Suppose a cell is suspended in a solution cor
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![Suppose a cell is suspended in a solution containing a solute of constant concentration \( C_s \). Suppose further that the cell has constant volume \( V \) and that the area of its permeable membrane is the constant \( A \). By Fick's law, the rate of change of its mass \( m \) is directly proportional to the area \( A \) and the difference \( C_s - C(t) \), where \( C(t) \) is the concentration of the solute inside the cell at time \( t \). Find \( C(t) \) if \( m = V \cdot C(t) \) and \( C(0) = C_0 \). See the figure below. (Use \( k > 0 \) as the proportionality constant.)
\[ C(t) = \]
---
**Diagram Explanation:**
The diagram illustrates a cell in a solution.
- The cell is represented as a circle with arrows pointing inwards, indicating the movement of solute molecules through the cell membrane.
- Inside the cell, the concentration is labeled as \( C(t) \).
- The external solution has a constant concentration labeled as \( C_s \).
- The diagram emphasizes the diffusion process, where solute molecules move from areas of higher concentration to lower concentration through the semi-permeable membrane of the cell.
This helps in understanding the application of Fick’s law in biological systems, particularly in how substances like nutrients or waste products move across cellular membranes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd608d232-a827-4526-b5b4-a763d0855f8e%2Fb9a4ac91-d34b-486f-8a92-ff5c38a6189b%2F6p3wnrh_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose a cell is suspended in a solution containing a solute of constant concentration \( C_s \). Suppose further that the cell has constant volume \( V \) and that the area of its permeable membrane is the constant \( A \). By Fick's law, the rate of change of its mass \( m \) is directly proportional to the area \( A \) and the difference \( C_s - C(t) \), where \( C(t) \) is the concentration of the solute inside the cell at time \( t \). Find \( C(t) \) if \( m = V \cdot C(t) \) and \( C(0) = C_0 \). See the figure below. (Use \( k > 0 \) as the proportionality constant.)
\[ C(t) = \]
---
**Diagram Explanation:**
The diagram illustrates a cell in a solution.
- The cell is represented as a circle with arrows pointing inwards, indicating the movement of solute molecules through the cell membrane.
- Inside the cell, the concentration is labeled as \( C(t) \).
- The external solution has a constant concentration labeled as \( C_s \).
- The diagram emphasizes the diffusion process, where solute molecules move from areas of higher concentration to lower concentration through the semi-permeable membrane of the cell.
This helps in understanding the application of Fick’s law in biological systems, particularly in how substances like nutrients or waste products move across cellular membranes.
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