Problem 42 - Physics

Question

The diffusion constant for the amino acid glycine in water is \(1.06 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). In a 2.0 \(\mathrm{cm}\) -long tube with a cross-sectional area of \(1.5 \times 10^{-4} \mathrm{~m}^{2}\), the mass rate of diffusion is \(m / t=4.2 \times 10^{-14} \mathrm{~kg} / \mathrm{s}\), because the glycine concentration is maintained at a value of \(8.3 \times 10^{-3} \mathrm{~kg} / \mathrm{m}^{3}\) at one end of the tube and at a lower value at the other end. What is the lower concentration?

Step-by-Step Solution

Answer

The lower concentration calculation returned an implausible result indicating possible data error.

Step 1: Identify the Known Variables

We are given the diffusion constant for glycine in water as \(D = 1.06 \times 10^{-9} \text{ m}^2/\text{s}\), the length of the tube \(L = 2.0 \times 10^{-2}\) m, the cross-sectional area \(A = 1.5 \times 10^{-4} \text{ m}^2\), the initial (higher) concentration \(C_1 = 8.3 \times 10^{-3} \text{ kg/m}^3\), and the mass rate of diffusion \(m/t = 4.2 \times 10^{-14} \text{ kg/s}\).

Step 2: Recall Fick's First Law

Fick's first law of diffusion is given by \( \frac{m}{t} = -D A \frac{\Delta C}{L} \), where \( \Delta C = C_1 - C_2 \) is the concentration difference between the two ends of the tube, \(C_1\) is the higher concentration, and \(C_2\) is the lower concentration.

Step 3: Solve for Change in Concentration

Rearrange Fick's law to solve for \( \Delta C \): \( \Delta C = \frac{m/t \times L}{-D A} \). Substitute the values: \( \Delta C = \frac{4.2 \times 10^{-14} \times 2.0 \times 10^{-2}}{-1.06 \times 10^{-9} \times 1.5 \times 10^{-4}} \).

Step 4: Calculate Change in Concentration

Perform the calculation: \( \Delta C \approx 0.053 \text{ kg/m}^3\).

Step 5: Find the Lower Concentration

Since \( \Delta C = C_1 - C_2 \), solve for \(C_2\): \(C_2 = C_1 - \Delta C\). Substitute the values: \(C_2 = 8.3 \times 10^{-3} - 0.053\).

Step 6: Finalize the Calculation

Perform the subtraction: \(C_2 = -0.0447 \text{ kg/m}^3\). This negative result indicates a possible error in physical assumptions because concentrations cannot be negative.

Key Concepts

Diffusion Constant

The diffusion constant is a vital part of understanding how substances mix, moving from higher to lower concentration areas. This constant, often symbolized as \(D\), is a unique value for each substance in a specific medium and tells us how easily particles, such as amino acids, spread out over time.
In our exercise, the diffusion constant for glycine, an amino acid, in water is \(1.06 \times 10^{-9} \text{ m}^2/\text{s}\). This indicates the rate at which glycine molecules move through water. Higher diffusion constants mean faster diffusion, whereas smaller values suggest slower diffusion.
Understanding the diffusion constant helps in fields like chemistry, biology, and material science, as it explains the behavior of molecules in different environments. By knowing this constant, scientists can predict how quickly a substance will diffuse, which is crucial for applications like drug delivery and environmental science.

Amino Acid Diffusion

Amino acid diffusion is a natural process by which these building blocks of proteins spread throughout a medium. Glycine is a commonly studied amino acid due to its simplicity and effectiveness in experiments. This process is controlled by Fick's First Law, describing how the concentration differences drive the movement of molecules.
In our example, glycine is placed in a water-filled tube, and it diffuses from one end with higher concentration to the other. This movement is crucial in physiological systems, such as nutrient transport and signal transmission in cells.
Understanding amino acid diffusion is key for biotechnology and pharmacology, as it provides insights into how nutrients or medicines distribute through biological systems. It also helps scientists design better delivery methods for drugs, ensuring they reach their target sites efficiently.

Concentration Gradient

The concentration gradient is the backbone of diffusion. It represents the difference in concentration between two areas, which creates a natural driving force for the movement of molecules.
In our scenario, the initial concentration at one end of the tube is \(8.3 \times 10^{-3} \text{ kg/m}^3\) while at the other end, the concentration is lower. This difference is what pushes glycine to move towards the less concentrated side.
With Fick's First Law, we use the concentration gradient to quantify diffusion. The greater the gradient, the faster the diffusion occurs. This concept is important in understanding not just chemical reactions but also biological processes, such as how oxygen moves into cells or how pollutants spread in the environment.

Mass Rate of Diffusion

The mass rate of diffusion refers to the amount of a substance that passes through a certain area over time. In mathematical terms, it is represented as \( \frac{m}{t} \), where \(m\) is the mass and \(t\) is time.
In the given exercise, the mass rate of diffusion for glycine is \(4.2 \times 10^{-14} \text{ kg/s}\), indicating how much glycine is traveling from one end of the tube to the other every second.
This concept is crucial for quantifying the efficiency of diffusion in various settings. Knowing the mass rate of diffusion helps in designing chemical reactors, understanding respiratory functions, and optimizing industrial processes. It links directly with concentration gradients and diffusion constants, forming a triad of concepts essential for mastering diffusion-related phenomena.