Final answer:
The concentration of the antibody in the middle of the cylindrical PLA after 24 hours is approximately 1.555 x 10⁻¹⁰ kg/m³.
Step-by-step explanation:
To calculate the concentration of the antibody in the middle of the cylindrical PLA after 24 hours, we can use Fick's second law of diffusion. Fick's law states that the rate of diffusion is proportional to the concentration gradient and the diffusion coefficient. Since the antibody is assumed to only diffuse radially, we can assume that the concentration gradient is only in the radial direction.
To calculate the concentration at the middle of the cylinder, we need to determine the diffusion distance and the time. The diffusion distance is the radius of the cylinder, which is half the diameter (10 mm/2 = 5 mm). The time is 24 hours, which we need to convert to seconds (24 hours * 60 minutes/hour * 60 seconds/minute = 86,400 seconds).
Using the equation for Fick's second law of diffusion, we can calculate the concentration at the middle of the cylinder:
C = (Co * t * D) / (4 * d²)
Where:
- C is the concentration at the middle of the cylinder
- Co is the initial concentration of the antibody (1 kg/m³)
- t is the time (86,400 seconds)
- D is the diffusion coefficient (3.6 * 10⁻¹⁰ m²/s)
- d is the diffusion distance (5 mm or 0.005 m)
Substituting the values into the equation gives:
C = (1 kg/m³ * 86,400 s * 3.6 * 10⁻¹⁰ m²/s) / (4 * (0.005 m)²)
Simplifying and converting the units:
C ≈ 1.555 x 10⁻¹⁰ kg/m³
Therefore, the concentration of the antibody in the middle of the cylindrical PLA after 24 hours is approximately 1.555 x 10⁻¹⁰ kg/m³.