentration difference: \( \frac{\Delta C}{\Delta x} = -\frac{\frac{dm}{dt}}{D \cdot A} \).

Step 3: Substitute Known Values

Substitute the known values into the equation: \( \frac{\Delta C}{\Delta x} = -\frac{1.7 \times 10^{-12}}{1.1 \times 10^{-5} \cdot 2.1 \times 10^{-9}} \).

Step 4: Calculate Concentration Gradient

Calculate the concentration gradient: \( \frac{\Delta C}{\Delta x} = -\frac{1.7 \times 10^{-12}}{2.31 \times 10^{-14}} = -73.5931 \mathrm{~kg}/\mathrm{m}^4 \).

Step 5: Solve for Concentration at Interior End

Use the formula for concentration difference: \( C_{0} - C_{L} = \frac{\Delta C}{\Delta x} \times L \). Rearrange to solve for \( C_{L} \): \( C_{L} = C_{0} - \frac{\Delta C}{\Delta x} \times L \). Substitute in the values: \( C_{L} = 0.28 + (73.5931 \times 1.9 \times 10^{-3}) \).

Step 6: Calculate Interior Concentration

Calculate \( C_{L} \): \( C_{L} = 0.28 + 0.13983 \approx 0.41983 \mathrm{~kg}/\mathrm{m}^3 \).

Key Concepts

Oxygen Diffusion

Oxygen diffusion is a natural process where oxygen spreads or moves from regions of higher concentration to regions of lower concentration. This movement is essential in biological systems for transporting oxygen into cells and organisms. Diffusion does not require energy, making it a passive transport mechanism. The key factors affecting diffusion include the concentration gradient, the diffusion constant, and the surface area available for diffusion. In insects, oxygen diffusion occurs through a specialized structure called the tracheae, ensuring that their tissues receive enough oxygen to survive and function efficiently.

Tracheae System

The tracheae system is a network of tiny tubes found in insects that facilitates the direct transport of oxygen to their tissues. Unlike mammals, insects do not rely on blood or another circulatory fluid to transport oxygen. Instead, the tracheae begin at small openings on an insect's body called spiracles. These tubes extend throughout the insect's body, enabling oxygen to diffuse directly into their cells. This system is highly efficient for smaller organisms because it minimizes diffusion distance and allows rapid oxygen exchange. When examining these systems, studying the dimensions of tracheae, such as length and cross-sectional area, is crucial as they play a vital role in determining how efficiently oxygen can reach the insect's tissues.

Concentration Gradient

The concentration gradient is a term describing the difference in concentration of a substance between two areas. In the context of diffusion, it acts as the driving force that prompts molecules to move from a region of high concentration to low concentration. A steeper gradient, where differences in concentration between areas are larger, will result in a faster rate of diffusion. In the case of oxygen diffusion through tracheae, the difference between the oxygen concentration outside the insect and inside the endpoint of the trachea creates this gradient. Calculating the gradient is essential to understanding how quickly and effectively oxygen will diffuse to meet the insect's metabolic needs.

Diffusion Constant

The diffusion constant, also known as the diffusion coefficient, is a crucial factor in understanding how substances diffuse through mediums. It is a measure of how easily a molecule can migrate through a specific substance and is influenced by factors such as temperature and the medium's properties. In Fick's Law, the diffusion constant is a parameter that helps predict the rate of diffusion. A higher diffusion constant signifies that molecules move more swiftly across the medium, while a lower constant indicates slower movement. In the case of the oxygen diffusion problem, the diffusion constant tells us how readily oxygen diffuses through the tracheae, impacting the speed and efficiency of oxygen delivery to the insect's tissues.