You want to make meatballs according to a recipe that says to marinate the meatballs in garlic and soy sauce before baking them in the oven. You prepare the marinade and immerse the meatballs, which are 6 cm in diameter, in it for 1 hour. When cutting into one of these rather large meatballs, you can clearly see how far the dark colored soy sauce has diffused in the white colored pork. You notice that only the outer 1 cm is marinated.

Question

You want to make meatballs according to a recipe that says to marinate the meatballs in garlic and soy sauce before baking them in the oven. You prepare the marinade and immerse the meatballs, which are 6 cm in diameter, in it for 1 hour. When cutting into one of these rather large meatballs, you can clearly see how far the dark colored soy sauce has diffused in the white colored pork. You notice that only the outer 1 cm is marinated.

The question is what is the diffusion coefficient of marinade in these meatballs cm2/sec

Answer 1

To determine the diffusion coefficient of the marinade in the meatballs, we can use Fick's second law of diffusion, which relates the rate of diffusion to the concentration gradient and the diffusion coefficient.

The equation is as follows:

J = -D * (dC/dx)

Where:
J is the diffusion flux (amount of substance diffusing per unit time per unit area),
D is the diffusion coefficient,
dC/dx is the concentration gradient (change in concentration per unit distance).

In this case, we know that the only the outer 1 cm of the meatball is marinated, which means that the concentration gradient starts from the surface (C = C_inner) and ends at a concentration of zero at the outer 1 cm (C = 0). So, we can rewrite the equation as:

J = -D * (dC/dx)
J = -D * ((C - C_inner)/distance)

Since the concentration gradient is linear, we can simplify further:

J = -D * (C_outer - C_inner)/distance

We also know that the concentration at the outer surface is that of the marinade, so C_outer = C_marinade.

Since we are given the diameter (6 cm), we can calculate the distance as the radius (3 cm). Now, we need to estimate J, the diffusion flux.

By observing that only the outer 1 cm is marinated in 1 hour, we can assume that the diffusion has reached a steady state. Therefore, the amount of marinade diffusing per unit time is constant.

To calculate J, we can use the volume of the outer 1 cm of the meatball as an estimate of the volume of marinade diffusing:

Volume of marinade diffusing = 4/3 * π * (3^3 - 2^3)

Now, we need to convert this volume into units of concentration per time per area (J).

Given that the marinade concentration is C_marinade and the volume diffusing is V_diffusing, we have:

J = V_diffusing * C_marinade / (time * surface area)

In this case, we immersed the meatballs in the marinade for 1 hour. Therefore, time = 1 hour = 3600 seconds.

The surface area of a sphere is given by:

Surface area = 4 * π * radius^2

Now we have all the values needed to calculate the diffusion coefficient (D).

D = -J * (time * surface area) / (C_marinade * (C_outer - C_inner))

Substitute the values you have and calculate the diffusion coefficient.